Weyl Transforms

Weyl Transforms

M.W. Wong

Springer

Preface

This book is an outgrowth of courses given by me for graduate students at York University in the past ten years. The actual writing of the book in this form was carried out at York University, Peking University, the Academia Sinica in Beijing, the University of California at Irvine, Osaka University, and the University of Delaware. The idea of writing this book was first conceived in the summer of 1989, and the protracted period of gestation was due to my daily duties as a professor at York University. I would like to thank Professor K.C. Chang, of Peking University; Professor Shujie Li, of the Academia Sinica in Beijing; Professor Martin Schechter, of the University of California at Irvine; Professor Michihiro Nagase, of Osaka University; and Professor M.Z. Nashed, of the University of Delaware, for providing me with stimulating environments for the exchange of ideas and the actual writing of the book. We study in this book the properties of pseudo-differential operators arising in quantum mechanics, first envisaged in [33] by Hermann Weyl, as bounded linear operators on L2 (Rn ). Thus, it is natural to call the operators treated in this book Weyl transforms. To be specific, my original plan was to supplement the standard graduate course in pseudo-differential operators at York University by writing a set of lecture notes on the derivation of a formula from first principles for the product of two Weyl transforms. This was achieved in the summer of 1990 when I was visiting Peking University and the Academia Sinica in Beijing. Chapters 2–6 of the book, which appeared then, albeit in embryonic form, already contained the formula for the product of two Weyl transforms obtained by Pool in [20]. Chapters 8 and 9 were written in the summer of 1993 at York University in order to get another formula for the product of two Weyl transforms using relatively new ideas, e.g.,

vi

Preface

the Heisenberg group and the twisted convolution, in noncommutative harmonic analysis developed by Folland in [6] and Stein in [26], among others. The result was an account, given in Chapter 9, of a formula for the product of two Weyl transforms in the paper [10] by Grossmann, Loupias, and Stein. A preliminary version of the derivations of the two formulas was written up for private circulation in the second quarter of 1994–95 at the University of California at Irvine. In the summer of 1994, I gave a course in special topics in pseudo-differential operators tailored to the needs of my Ph.D. students at York University. I chose to study the criteria in terms of the symbols for the boundedness and compactness of the Weyl transforms. Two sets of results were presented. The first set was about the compactness of a Weyl transform with symbol in Lr (R2n ), 1 ≤ r ≤ ∞, and the second set, inspired by the book [29] by Thangavelu, was concerned with the criteria for the boundedness and compactness of Weyl transforms in terms of symbols evaluated at Wigner transforms of Hermite functions. The two sets of results can be found in, respectively, Chapters 11–14 and Chapters 24–27. Chapter 28 is devoted to the study of the eigenvalues and eigenfunctions of a Weyl transform of which the symbol is a Dirac delta on a disk in R2 . The preliminary version of the formulas for the product of two Weyl transforms and the lecture notes of the topics course given in the summer of 1994 were then put together, simplified, polished, and supplemented with background materials at Osaka University and the University of Delaware in the winter of 1997. To this end, I found it instructive to add new chapters, i.e., Chapters 15–17, on localization operators initiated by Daubechies in [3, 4] and Daubechies and Paul in [5], and the closely related theory of square-integrable group representations studied by Grossmann, Morlet, and Paul in [11, 12]. The final two chapters were added in an attempt to make explicit the role of the symplectic group in the study of Weyl transforms. The connections of the Weyl transforms with quantization in physics, highlighted in this book, can be found in the references [6, 10, 20, 26, 33] already cited, the book [2] by Berezin and Shubin, the paper [18] by Iancu and Wong, and the papers [37, 38] by Wong. All the topics in this book should be accessible to a first-year graduate student. The book is a natural sequel to a first course in pseudo-differential operators, but no familiarity with even the basics of pseudo-differential operators is required for a good understanding of the entire book. The only essential prerequisites are the elementary properties of the Fourier transform and tempered distributions given in the beginning chapters of, say, the book [8] by Goldberg, the book [27] by Stein and Weiss, and the book [36] by Wong, and these are collected in Chapter 1. Of course, a nodding acquaintance with basic functional analysis is necessary for an intelligent reading of this book. Finally, it must be emphasized that this book is far from being a definitive treatise on Weyl transforms. Thus, the choice of topics in this book was guided by personal predilections, and the references at the end of the book are limited to those that have been instrumental in my understanding of Weyl transforms.

Contents

Preface

v

1

Prerequisite Topics in Fourier Analysis

1

2

The Fourier–Wigner Transform

9

3

The Wigner Transform

13

4

The Weyl Transform

19

5

Hilbert–Schmidt Operators on L2 (Rn )

25

6

The Tensor Product in L2 (Rn )

29

7 H ∗ -Algebras and the Weyl Calculus

33

8

The Heisenberg Group

37

9

The Twisted Convolution

43

10 The Riesz–Thorin Theorem

47

11 Weyl Transforms with Symbols in Lr (R2n ), 1 ≤ r ≤ 2

55

12 Weyl Transforms with Symbols in L∞ (R2n )

59

viii

Contents

13 Weyl Transforms with Symbols in Lr (R2n ), 2 < r < ∞

63

14 Compact Weyl Transforms

71

15 Localization Operators

75

16 A Fourier Transform

79

17 Compact Localization Operators

83

18 Hermite Polynomials

87

19 Hermite Functions

93

20 Laguerre Polynomials

95

21 Hermite Functions on C

101

22 Vector Fields on C

103

23 Laguerre Formulas for Hermite Functions on C

107

24 Weyl Transforms on L2 (R) with Radial Symbols

113

25 Another Fourier Transform

119

26 A Class of Compact Weyl Transforms on L2 (R)

123

27 A Class of Bounded Weyl Transforms on L2 (R)

127

28 A Weyl Transform with Symbol in S  (R2 )

131

29 The Symplectic Group

135

30 Symplectic Invariance of Weyl Transforms

145

Notation Index

155

Index

157

1 Prerequisite Topics in Fourier Analysis

The basic topics in Fourier analysis that we need for a good understanding of the book are collected in this chapter. In view of the fact that these topics can be found in many books on Fourier analysis, e.g., [8] by Goldberg, [27] by Stein and Weiss, and [36] by Wong, among others, we provide only the proofs of the key results in the study of the Weyl transform. Another important role played by this chapter is to fix the notation used throughout the book. Let Rn  {(x1 , x2 , . . . , xn ) : xj real numbers}. Points in Rn are denoted by x, y, ξ, η, etc. Let x  (x1 , x2 , . . . , xn ) and y  (y1 , y2 , . . . , yn ) be in Rn . The inner product x · y of x and y is defined by x·y 

n


xj yj ,

j 1

and the norm |x| of x is defined by |x| 

 n
j 1

 21 xj2

.

We denote the differential operators ∂x∂ 1 , ∂x∂ 2 , . . . , ∂x∂ n on Rn by ∂1 , ∂2 , . . . , ∂n , respectively, and the differential operators −i∂1 , −i∂2 , . . . , −i∂n on Rn by D1 , D2 , . . . , Dn , respectively, where i 2  −1. A reason for using the factor of −i is to make some formulas, e.g., Proposition 1.8, look better, but the main justification for its appearance lies in the fact that the quantum-mechanical momentum observable in the direction of the j th coordinate is represented by Dj if we choose to work with units that give the value 1 to Planck’s constant. More de-

2

1. Prerequisite Topics in Fourier Analysis

tails are given in the discussion at the end of Chapter 4. A linear partial differential operator P (x, D) on Rn is given by
aα (x)D α , x ∈ Rn , P (x, D)  |α|≤m

, α2 , . . . , αn ) is a multi-index, i.e., an n-tuple of nonnegative inwhere α  (α1 n α tegers; |α|   D α1 D α2 · · · D αn , and aα is j 1 αj is the length of α; D n a measurable complex-valued function on R for |α| ≤ m. The symbol of the differential operator P (x, D) is the function on R2n defined by
aα (x)ξ α , x, ξ ∈ Rn , P (x, ξ )  |α|≤m

ξ1α1 ξ2α2

· · · ξnαn . where ξ α  The differential operator ∂ α , for any multi-index α, defined by ∂ α  ∂1α1 ∂2α2 · · · ∂nαn , will also be used frequently in the book. We write ∂xα (or ∂ξα ) for ∂ α , and Dxα (or Dξα ) for D α , when we need to specify the variable with respect to which we differentiate. Let f and g be infinitely differentiable functions on Rn . Then we have the Leibnitz formula
α  α (D β f )(D α−β g) D (fg)  β β≤α for all multi-indices α, and the more general Leibnitz formula
P (D)(f g)  (P (µ) (D)f )(D µ g) |µ|≤m

 for any linear partial differential operator P (D)  |α|≤m aα D α with constant        coefficients, where β ≤ α means βj ≤ αj , j  1, 2, · · · , n, βα  αβ11 αβ22 . . . αβnn , µ!  µ1 !µ2 ! · · · µn !, and P (µ) (D) is the linear partial differential operator with symbol P (µ) on Rn given by P (µ) (ξ )  (∂ µ P )(ξ ),

ξ ∈ Rn .

Now we let C0∞ (Rn ) be the set of all infinitely differentiable functions on Rn with compact supports, and we let S(Rn ) be the set of all infinitely differentiable functions on Rn such that sup |x α (∂ β ϕ)(x)| < ∞

x∈Rn

for all multi-indices α and β. Theorem 1.1. C0∞ (Rn ) and S(Rn ) are dense in Lr (Rn ), 1 ≤ r < ∞. Theorem 1.1 can be proved using the convolution of functions on Rn .

1. Prerequisite Topics in Fourier Analysis

3

Theorem 1.2. (Young’s Inequality) Let f ∈ L1 (Rn ) and g ∈ Lr (Rn ), 1 ≤ r ≤ ∞. Then the integral f (x − y)g(y)dy Rn

n

exists for almost all x in R . If we denote the value of the integral by (f ∗ g)(x), then f ∗ g ∈ Lr (Rn ) and f ∗ g Lr (Rn ) ≤ f L1 (Rn ) g Lr (Rn ) . Remark 1.3. The function f ∗ g in Theorem 1.2 is usually called the convolution of f and g. The formulation of Young’s inequality for the convolution of sequences ∞ {ak }∞ k−∞ and {bk }k−∞ of complex numbers is left as an exercise. It is useful to have the following result. Proposition 1.4. Let f and g be in S(Rn ). Then f ∗ g ∈ S(Rn ). Our next result is a technique of regularization to be used in the proofs of Theorems 1.11, 3.1, and 16.1.

Theorem 1.5. Let ϕ ∈ L1 (Rn ) be such that Rn ϕ(x)dx  a. For any positive number ε, we define the function ϕε on Rn by x ϕε (x)  ε−n ϕ , x ∈ Rn . ε Then, for any bounded function f on Rn that is continuous on an open subset V of Rn , f ∗ ϕε → af uniformly on compact subsets of V as ε → 0. Proof. Without loss of generality, we can assume that ϕ(x)  0 for almost all x in Rn . Since ϕε (x)dx  a Rn

for any positive number ε, it follows that (f ∗ ϕε )(x) − af (x)  {f (x − εy) − f (x)}ϕ(y)dy

(1.1)

Rn

for all x in Rn . Let K be a compact subset of V and δ be a positive number. Let K1 be a compact subset of V such that K is properly contained in K1 . Then there exists a positive number δ1 such that δ (1.2) ϕ −1 L1 (Rn ) 2 for all x and y in K1 with |x − y| < δ1 . Let U be an open subset of Rn such that K ⊂ U ⊂ K1 . Let δ2 be the distance between K and the complement of U in Rn and let δ0  min(δ1 , δ2 ). Then

K⊂ B(x, δ0 ) ⊂ U, (1.3) |f (x) − f (y)| <

x∈K

4

1. Prerequisite Topics in Fourier Analysis

where B(x, δ0 ) is the open ball with center x and radius δ0 . Since ϕ ∈ L1 (Rn ), there exists a positive number R such that δ , (1.4) |ϕ(y)|dy < 4M |y|≥R where M  sup |f (x)|. Hence, for any positive number ε, we get, by (1.1) and (1.4),

x∈Rn

|(f ∗ ϕε )(x) − af (x)| ≤ |f (x − εy) − f (x) ϕ(y)|dy + |f (x − εy) − f (x) ϕ(y)|dy |y|≥R |y|
δ0 . R

(1.6)

Thus, by (1.2), (1.3), (1.5), and (1.6), |(f ∗ ϕε )(x) − af (x)| < δ, x ∈ K,

whenever ε <

δ0 , R

✷

and the proof is complete.

Of fundamental importance in this book is the Fourier transform, which we now define. The Fourier transform of a function in L1 (Rn ) is the function fˆ, sometimes denoted by Ff , on Rn defined by −n/2 ˆ e−ix·ξ f (x)dx, ξ ∈ Rn . f (ξ )  (2π ) Rn

We give in the following proposition some useful, albeit simple, properties of the Fourier transform. Proposition 1.6. (The Riemann–Lebesgue Lemma) Let f ∈ L1 (Rn ). Then fˆ is a continuous function on Rn such that lim|ξ |→∞ fˆ(ξ )  0. Proposition 1.7. Let f and g be in L1 (Rn ). Then ˆ ), (f ∗ g)ˆ(ξ )  (2π)n/2 fˆ(ξ )g(ξ

ξ ∈ Rn .

Proposition 1.8. Let ϕ ∈ S(Rn ). Then, for all multi-indices α, (i) (ii)

ˆ ), ξ ∈ Rn , (D α ϕ)ˆ(ξ )  ξ α ϕ(ξ α (D ϕ)(ξ ˆ )  ((−x)α ϕ)ˆ(ξ ), ξ ∈ Rn .

Proposition 1.9. Let ϕ ∈ L1 (Rn ). Then (i) (Ty f )ˆ(ξ )  (My fˆ)(ξ ), ξ ∈ Rn , (ii) (My f )ˆ(ξ )  (T−y fˆ)(ξ ), ξ ∈ Rn , (iii) (Da f )ˆ(ξ )  |a|−n (D 1 fˆ)(ξ ), ξ ∈ Rn , a

1. Prerequisite Topics in Fourier Analysis

5

where (Ty f )(x)  f (x + y),

x ∈ Rn ,

(My f )(x)  eix·y f (x),

x ∈ Rn ,

and (Da f )(x)  f (ax),

x ∈ Rn ,

for all y in Rn and all nonzero real numbers a. Ty , My , and Da in Proposition 1.9 are respectively the translation operator, the modulation operator, and the dilation operator on Rn . Proposition 1.10. (The Adjoint Formula) Let f and g be in L1 (Rn ). Then ˆ f (x)g(x)dx. ˆ f (x)g(x)dx  Rn

Rn

Important and less superficial properties of the Fourier transform are given in the following two theorems. Theorem 1.11. (The Fourier Inversion Formula) The Fourier transform is a one to one and onto mapping from S(Rn ) into S(Rn ). Moreover, (fˆ)ˇ  f, where g(x) ˇ  (2π)−n/2

f ∈ S(Rn ),

Rn

eix·ξ g(ξ )dξ, x ∈ Rn ,

for all g in S(Rn ). Proof. Let f ∈ S(Rn ). Then, for any positive number ε, we define the function Iε on Rn by ε 2 |ξ |2 −n/2 Iε (x)  (2π) eix·ξ − 2 fˆ(ξ )dξ, x ∈ Rn . (1.7) Rn

Then, by (1.7), Propositions 1.9, 1.10, and the well-known fact that the Fourier transform ϕˆ of the function ϕ on Rn given by ϕ(x)  e−

|x|2 2

,

x ∈ Rn ,

(1.8)

is equal to ϕ, we get Iε (x)  (2π )−n/2 (f ∗ ϕε )(x), x ∈ Rn , (1.9)   where ϕε (x)  ε −n ϕ xε for all x in Rn . Since f ∈ S(Rn ), it follows from (1.8), (1.9), and Theorem 1.5 that   2 −n/2 − |x|2 Iε → (2π) e dx f  f (1.10) Rn

6

1. Prerequisite Topics in Fourier Analysis

uniformly on compact subsets of Rn as ε → 0. By (1.7) and the Lebesgue dominated convergence theorem, Iε (x) → (2π)−n/2 (1.11) eix·ξ fˆ(ξ )dξ Rn

for all x in Rn . Hence, by (1.10) and (1.11), (fˆ)ˇ  f . That the Fourier transform from S(Rn ) into S(Rn ) is one-to-one and onto is then an easy consequence. ✷ Remark 1.12. The function gˇ in Theorem 1.11 is called the inverse Fourier transform of g and is sometimes denoted by F −1 g. Theorem 1.13. (The Plancherel Theorem) The mapping F : S(Rn ) → S(Rn ) can be extended uniquely to a unitary operator on L2 (Rn ). Proof.

By Theorems 1.1 and 1.11, it is sufficient to prove that ϕ ˆ L2 (Rn )  ϕ L2 (Rn ) ,

ϕ ∈ S(Rn ).

(1.12)

To this end, let ϕ ∈ S(Rn ) and let ψ be the function on Rn defined by ψ(x)  ϕ(−x),

x ∈ Rn .

(1.13)

n

Then ψ ∈ S(R ), and an easy computation gives ˆ )  ϕ(ξ ψ(ξ ˆ ), Thus, by (1.13), ϕ 2L2 (Rn )



Rn

ξ ∈ Rn .

(1.14)

ϕ(x)ϕ(x)dx 

Rn

ϕ(x)ψ(−x)dx  (ϕ ∗ ψ)(0).

(1.15)

By Proposition 1.4, ϕ∗ψ ∈ S(Rn ). Hence, by (1.14), Proposition 1.7, and Theorem 1.11, −n/2 ˆ ˆ )dξ (ϕ ∗ ψ)(0)  (2π) (ϕ ∗ ψ) (ξ )dξ  ϕ(ξ ˆ )ψ(ξ Rn Rn ϕ(ξ ˆ )ϕ(ξ ˆ )dξ  ϕ ˆ 2L2 (Rn ) . (1.16)  Rn

Hence, by (1.15) and (1.16), (1.12) follows.

✷

Remark 1.14. In view of the Plancherel theorem, we can define the Fourier transform of a function f in L2 (Rn ), again denoted by fˆ or Ff . The inverse Fourier transform of a function f in L2 (Rn ) is denoted by fˇ or F −1 f . Let us now review the very basic notions of tempered distributions used in this book. n Let {ϕj }∞ j 1 be a sequence of functions in S(R ) such that for all multi-indices α and β, sup |x α (∂ β ϕj )(x)| → 0

x∈Rn

1. Prerequisite Topics in Fourier Analysis

7

as j → ∞. Then we say that ϕj → 0 in S(Rn ) as j → ∞. A linear functional T on S(Rn ) is called a tempered distribution on Rn if T (ϕj ) → 0 n as j → ∞ for every sequence {ϕj }∞ j 1 of functions in S(R ) such that ϕj → 0 in n S(R ) as j → ∞. The collection of all tempered distributions on Rn is denoted  n by S  (Rn ), and a sequence {Tj }∞ j 1 of tempererd distributions in S (R ) is said to  n  n converge to zero in S (R ), denoted by Tj → 0 in S (R ), as j → ∞ if

Tj (ϕ) → 0,

ϕ ∈ S(Rn ),

as j → ∞. The most important tempered distributions to us are given by tempered functions on Rn . Let us recall that a measurable function f on Rn is said to be tempered if |f (x)| dx < ∞ n (1 + |x|)N R for some positive integer N. All functions in Lr (Rn ), 1 ≤ r ≤ ∞, are therefore tempered. Proposition 1.15. Let f be a tempered function on Rn . Then the linear functional Tf on S(Rn ) defined by f (x)ϕ(x)dx, ϕ ∈ S(Rn ), Tf (ϕ)  Rn

is a tempered distribution. Remark 1.16. It is customary to identify the tempered distribution Tf with the function f . We leave it as an exercise to prove the following proposition, which will be used in Chapter 28. Proposition 1.17. Let δ : S(R2 ) → C be the linear mapping defined by 2π ϕ(ρeie )ρdθ, ρ > 0. δ(ϕ)  0 

Then δ ∈ S (R ). 2

Another exercise is to prove the following proposition. Proposition 1.18. S(Rn ) is dense in S  (Rn ). We can now introduce the class of pseudo-differential operators studied in the book [36] by Wong. They will be used in Chapter 4 to motivate the definition of the Weyl transform. Familiarity of pseudo-differential operators is desirable, but not necessary, for a good understanding of the materials in this book. Let m be any real number. Then we define S m to be the set of all infinitely differentiable functions on R2n such that for all multi-indices α and β, there is a

8

1. Prerequisite Topics in Fourier Analysis

positive constant Cα,β , depending on α and β only, for which β

|(Dxα Dξ σ )(x, ξ )| ≤ Cα,β (1 + |ξ |)m−|β| , x, ξ ∈ Rn .  We call any σ ∈ m∈R S m a symbol. Let σ be a symbol. Then we define the pseudo-differential operator Tσ corresponding to the symbol σ by (Tσ ϕ)(x)  (2π )−n/2 eix·ξ σ (x, ξ )ϕ(ξ ˆ )dξ, x ∈ Rn , Rn

n

for all ϕ in S(R ). Using Proposition 1.8 and the Fourier
inversion formula, it can be shown that a linear partial differential operator aα (x)D α on Rn , where aα |α|≤m

is an infinitely differentiable function on Rn such that

sup |(D β aα )(x)| < ∞, |α| ≤ m, x∈R

for all multi-indices β, is a pseudo-differential operator corresponding to the symbol σ in S m given by
σ (x, ξ )  aα (x)ξ α , x, ξ ∈ Rn . |α|≤m

Proposition 1.19. Let σ be a symbol. Then the pseudo-differential operator Tσ maps S(Rn ) continuously into S(Rn ).

2 The Fourier–Wigner Transform

A basic tool we use in the study of the Weyl transform is the Wigner transform. We find it convenient to introduce first a related transform, which we call the Fourier–Wigner transform. Let q and p be in Rn , and let f be a measurable function on Rn . We define the function ρ(q, p)f on Rn by (ρ(q, p)f )(x)  eiq·x+ 2 iq·p f (x + p), 1

x ∈ Rn .

(2.1)

Proposition 2.1. ρ(q, p) : L2 (Rn ) → L2 (Rn ) is a unitary operator for all q and p in Rn . Proof.

We only need to prove that f ∈ L2 (Rn ),

ρ(q, p)f L2 (Rn )  f L2 (Rn ) ,

and ρ(q, p) is onto for all q and p in Rn . But, by (2.1), ρ(q, p)f 2L2 (Rn ) 



Rn



Rn

 

Rn

|eiq·x+ 2 iq·p f (x + p)|2 dx 1

|f (x + p)|2 dx |f (x)|2 dx

 f 2L2 (Rn ) ,

f ∈ L2 (Rn ),

10

2. The Fourier–Wigner Transform

for all q and p in Rn . To prove that ρ(q, p) is onto, we let g ∈ L2 (Rn ) and define the function f on Rn by f (x)  e−iq·x+ 2 iq·p g(x − p), 1

x ∈ Rn .

(2.2)

Then f is obviously in L2 (Rn ), and by (2.1) and (2.2), (ρ(q, p)f )(x)  eiq·x+ 2 iq·p f (x + p) 1

 eiq·x+ 2 iq·p e−iq·(x+p)+ 2 iq·p g(x) 1

 g(x),

1

x ∈ Rn .

✷ −1

 Remark 2.2. It is clear from the proof of Proposition 2.1 that ρ(q, p) ρ(−q, −p), q, p ∈ Rn . In fact, ρ is a projective representation, i.e., a unitary representation up to phase factors, of the phase space R2n on L2 (Rn ), and it is closely related to the Schr¨odinger representation R 1 of the Heisenberg group H n on L2 (Rn ) to be studied in Chapter 8. The elucidation of the connection between ρ and R 1 is given in Remark 8.8. Let f and g be in S(Rn ). Then we define the function V (f, g) on R2n by V (f, g)(q, p)  (2π )−n/2 ρ(q, p)f, g,

q, p ∈ Rn ,

(2.3)

where  ,  is the inner product in L2 (Rn ). We call V (f, g) the Fourier–Wigner transform of f and g. The notation  ,  will also be used to denote the inner product in L2 (R2n ). Proposition 2.3. Let f and g be in S(Rn ). Then  p  p dy eiq·y f y + g y− V (f, g)(q, p)  (2π)−n/2 2 2 Rn

(2.4)

for all q and p in Rn . Proof.

By (2.1) and (2.3), V (f, g)(q, p)  (2π)−n/2 ρ(q, p)f, g 1 −n/2 eiq·x+ 2 iq·p f (x + p)g(x)dx  (2π)

(2.5)

Rn

p in (2.5). Then we get (2.4) immediately. 2 n Proposition 2.4. V : S(R ) × S(Rn ) → S(R2n ) is a bilinear mapping. for all q and p in Rn . Let x  y −

✷

Recall that if X and Y are complex vector spaces, then a mapping f : X×X → Y is said to be bilinear if for all α1 and α2 in C and all x1 and x2 in X, we have f (α1 x1 + α2 x2 , x)  α1 f (x1 , x) + α2 f (x2 , x) and f (x, α1 x1 + α2 x2 )  α¯1 f (x, x1 ) + α¯2 f (x, x2 ) for all x in X.

2. The Fourier–Wigner Transform

11

The bilinearity in Proposition 2.4 is easy to check. To prove that V maps S(Rn )× S(Rn ) into S(R2n ), we need a lemma. Lemma 2.5. Let ϕ ∈ S(R2n ). Then the function 5 on R2n defined by 5(q, p)  eiq·y ϕ(y, p)dy, q, p ∈ Rn ,

(2.6)

Rn

is also in S(R2n ). We assume Lemma 2.5 for a moment and use it to complete the proof of Proposition 2.4. To do this, we note that for all f and g in S(Rn ), the function ϕ on S(R2n ) defined by ϕ(y, p)  f (y)g(p),

y, p ∈ Rn ,

is obviously in S(R2n ). Hence the function ψ on R2n defined by  p  p ψ(y, p)  f y + , y, p ∈ Rn , g y− 2 2

(2.7)

is also in S(R2n ). Therefore, by (2.4), (2.7), and Lemma 2.5, V (f, g) ∈ S(R2n ). The proof that ψ is in S(R2n ) is left as an exercise. Proof of Lemma 2.5. Let α, β, γ , and δ be multi-indices. Then, by (2.6), (iy)γ eiq·y (∂pδ ϕ)(y, p)dy q α p β (∂qγ ∂pδ 5)(q, p)  q α p β Rn 1 |γ | α iq·y γ β δ  i (∂y e )y p (∂p ϕ)(y, p)dy |α| n i R  (−1)|α| i |γ |−|α| eiq·y ∂yα {y γ p β (∂pδ ϕ)(y, p)}dy (2.8) Rn

n

for all q and p in R . Now, there exists a positive constant Cαβγ δ , depending on α, β, γ , and δ only, such that |∂yα {y γ p β (∂pδ ϕ)(y, p)}| ≤ Cαβγ δ (1 + |y|2 )−N ,

y ∈ Rn ,

(2.9)

n where N is some positive integer greater than . Hence, by (2.8) and (2.9), 2 sup |q α p β (∂qγ ∂pδ 5)(q, p)| ≤ Cαβγ δ (1 + |y|2 )−N dy, q,p∈Rn

and the proof is complete.

Rn

✷

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3 The Wigner Transform

In this chapter, we introduce the Wigner transform and study some of its very basic properties. The Wigner transform W (f ) of a function f in L2 (Rn ), introduced by Wigner in [35], is a tool for the study of the nonexisting joint probability distribution of position and momentum in the state f . In order to study the Weyl transform, it is necessary to have the notion of the Wigner transform of two arbitrary functions in L2 (Rn ). To do this, we begin by computing the Fourier transform of the Fourier–Wigner transform. Theorem 3.1. Let f and g be in S(Rn ). Then ˆ

−n/2

V (f, g) (x, ξ )  (2π )

Proof.



 p  p e−iξ ·p f x + g x− dp, x, ξ ∈ Rn . 2 2 Rn (3.1)

For any positive number ε, we define the function Iε on R2n by

Iε (x, ξ ) 

Rn

Rn

e−

ε 2 |q|2 2

e−ix·q−iξ ·p V (f, g)(q, p)dq dp, x, ξ ∈ Rn .

(3.2)

Then, using Fubini’s theorem and the fact that the Fourier transform of the function ϕ given by ϕ(x)  e−

|x|2 2

,

x ∈ Rn ,

(3.3)

14

3. The Wigner Transform

is equal to ϕ, we get, by (3.2), Iε (x, ξ )

ε 2 |q|2 e− 2 e−ix·q−iξ ·p  (2π )−n/2 n n  R R   p  p iq·y × e f y+ g y− dy dq dp n 2 2 R e−iξ ·p  (2π)−n/2 Rn      ε 2 |q|2 p  p × e−i(x−y)·q e− 2 dq f y + g y− dy dp 2 2 Rn Rn     |x−y|2 p p e−iξ ·p ε −n e− 2ε2 f y + g y− dy dp,  n n 2 2 R R x, ξ ∈ Rn . (3.4)

Now, for each p in Rn , we define the function Fp on Rn by  p  p g y− , y ∈ Rn . Fp (y)  f y + 2 2 Then, by (3.4) and (3.5), Iε (x, ξ )  e−iξ ·p (Fp ∗ ϕε )(x)dp, x, ξ ∈ Rn ,

(3.5)

(3.6)

Rn

where ϕε (x)  ε−n ϕ

x

, x ∈ Rn . ε Note that, for each fixed p in Rn , by (3.3) and (3.5),   n Fp ∗ ϕε → ϕ(x)dx Fp  (2π) 2 Fp

(3.7)

(3.8)

Rn

uniformly on compact subsets of Rn as ε → 0. Let N be any positive integer. Then, by (3.3), (3.5), and (3.7), there exists a positive constant CN such that |(Fp ∗ ϕε )(x)| ≤ Fp L∞ (Rn ) ϕε L1 (Rn )  Fp L∞ (Rn ) ϕ L1 (Rn )   p  p   ≤ (2π)n/2 sup f y + g y−  2 2 y∈Rn ≤ CN (1 + |p|2 )−N ,

x, p ∈ Rn ,

(3.9)

for all positive numbers ε. So, by (3.6), (3.8), (3.9), and the Lebesgue dominated convergence theorem,  p  p n/2 lim Iε (x, ξ )  (2π) dp, x, ξ ∈ Rn . e−iξ ·p f x + g x− ε→0 2 2 Rn (3.10)

3. The Wigner Transform

15

But, using (3.2) and again the Lebesgue dominated convergence theorem, e−ix·q−iξ ·p V (f, g)(q, p)dq dp lim Iε (x, ξ )  ε→0

Rn

Rn n

 (2π) V (f, g)ˆ(x, ξ ),

x, ξ ∈ Rn .

(3.11) ✷

So, by (3.10) and (3.11), (3.1) is valid. n

We can now define the Wigner transform of two functions in S(R ). To do this, let f and g be in S(Rn ). Then the function W (f, g) on R2n , defined by  p  p dp, x, ξ ∈ Rn , e−iξ ·p f x + g x− W (f, g)(x, ξ )  (2π)−n/2 2 2 Rn (3.12) is called the Wigner transform of f and g, and can be used, as in the paper [19] by Moyal, to interpret quantum mechanics as a form of nondeterministic statistical dynamics. Some of the most basic properties of the Wigner transform are given in this chapter. Theorem 3.2. (The Moyal Identity) For all f1 , g1 , f2 , and g2 in S(Rn ), we have W (f1 , g1 ), W (f2 , g2 )  f1 , f2 g1 , g2 .

(3.13)

We define W˜ : S(R ) → S(R ) by  p p −n/2 ˜ dp, x, ξ ∈ Rn , (3.14) e−iξ ·p F x + , x − (W F )(x, ξ )  (2π) 2 2 Rn

Proof.

2n

2n

for all F in S(R2n ). Then, by (3.14) and the Plancherel theorem, (W˜ F1 )(x, ξ )(W˜ F2 )(x, ξ )dx dξ W˜ F1 , W˜ F2   Rn Rn   (W˜ F1 )(x, ξ )(W˜ F2 )(x, ξ )dξ dx  Rn Rn    p p p  p F1 x + , x − F2 x + , x − dp dx  n n 2 2 2 2 R R  p p p  p F1 x + , x − F2 x + , x − dp dx (3.15)  2 2 2 2 Rn Rn p p for all F1 and F2 in S(R2n ). Let u  x + and v  x − . Then, by (3.15), we 2 2 get F1 (u, v)F2 (u, v)du dv W˜ F1 , W˜ F2   Rn

Rn

 F1 , F2 ,

F1 , F2 ∈ S(R2n ).

(3.16)

Now, let f1 , g1 , f2 , and g2 be in S(Rn ). Let F1 and F2 be functions on R2n defined by F1 (u, v)  f1 (u)g1 (v),

u, v ∈ Rn ,

(3.17)

16

3. The Wigner Transform

and F2 (u, v)  f2 (u)g2 (v),

u, v ∈ Rn .

(3.18)

Then, by (3.12), (3.14), and (3.16)–(3.18), W (f1 , g1 ), W (f2 , g2 )  W˜ F1 , W˜ F2   F1 , F2   F1 (u, v)F2 (u, v)du dv n n R R  f1 (u)g1 (v)f2 (u)g2 (v)du dv Rn Rn      f1 (u)f2 (u)du g1 (v)g2 (v)dv Rn

Rn

 f1 , f2 g1 , g2 . ✷ Corollary 3.3. The Moyal identity is also true for the Fourier–Wigner transform V. Corollary 3.4. W : S(Rn ) × S(Rn ) → S(R2n ) can be extended uniquely to a bilinear operator W : L2 (Rn ) × L2 (Rn ) → L2 (R2n ) such that W (f, g) L2 (R2n )  f L2 (Rn ) g L2 (Rn ) for all f and g in L2 (Rn ). Corollary 3.5. The preceding corollary is also true for the Fourier–Wigner transform V . Proposition 3.6. (i)

Let t ∈ R − {0} and let f be any measurable function on Rn . Let f t be the function on Rn defined by n

f t (x)  |t| 2 f (tx),

x ∈ Rn .

Then, for all f and g in S(Rn ), W (f t , g t )(x, ξ )  W (f, g)(tx, t −1 ξ ), (ii)

x, ξ ∈ Rn .

Let a, b, c, and d be in Rn and let f and g be in S(Rn ). Then W (ρ(a, b)f, ρ(c, d)g)(x, ξ ) e

i{(a−c)·x+(b−d)·ξ }

for all x and ξ in Rn . (iii) W (g, f )  W (f, g),

e

1 2 i(a·d−b·c)

  b+d a+c W (f, g) x + ,ξ − 2 2

f, g ∈ S(Rn ).

3. The Wigner Transform

17

Corollary 3.7. Let W (f )  W (f, f ), f ∈ L2 (Rn ). Then (i) (ii)

W (ρ(a, b)f )(x, ξ )  W (f )(x + b, ξ − a), a, b, x, ξ ∈ Rn , W (f ) is real-valued.

The proofs of Corollaries 3.3, 3.4, 3.5, Proposition 3.6, and Corollary 3.7 are left as exercises. Proposition 3.8. Let f ∈ L2 (Rn ) and G ∈ L2 (R2n ). Then (W (f ) ∗ G)(x, ξ )  W (ρ(ξ, −x)f˜), G,

x, ξ ∈ Rn ,

(3.19)

where f˜(x)  f (−x), x ∈ Rn . Proof.

We begin by noting that (W (f ) ∗ G)(x, ξ )  W (f )(x − y, ξ − η)G(y, η)dy dη, x, ξ ∈ Rn . Rn

Rn

(3.20)

But, by part (i) of Proposition 3.6 and part (i) of Corollary 3.7, W (f )(x − y, ξ − η)  W (f˜)(y − x, η − ξ )  W (ρ(ξ, −x)f˜)(y, η), x, y, ξ, η ∈ Rn . So, by (3.20) and (3.21), we get (W (f ) ∗ G)(x, ξ )  Rn

Rn

(3.21)

W (ρ(ξ, −x)f˜)(y, η)G(y, η)dy dη

 W (ρ(ξ, −x)f˜), G for all x and ξ in Rn .

✷ n

Corollary 3.9. Let f and g be in S(R ). Then (W (f ) ∗ W (g))(x, ξ )  (2π)n |V (f˜, g)(ξ, −x)|2 ,

x, ξ ∈ Rn .

Proof. Let G  W (g). Then, by part (ii) of Corollary 3.6, G is real-valued. Thus, by (3.13) and (3.19), (W (f ) ∗ W (g))(x, ξ )  W (ρ(ξ, −x)f˜), Wg  ρ(ξ, −x)f˜, gρ(ξ, −x)f˜, g  |ρ(ξ, −x)f˜, g|2 , x, ξ ∈ Rn . Therefore, by (2.3) and (3.22), we complete the proof.

(3.22) ✷

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4 The Weyl Transform

We can now introduce the Weyl transform and explicate its beautiful connection with the Wigner transform. It is instructive, though absolutely unnecessary, to motivate the definition of the Weyl transform by means of pseudo-differential operators briefly described in Chapter 1. The role of the Weyl transform in quantization is given at the end of the chapter as an impetus for further development of the subject. Let σ ∈ S m , m ∈ R. Then, for any function ϕ in S(Rn ), the function Tσ ϕ on Rn can be written as ei(x−y)·ξ σ (x, ξ )ϕ(y)dy dξ, x ∈ Rn , (4.1) (Tσ ϕ)(x)  (2π)−n Rn

Rn

where the integral in (4.1) is understood to be an iterated integral. There is another way of associating a linear operator from S(Rn ) into S(Rn ) with a symbol σ in S m . Note that for all ϕ in S(Rn ), we can define another function Wσ ϕ on Rn by   x+y ei(x−y)·ξ σ , ξ ϕ(y)dy dξ, x ∈ Rn , (4.2) (Wσ ϕ)(x)  (2π)−n 2 Rn Rn where, again, the integral in (4.2) is understood to be an iterated integral. In order to obtain another representation of Wσ ϕ, we let θ be any function in C0∞ (Rn ) such that θ(0)  1. Then we get the following result. Lemma 4.1. The limit lim (2π)

ε→0+

−n



Rn

Rn

θ(εξ )e

i(x−y)·ξ

 σ

 x+y , ξ ϕ(y)dy dξ 2

20

4. The Weyl Transform

exists and is independent of the choice of the function θ. Moreover, the convergence is uniform with respect to x on Rn . Proof.

Note that for any positive integer N , (1 − >y )N {ei(x−y)·ξ }  (1 + |ξ |2 )N ei(x−y)·ξ , x, y, ξ ∈ Rn .

(4.3)

So, by (4.3) and integration by parts, we get   x+y (2π)−n θ (εξ )ei(x−y)·ξ σ , ξ ϕ(y)dy dξ 2 Rn Rn   x+y  (2π)−n θ (εξ )(1 + |ξ |2 )−N ((1 − >y )N {ei(x−y)·ξ })σ , ξ ϕ(y)dy dξ 2 Rn Rn     x + y  (2π)−n θ (εξ )(1 + |ξ |2 )−N ei(x−y)·ξ (1 − >y )N σ , ξ ϕ(y) dy dξ 2 Rn Rn (4.4) for all x in Rn and all positive numbers ε. Let P (D)  (1 − >)N . Then, by (4.4) and Leibnitz’s formula,   x+y −n i(x−y)·ξ (2π) θ(εξ )e σ , ξ ϕ(y)dy dξ 2 Rn Rn
1  θ(εξ )(1 + |ξ |2 )−N ei(x−y)·ξ (2π)−n n n µ! R R µ   1 x + y × |µ| (D µ σ ) (4.4) , ξ (P (µ) (D)ϕ)(y)dy dξ 2 2 for all x in Rn and all positive numbers ε, where P (µ) (D) is the partial differential operator with symbol P (µ) , where P (µ) (ξ )  (∂ µ P )(ξ ), ξ ∈ Rn . Now, for each fixed x in Rn ,   x+y θ (εξ )(1 + |ξ |2 )−N ei(x−y)·ξ (D µ σ ) , ξ (P (µ) (D)ϕ)(y) 2   x+y −→ (1 + |ξ |2 )−N ei(x−y)·ξ (D µ σ ) (4.5) , ξ (P (µ) (D)ϕ)(y) 2 for all y and ξ in Rn as ε → 0+. Furthermore, there exists a positive constant C such that       θ (εξ )(1 + |ξ |2 )−N ei(x−y)·ξ (D µ σ ) x + y , ξ (P (µ) (D)ϕ)(y)   2 ≤ C(1 + |ξ |2 )−N (1 + |ξ |)m |(P (µ) (D)ϕ)(y)|,

y, ξ ∈ Rn .

(4.6)

Since (1 + |ξ |2 )−N (1 + |ξ |)m |(P (µ) (D)ϕ)(y)| is in L1 (R2n ) as a function of (y, ξ ) on R2n if 2N − m > n, i.e., N > m+n , it 2 follows from (4.4), (4.5), (4.6), and the Lebesgue dominated convergence theorem

4. The Weyl Transform

that lim (2π)−n

ε→0



Rn

Rn

θ(εξ )ei(x−y)·ξ σ



21

 x+y , ξ ϕ(y)dy dξ 2

exists and is independent of the choice of θ. It can also be checked that the convergence is uniform with respect to x on Rn . ✷ From the proof of Lemma 4.1, we can get another formula for Wσ ϕ when ϕ is in S(Rn ). Indeed, (Wσ ϕ)(x)  (2π )−n (1 + |ξ |2 )−N ei(x−y)·ξ (1 − >y )N Rn Rn     x+y × σ , ξ ϕ(y) dy dξ 2 . for all x in Rn , where N is any positive integer greater than m+n 2 The proof of the following proposition is left as an exercise. Proposition 4.2. Let σ ∈ S m , m ∈ R. Then Wσ : S(Rn ) → S(Rn ) is continuous. We call the linear operator Wσ the Weyl transform associated with the symbol σ . The following result illuminates the relationship between the Weyl transform and the Wigner transform and plays a major role in the development of the theory of the Weyl transform in this book. Theorem 4.3. Let σ ∈ S m , m ∈ R. Then Wσ f, g  (2π)−n/2 σ (x, ξ )W (f, g)(x, ξ )dx dξ, f, g ∈ S(Rn ). Rn

Rn

Proof. Let θ be any function in C0∞ (Rn ) such that θ(0)  1. Then, by (3.12), Lemma 4.1, the Lebesgue dominated convergence theorem, and Fubini’s theorem, σ (x, ξ )W (f, g)(x, ξ )dx dξ Rn Rn  lim θ(εξ )σ (x, ξ )W (f, g)(x, ξ )dx dξ ε→0+ Rn Rn  lim (2π)−n/2 θ(εξ )σ (x, ξ ) ε→0+ Rn Rn    p  p × e−iξ ·p f x + g x− dp dx dξ 2 2 Rn  lim (2π)−n/2 θ(εξ ) ε→0+ Rn    p  p −iξ ·p × dp dx dξ. σ (x, ξ )e f x+ g x− 2 2 Rn Rn (4.7)

22

4. The Weyl Transform

Let u  x + p2 and v  x − p2 in the last term in (4.7). Then, by Lemma 4.1, Fubini’s theorem, and the Lebesgue dominated convergence theorem, (4.7) becomes σ (x, ξ )W (f, g)(x, ξ )dx dξ Rn Rn     u+v −n/2 i(v−u)·ξ θ (εξ ) ,ξ e σ f (u)g(v)du dv dξ  lim (2π) ε→0+ n n 2 Rn  R R    u+v −n/2 i(v−u)·ξ g(v) θ(εξ )σ f (u)du dξ dv ,ξ e  lim (2π) ε→0+ 2 Rn Rn Rn ✷ g(v)(Wσ f )(v)dv  (2π)n/2 Wσ f, g.  (2π)−n/2 Rn

We denote the C ∗ -algebra of all bounded linear operators from L2 (Rn ) into 2 L (Rn ) by B(L2 (Rn )). Theorem 4.4. There exists a unique bounded linear operator Q : L2 (R2n ) → B(L2 (Rn )) such that σ (x, ξ )W (f, g)(x, ξ )dx dξ (4.8) (Qσ )f, g  (2π )−n/2 Rn

Rn

and Qσ ∗ ≤ (2π)−n/2 σ L2 (R2n ) n

(4.9)

for all f and g in L (R ) and σ in L (R ), where ∗ denotes the norm in B(L2 (Rn )). 2

Proof.

2

2n

Let σ ∈ S(R2n ). Then, for any f in S(Rn ), we define (Qσ )f by (Qσ )f  Wσ f.

(4.10)

Then, by Theorem 4.3 and (4.10), (Qσ )f, g  Wσ f, g  (2π)−n/2

Rn

Rn

σ (x, ξ )W (f, g)(x, ξ )dx dξ

(4.11)

for all f and g in S(Rn ). Therefore, by Theorem 3.2 and (4.11), |(Qσ )f, g| ≤ (2π)−n/2 σ L2 (R2n ) W (f, g) L2 (R2n )  (2π)−n/2 σ L2 (R2n ) f L2 (Rn ) g L2 (Rn )

(4.12)

n

for all f and g in S(R ). Hence, by (4.12), (Qσ )f L2 (Rn ) ≤ (2π)−n/2 σ L2 (R2n ) f L2 (Rn )

(4.13)

n

for all f in S(R ). Therefore, by (4.13), Qσ ∗ ≤ (2π)−n/2 σ L2 (R2n ) ,

σ ∈ S(R2n ).

(4.14)

2n Now, let σ ∈ L2 (R2n ). Let {σk }∞ k1 be a sequence of functions in S(R ) such that 2 2n σk → σ in L (R ) as k → ∞. Then, by (4.14),

Qσk − Qσl ∗ ≤ (2π)−n/2 σk − σl L2 (R2n ) → 0

4. The Weyl Transform

23

2 n as k, l → ∞. Thus, {Qσk }∞ k1 is a Cauchy sequence in B(L (R )). We define Qσ to be the limit in B(L2 (Rn )) of the sequence {Qσk }∞ . This definition is k1 ∞ independent of the choice of the sequence {σk }∞ . Indeed, let {τ } k k1 be another k1 sequence of functions in S(R2n ) such that τk → σ in L2 (R2n ) as k → ∞. Then, again, by (4.14),

Qσk − Qτk ∗ ≤ (2π)−n/2 σk − τk L2 (R2n ) → 0 ∞ as k → ∞. Thus, the limits in B(L2 (Rn )) of {Qσk }∞ k1 and {Qτk }k1 are equal. ∞ 2 2n Next, let σ ∈ L (R ), and let {σk }k1 be a sequence of functions in S(R2n ) such that σk → σ in L2 (R2n ) as k → ∞. Then, by (4.14),

Qσ ∗  lim Qσk ∗ ≤ (2π)−n/2 lim σk L2 (R2n )  (2π )−n/2 σ L2 (R2n ) , k→∞

k→∞

and (4.9) is proved. Now, if f and g are in S(Rn ), then, by (4.1), (Qσ )f, g  lim (Qσk )f, g k→∞  lim (2π )−n/2 σk (x, ξ )W (f, g)(x, ξ )dx dξ k→∞ n n R R  (2π )−n/2 σ (x, ξ )W (f, g)(x, ξ )dx dξ. Rn

Rn

∞ Finally, let f and g be in L2 (Rn ). Then we pick sequences {fk }∞ k1 and {gk }k1 in n 2 n 2 n S(R ) such that fk → f in L (R ) and gk → g in L (R ) as k → ∞. We have

(Qσ )f, g  lim (Qσ )fk , gk  k→∞  lim (2π )−n/2 σ (x, ξ )W (fk , gk )(x, ξ )dx dξ k→∞ Rn Rn  (2π)−n/2 σ (x, ξ )W (f, g)(x, ξ )dx dξ. Rn

Rn

It is obvious that Q : L (R ) → B(L2 (Rn )) is the only bounded linear operator satisfying (4.8) for all f and g in L2 (Rn ) and σ in L2 (R2n ). ✷ 2

2n

Remark 4.5. From now on, we shall denote Qσ by Wσ for any function σ in L2 (R2n ). In classical mechanics, the phase space used to describe the motion of a particle moving in Rn is given by R2n  {(x, ξ ) :

x, ξ ∈ Rn },

where the variables x and ξ are used to denote the position and momentum of the particle, respectively. The observables of the motion are given by real-valued tempered distributions on R2n . The rules of quantization, with Planck’s constant adjusted to 1, say that a quantum-mechanical model of the motion can be set up using the Hilbert space L2 (Rn ) for the phase space, the multiplication operator on L2 (Rn ) by the function xj for the position variable xj , and the differential operator Dj for the momentum variable ξj . Thus, the quantum-mechanical analogue of

24

4. The Weyl Transform

the classical mechanical observable σ (x, ξ ) should be the linear operator σ (x, D) obtained by direct substitution, provided that D is understood to be the “vector” (D1 , D2 , . . . , Dn ). The mathematical problem then is to define σ (x, D) for, at least, a good class of functions σ on R2n . Let σ ∈ S m , m ≤ 0, say. Can we use the (bounded) pseudo-differential operator Tσ on L2 (Rn ) for σ (x, D)? To answer this question, let us note that in quantum mechanics, observables must be represented by self-adjoint operators. Unfortunately, the pseudo-differential operator Tσ fails in general to be self-adjoint, despite the fact that σ is real-valued. Hence, Tσ is not the correct definition for σ (x, D), and our immediate task is to develop a functional calculus for the Weyl transform. The resulting Weyl calculus, given in Theorem 7.5, shows that σ (x, D) should be defined to be Wσ . Good accounts of the mathematical foundations of quantum mechanics based on self-adjoint operators on Hilbert spaces, pioneered by von Neumann in [32] and adopted in this book, can be found in, e.g., the book [2], by Berezin and Shubin; the book [21], by Prugoveˇcki; and the book [22], by Schechter.

5 Hilbert–Schmidt Operators on L2(Rn)

We give in this chapter a self-contained treatment of Hilbert–Schmidt operators on L2 (Rn ), which play a central role in the development of a functional calculus for the Weyl transform. Let h ∈ L2 (R2n ). Then we define the integral operator Sh : L2 (Rn ) → L2 (Rn ) by (Sh f )(x)  h(x, y)f (y)dy, x ∈ Rn , (5.1) Rn

for all f in L2 (Rn ). To check that Sh f is in L2 (Rn ), we note that by (5.1),  |(Sh f )(x)| dx 2

Rn

 21

 

R

|h(x, y)f (y)| dx 2

Rn

Rn



R

2  21  h(x, y)f (y)dy  dx n



≤

    n

Rn



|f (y)|

 ≤

Rn

 f

Rn

|f (y)|2 dy

L2 (Rn )

 21

h

|h(x, y)|2 dx  21 

L2 (R2n )

dy

 21

dy

Rn

Rn

|h(x, y)|2 dx dy

.

We call Sh the Hilbert–Schmidt operator corresponding to the kernel h.

 21 (5.2)

5. Hilbert–Schmidt Operators on L2 (Rn )

26

Let g and h be in L2 (R2n ). Then we define the function g ◦ h on R2n by (g ◦ h)(x, y)  g(x, z)h(z, y)dz, x, y ∈ Rn . (5.3) Rn

Lemma 5.1. Let g and h be in L2 (R2n ). Then g ◦ h ∈ L2 (R2n ). Proof.

By (5.3),



|(g ◦ h)(x, y)|2 ≤

Rn

  |g(x, z)|2 dz

Rn

 |h(z, y)|2 dz

for all x and y in Rn . By (5.4), |(g ◦ h)(x, y)|2 dx dy Rn Rn     2 2 ≤ |g(x, z)| dz |h(z, y)| dz dx dy Rn Rn Rn Rn       2 2  |h(z, y)| dz |g(x, z)| dz dx dy Rn Rn Rn Rn    |h(z, y)|2 dz dy g 2L2 (R2n )  g 2L2 (R2n ) h 2L2 (R2n ) . Rn

(5.4)

✷

Rn

Theorem 5.2. Let Sg and Sh be the Hilbert–Schmidt operators corresponding to kernels g and h respectively. Then (i) (ii)

Sg Sh  Sg◦h ; Sh∗  Sh∗ ,

where Sh∗ is the adjoint of Sh and h∗ is the function on R2n defined by h∗ (x, y)  h(y, x), Proof.

x, y ∈ Rn .

(5.5)

By (5.3), ((Sg Sh )f )(x)  (Sg (Sh f ))(x)  g(x, z)(Sh f )(z)dz Rn    g(x, z) h(z, y)f (y)dy dz Rn Rn    g(x, z)h(z, y)dz f (y)dy n Rn R  (g ◦ h)(x, y)f (y)dy  (Sg◦h f )(x) Rn

n

n

for all x in R and f in L (R ), provided that we can justify the interchange of the order of integration. To do this, we let I (x)  |g(x, z) h(z, y) f (y)|dy dz, x ∈ Rn , (5.6) 2

Rn

Rn

5. Hilbert–Schmidt Operators on L2 (Rn )

27

and we only need to show that I (x) < ∞ for almost all x in Rn . But, by Lemma 5.1 and (5.6),   I (x)  |g(x, z) h(z, y)|dz |f (y)|dy Rn

Rn



≤ f L2 (Rn )

 Rn

 21

2 Rn

|g(x, z) h(z, y)|dz

dy

 f L2 (Rn ) (|g| ◦ |h|)(x, ·) L2 (Rn ) < ∞ for almost all x in Rn . Next, let f and g be in L2 (Rn ). Then Sh f, g  (Sh f )(x)g(x)dx n  R   h(x, y)f (y)dy g(x)dx Rn Rn   

Rn

f (y) 



Rn

f (x)

Rn

Rn

h(x, y)g(x)dx dy  h(y, x)g(y)dy dx  f, Sh∗ g,

provided that we can interchange the order of integration. But, |h(x, y) f (y) g(x)|dx dy Rn Rn    |g(x)| |h(x, y) f (y)|dy dx Rn

≤ g L2 (Rn )



Rn

Rn



1/2

2 Rn

|h(x, y) f (y)|dy

dx

 g L2 (Rn ) S|h| |f | L2 (Rn ) < ∞, and hence the order of integration can be interchanged.

✷

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6 The Tensor Product in L2(Rn)

In this chapter, we use some basic facts about tensor products of functions in L2 (Rn ) to prove that the set of Weyl transforms with symbols in L2 (R2n ) is equal to the set of Hilbert–Schmidt operators on L2 (Rn ). Let f and g be in L2 (Rn ). Then we define the function f ⊗ g on R2n by (f ⊗ g)(x, y)  f (x)g(y),

x, y ∈ Rn .

(6.1)

We call f ⊗ g the tensor product of f and g. Proposition 6.1. Let f and g be in L2 (Rn ). Then f ⊗ g ∈ L2 (R2n ), and f ⊗ g L2 (R2n )  f L2 (Rn ) g L2 (Rn ) .

(6.2)

Proof. By (6.1), |(f ⊗ g)(x, y)|2 dx dy  |f (x)g(y)|2 dx dy Rn Rn Rn Rn     2 2  |f (x)| dx |g(y)| dy 

Rn Rn 2 2 f L2 (Rn ) g L2 (Rn ) .

Therefore, by (6.3), f ⊗ g ∈ L2 (R2n ), and (6.2) follows. Proposition 6.2. The set of all finite linear combinations of the form m
k1

is dense in L2 (R2n ).

ak fk ⊗ gk , fk , gk ∈ L2 (Rn ),

ak ∈ C,

(6.3) ✷

6. The Tensor Product in L2 (Rn )

30

The proof of Proposition 6.2 is left as an exercise. Let f be in L2 (R2n ). Then |f (x, y)|2 dx dy < ∞. Rn

(6.4)

Rn

Therefore, by (6.4), f (·, y) is in L2 (Rn ) for almost all y in Rn . Similarly, f (x, ·) is in L2 (Rn ) for almost all x in Rn . We denote by F1 f and F2 f the Fourier transforms of f with respect to the “first” and “second” variables, respectively. The inverse Fourier transforms of f with respect to the “first” and “second” variables are denoted by F1−1 f and F2−1 f , respectively. Proposition 6.3. F1 and F2 are unitary operators on L2 (R2n ). Proposition 6.4. Let f and g be in L2 (Rn ). Then (i) (ii)

F1 (f ⊗ g)  (Ff ) ⊗ g, F2 (f ⊗ g)  f ⊗ (Fg).

The proofs of Propositions 6.3 and 6.4 are left as exercises. Remark 6.5. We can obtain analogues of Propositions 6.3 and 6.4 if F1 , F2 , and F are replaced by F1−1 , F2−1 , and F −1 , respectively. We define the linear operator T : L2 (R2n ) → L2 (R2n ) by  y y , x, y ∈ Rn , (Tf )(x, y)  f x + , x − 2 2

(6.5)

for all f in L2 (R2n ). We call T the twisting operator on L2 (R2n ). Proposition 6.6. The twisting operator T : L2 (R2n ) → L2 (R2n ) is unitary, and   x+y −1 (6.6) , x − y , x, y ∈ Rn , (T f )(x, y)  f 2 for all f in L2 (R2n ). The proof of Proposition 6.6 is also left as an exercise. Now we define the linear operator K : L2 (R2n ) → L2 (R2n ) by (Kf )(x, y)  (T −1 F2 f )(y, x),

x, y ∈ Rn ,

(6.7)

for all f in L2 (R2n ). Proposition 6.7. The linear operator K on L2 (R2n ) defined by (6.7) has the following properties: (i) (ii) (iii) (iv)

K : L2 (R2n ) → L2 (R2n ) is a unitary operator. K  T −1 F2−1 . K f¯  (Kf )∗ , f ∈ L2 (R2n ). W (f, g)  K −1 (f ⊗ g), ¯ f, g ∈ L2 (Rn ).

6. The Tensor Product in L2 (Rn )

31

Proof. Part (i) is obvious. For part (ii), we note that in view of Proposition 6.2, it is enough to prove that K(f ⊗ g)  T −1 F2−1 (f ⊗ g),

f, g ∈ L2 (Rn ).

(6.8)

n

But for all f and g in L (R ), by (6.7) and Propositions 6.4 and 6.6, 2

K(f ⊗ g)(x, y)  T −1 F2 (f ⊗ g)(y, x)  T −1 (f ⊗ Fg)(y, x)   y+x (Fg)(y − x), f 2

x, y ∈ Rn ,

(6.9)

and T −1 F2−1 (f ⊗ g)(x, y)  T −1 (f ⊗ F −1 g)(x, y)   x+y f (F −1 g)(x − y) 2   x+y (Fg)(y − x), x, y ∈ Rn . (6.10) f 2 Therefore, by (6.9) and (6.10), (6.8) is true, and part (ii) is proved. To prove part (iii), again, in view of Proposition 6.2, it is enough to prove the formula for functions of the form f ⊗ g, f, g ∈ L2 (Rn ). But for all x and y in Rn , we get, by (6.7) and Propositions 6.4 and 6.6, K(f ⊗ g)(x, y)  K(f ⊗ g)(x, y)  T −1 F2 (f ⊗ g)(y, x)  T −1 (f ⊗ Fg)(y, x)   y+x (Fg)(y − x), f 2

x, y ∈ Rn .

Next, for all x and y in Rn , we get, by (5.5), part (ii), and (6.6),   x+y ∗ (Fg)(y − x). (K(f ⊗ g)) (x, y)  K(f ⊗ g)(y, x)  f 2

(6.11)

(6.12)

Thus, by (6.11) and (6.12), the proof of part (iii) is complete. Finally, by (3.12), (6.1), and (6.5),  p  p −n/2 e−iξ ·p f x + g x− dp W (f, g)(x, ξ )  (2π) n 2 2 R e−iξ ·p T (f ⊗ g)(x, p)dp  (2π)−n/2 Rn

 F2 T (f ⊗ g)(x, ξ ),

x, ξ ∈ Rn ,

(6.13)

n

for all f and g in S(R ). So, by Corollary 3.4 and (6.13), W (f, g)  F2 T (f ⊗ g),

f, g ∈ L2 (Rn ).

(6.14)

6. The Tensor Product in L2 (Rn )

32

But by part (ii) and (6.14), K  T −1 F2−1 ⇒ K −1  F2 T ⇒ W (f, g)  K −1 (f ⊗ g) for all f and g in L2 (Rn ), and hence, by Proposition 6.2, part (iv) is proved.

✷

We can now give the following important property of the Weyl transform. Theorem 6.8. Let σ ∈ L2 (R2n ). Then Wσ : L2 (Rn ) → L2 (Rn ) is a Hilbert– Schmidt operator with kernel (2π)−n/2 Kσ . Proof. Let f and g be in L2 (Rn ). Then, by (4.8), Remark 4.5 and parts (i), (iii), and (iv) of Proposition 6.7, we get −n/2 σ (x, ξ )W (f, g)(x, ξ )dx dξ Wσ f, g  (2π) Rn

Rn

 (2π)−n/2 W (f, g), σ   (2π)−n/2 K −1 (f ⊗ g), σ   (2π)−n/2 f ⊗ g, (Kσ )∗ .

(6.15)

Therefore, by (5.1), (5.5), (6.1), and (6.15), −n/2 f (x)g(y)(Kσ )(y, x)dx dy Wσ f, g  (2π) n Rn  R  −n/2 (Kσ )(y, x)f (x)dx g(y)dy  (2π) Rn

Rn

f, g ∈ L2 (Rn ),

 Sk f, g,

where k is the function on R2n defined by k(x, y)  (2π)−n/2 (Kσ )(x, y),

x, y ∈ Rn , ✷

and the proof is complete.

We now know that if σ ∈ L2 (R2n ), then Wσ is a Hilbert–Schmidt operator with kernel (2π )−n/2 Kσ . Suppose that A is an arbitrary Hilbert–Schmidt operator. Is it necessarily of the form Wσ for some σ in L2 (R2n )? The answer to the question is yes. Indeed, if A  Sh

(6.16)

for some h in L (R ), then we let σ be the function on R 2

2n

n/2

σ  (2π)

K

−1

h.

2n

given by (6.17)

Then, by (6.17) and Theorem 6.8, Wσ  Sk ,

(6.18)

k  (2π)−n/2 Kσ  h.

(6.19)

where

Therefore, by (6.16), (6.18), and (6.19), Wσ  Sh  A.

7 H ∗-Algebras and the Weyl Calculus

This chapter is an account, based on the paper [20] by Pool, of a functional calculus for the Weyl transform with symbol in L2 (R2n ). In addition to the identification of Weyl transforms with symbols in L2 (R2n ) with Hilbert–Schmidt operators on L2 (Rn ) proved in the preceding chapter, we need the notion of an H ∗ -algebra studied by Ambrose in [1]. Let H be a complex and separable Hilbert space in which the norm and inner product are denoted by and  , , respectively. Suppose that we are given two operations in H , a, b a

→ ab, → a ∗ ,

satisfying the following conditions. (i) a(b + c)  ab + ac, (a + b)c  ac + bc, (ii) λ(ab)  (λa)b  a(λb), (iii) a(bc)  (ab)c, (iv) a ∗∗  a, (v) (a + b)∗  a ∗ + b∗ , (vi) (ab)∗  b∗ a ∗ , ¯ ∗, (vii) (λa)∗  λa (viii) a ∗  a , (ix) ab ≤ a b , (x) ab, c  b, a ∗ c, for all a, b, and c in H , and λ in C. Then we call H an H ∗ -algebra with respect to the given operations.

34

7. H ∗ -Algebras and the Weyl Calculus

It is easy to prove the following proposition. Proposition 7.1. The Hilbert space L2 (R2n ) equipped with the two operations f, g f

→ f ◦ g, → f ∗ ,

defined by (5.3) and (5.5), respectively, is an H ∗ -algebra. The H ∗ -algebra so obtained is denoted by (L2 (R2n ), ◦, ∗). We denote by H S(L2 (Rn )) the set of all Hilbert–Schmidt operators on L2 (Rn ). Let A and B be in H S(L2 (Rn )). Then we define (A, B)H S by (A, B)H S 

∞


Aϕk , Bϕk ,

(7.1)

k1

where {ϕk : k  1, 2, . . .} is an orthonormal basis for L2 (Rn ). It can be shown that the definition of (A, B)H S given by (7.1) is independent of the choice of the orthonormal basis {ϕk : k  1, 2, . . .}. It can also be shown that ( , )H S given by (7.1) is an inner product in H S(L2 (Rn )). It is easy to prove the following proposition. Proposition 7.2. The space H S(L2 (Rn )) equipped with the inner product ( , )H S given by (7.1) and the operations of taking the usual products and adjoints is an H ∗ -algebra. The H ∗ -algebra in Propostion 7.2 is again denoted by H S(L2 (Rn )). The following proposition will be used later. Proposition 7.3. The H ∗ -algebra (L2 (R2n ), ◦, ∗) is isometrically ∗-isomorphic to the H ∗ -algebra H S(L2 (Rn )) under the mapping h → Sh , where Sh is given by (5.1). The proofs of Propositions 7.1, 7.2, and 7.3 are left as exercises. Let f and g be in L2 (R2n ). Then we define the function f × g on R2n by f × g  K −1 (Kf ◦ Kg),

(7.2)

where K : L (R ) → L (R ) is given by (6.7). The following proposition provides us with another H ∗ -algebra structure on L2 (R2n ). 2

2n

2

2n

Proposition 7.4. The Hilbert space L2 (R2n ) equipped with the two operations f, g

→

f × g,

f

→

f,

is an H ∗ -algebra. Moreover, the linear operator K : L2 (R2n ) → L2 (R2n ) defined by (6.7) is an isometric ∗-isomorphism of the H ∗ -algebra (L2 (R2n ), ×, −) onto the H ∗ -algebra (L2 (R2n ), ◦, ∗). Proposition 7.4 is an immediate consequence of (7.2) and parts (i) and (ii) of Proposition 6.7. The task of providing the details is left as an exercise.

7. H ∗ -Algebras and the Weyl Calculus

35

A functional calculus for the Weyl transform is provided by the following theorem. Theorem 7.5. The mapping n

L2 (R2n )  σ → (2π) 2 Wσ ∈ H S(L2 (Rn )) is an isometric ∗-isomorphism of the H ∗ -algebra (L2 (R2n ), ×, −) onto the H ∗ algebra H S(L2 (R2n )). Consequently, we have (i) (ii) (iii) (iv) (v)

Wσ∗  Wσ¯ , Wσ Wτ  W(2π )−n/2 σ ×τ , Wσ + Wτ  Wσ +τ , λWσ  Wλσ , Wσ ∗ ≤ (2π )−n/2 σ L2 (R2n )  Wσ H S ,

where H S is the norm in H S(L2 (Rn )) induced by the inner product ( , )H S given by (7.1). Proof.

We have K

(L2 (R2n ), ×, −)

✲

(L2 (R2n ), ◦, ∗)

S ❥ Fig. 1

❄ H S(L2 (Rn ))

where S is the mapping defined by Sh  Sh ,

h ∈ L2 (R2n ),

and Sh is given by (5.1). Then it is clear from Fig. 1 that L2 (R2n )  σ → SKσ ∈ H S(L2 (Rn )) is an isometric ∗-isomorphism from (L2 (R2n ), ×, −) onto H S(L2 (Rn )). By Theorem 6.8, we get SKσ  (2π)n/2 Wσ , and hence complete the proof of the theorem.

✷

By part (i) of Theorem 7.5, we see that Wσ is self-adjoint if and only if σ is real-valued. Thus, a good model for the quantization σ (x, D) of the classical mechanical observable σ (x, ξ ), discussed at the end of Chapter 4, is Wσ instead of Tσ .

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8 The Heisenberg Group

Although the formula provided by part (ii) of Theorem 7.5 is very useful for the actual computation of the symbol of the product of two Weyl transforms with symbols in L2 (R2n ), it is inadequate in revealing its connections with some fundamental notions of quantum mechanics. In order to obtain a conceptual formula for the symbol of the product of two Weyl transforms Wσ and Wτ , where σ and τ are in L2 (R2n ), we use the Heisenberg group and the twisted convolution studied, respectively, in this and the next chapter. We identify any point (q, p) in R2n with the point z  q + ip in Cn . We define the symplectic form [ , ] on Cn by [z, w]  2Im(z · w),

z, w ∈ Cn ,

where z  (z1 , z2 , . . . , zn ), w  (w1 , w2 , . . . , wn ), and z · w¯ 

n


zj w¯ j .

j 1

In the following proposition, we give some easy properties of [ , ]. Proposition 8.1. Let ζ, z, and w be in Cn , and let λ and µ be in R. Then (i) (ii)

[λζ + µz, w]  λ[ζ, w] + µ[z, w], [ζ, λz + µw]  λ[ζ, z] + µ[ζ, w],

(8.1)

38

8. The Heisenberg Group

(iii) [z, w]  −[w, z], (iv) [z, z]  0. The proof of Proposition 8.1 is easy and left as an exercise. We define the multiplication · on Cn × R by (z, t) · (w, s)  (z + w, t + s + [z, w]) n

(8.2) n

for all (z, t) and (w, s) in C × R, where [ , ] is the symplectic form on C defined by (8.1). It is obvious that · is a binary operation on Cn × R. Proposition 8.2. Cn × R is a group with respect to the multiplication defined by (8.2). Proof.

Let (ζ, u), (z, t), and (w, s) be in Cn × R. Then, by (8.2),

((ζ, u) · (z, t)) · (w, s)  (ζ + z, u + t + [ζ, z]) · (w, s)  (ζ + z + w, u + t + [ζ, z] + s + [ζ + z, w])  (ζ + z + w, u + t + s + [ζ, z] + [ζ, w] + [z, w]). (8.3) Also, by (8.2), (ζ, u) · ((z, t) · (w, s))  (ζ, u) · (z + w, t + s + [z, w])  (ζ + z + w, u + t + s + [z, w] + [ζ, z + w])  (ζ + z + w, u + t + s + [z, w] + [ζ, z] + [ζ, w]). (8.4) Therefore, by (8.3) and (8.4), the associative law is valid. Now, for all (z, t) in Cn × R, we have, by (8.2), (z, t) · (0, 0)  (z, t + [z, 0])  (z, t), and (0, 0) · (z, t)  (z, t + [0, z])  (z, t). Therefore, (0, 0) is the identity element in Cn × R. Finally, for all (z, t) in Cn × R, we get, by (8.2), parts (ii) and (iv) of Proposition 8.1, (z, t) · (−z, −t)  (0, 0 + [z, −z])  (0, 0), and, by (8.2), parts (i) and (iv) of Proposition 8.1, (−z, −t) · (z, t)  (0, 0 + [−z, z])  (0, 0). Therefore, the inverse of any element (z, t) in Cn × R is equal to (−z, −t).

✷

The group Cn × R with respect to the multiplication defined by (8.2) is denoted by H n and is called the Heisenberg group. This terminology stems from the fact that the structure equations of the Lie algebra of H n , i.e., the vector space of all left-invariant vector fields on H n equipped with the bracket operation of two left-invariant vector fields given by their commutator, satisfy the canonical commutation relations, due to Heisenberg, in quantum mechanics.

8. The Heisenberg Group

39

For each real number λ, we define the mapping R λ from H n into the group G of all unitary operators on L2 (Rn ) by (R λ (z, t)f )(x)  eiλ(q·x+ 2 q·p+ 4 t) f (x + p), 1

n

x ∈ Rn ,

1

n

(8.5) λ

for all (z, t) in H and f in L (R ), where z  q + ip. It is easy to see that R (z, t) is indeed a unitary operator on L2 (Rn ) for all (z, t) in H n . 2

Proposition 8.3. Let λ be a fixed real number. Then (i) (ii)

R λ : H n → G is a group homomorphism, R λ (z, t)f → f in L2 (Rn ) as (z, t) → (0, 0).

Remark 8.4. For any fixed real number λ, R λ is a unitary representation of H n on L2 (Rn ). We call it the Schr¨odinger representation of H n on L2 (Rn ) with parameter λ. Proof of Proposition 8.3. Let (z, t)  (q + ip, t) and (z , t  )  (q  + ip , t  ) be in H n , and let f be in L2 (Rn ). Then, by (8.5), (R λ (z, t)R λ (z , t  )f )(x)  eiλ(q·x+ 2 q·p+ 4 t) (R λ (z , t  )f )(x + p) 1

1







1 

 eiλ(q·x+ 2 q·p+ 4 t) eiλ(q ·(x+p)+ 2 q ·p + 4 t ) f (x + p + p  ) 1

1

1

(8.6) for all x in Rn . We also have, by (8.2) and (8.5), (R λ ((z, t) · (z , t  ))f )(x)  (R λ (z + z , t + t  + [z, z ])f )(x) 









 eiλ((q+q )·x+ 2 (q+q )·(p+p )+ 4 (t+t +[z,z ])) f (x + p + p  ) 1

1

(8.7) for all x in Rn . But, by (8.1), [z, z ]  2Im(q + ip) · (q  − ip  )  2(q  · p − q · p  ).

(8.8)

Therefore, by (8.7) and (8.8), (R λ ((z, t) · (z , t  ))f )(x) 















 eiλ((q+q )·x+ 2 q·p+ 2 q·p + 2 q ·p+ 2 q ·p + 4 (t+t )+ 2 (q ·p−q·p )) f (x + p + p  ) (8.9) 1

1

1

1

1

1

for all x in Rn . Hence, by (8.6) and (8.9), R λ (z, t)R λ (z , t  )  R λ ((z, t) · (z , t  )), and part (i) is proved. To prove part (ii), let f be in L2 (Rn ). Then, for all (z, t)  (q + ip, t) in H n , we get, by (8.5), R λ (z, t)f − f 2L2 (Rn )  |(R λ (z, t)f )(x) − f (x)|2 dx Rn 1 1  |eiλ(q·x+ 2 q·p+ 4 t) f (x + p) − f (x)|2 dx Rn

40

8. The Heisenberg Group



|eiλ(q·x+ 2 q·p+ 4 t) {f (x + p) − f (x)} 1

Rn

1

+ eiλ(q·x+ 2 q·p+ 4 t) f (x) − f (x)|2 dx |f (x + p) − f (x)|2 dx ≤2 Rn 1 1 |eiλ(q·x+ 2 q·p+ 4 t) f (x) − f (x)|2 dx. +2 1

1

(8.10)

Rn

By the L2 -continuity of translations, |f (x + p) − f (x)|2 dx → 0

(8.11)

Rn

as p → 0. For almost all x in Rn , |eiλ(q·x+ 2 q·p+ 4 t) f (x) − f (x)|2 → 0

(8.12)

|eiλ(q·x+ 2 q·p+ 4 t) f (x) − f (x)|2 ≤ 4|f (x)|2 .

(8.13)

1

1

as (z, t) → (0, 0), and 1

1

Hence, by (8.12), (8.13), and the Lebesgue dominated convergence theorem, 1 1 |eiλ(q·x+ 2 q·p+ 4 t) f (x) − f (x)|2 dx → 0 (8.14) Rn

as (z, t) → (0, 0). Therefore, by (8.10), (8.11), and (8.14), R λ (z, t)f − f L2 (Rn ) → 0 as (z, t) → (0, 0), and the proof is complete.

✷

Proposition 8.5. Let λ be a fixed real number. Then the unitary representation R λ of H n on L2 (Rn ) is irreducible in the sense that the only closed subspaces of L2 (Rn ) that are invariant under all the operators R λ (z, t), (z, t) ∈ H n , are {0} and L2 (Rn ). To prove Proposition 8.5, it is enough to prove that any bounded linear operator on L2 (Rn ) that commutes with R λ (z, t) for all (z, t) in H n is a scalar multiple of the identity operator on L2 (Rn ). For then, if M is a closed subspace of L2 (Rn ) that is invariant under all the operators R λ (z, t), (z, t) ∈ H n , then so is the orthogonal complement M ⊥ of M. Indeed, if f ∈ M ⊥ and g ∈ M, then, using the fact that R λ is a unitary representation of H n on L2 (Rn ) and the fact that (−z, −t) is the inverse of (z, t) for all (z, t) in H n , we get R λ (z, t)f, g  f, R λ (−z, −t)g  0, (z, t) ∈ H n . Now let P be the orthogonal projection of L2 (Rn ) onto M. Then, for all (z, t) in H n and f in L2 (Rn ), P R λ (z, t)f  P R λ (z, t)(f1 + f2 ),

(8.15)

8. The Heisenberg Group

41

where f  f1 + f2 , f1 ∈ M, and f2 ∈ M ⊥ . Thus, using (8.15) and the fact that M and M ⊥ are both invariant under all the operators R λ (z, t), (z, t) ∈ H n , we get P R λ (z, t)f  P R λ (z, t)f1  R λ (z, t)f1 ,

(z, t) ∈ H n .

(8.16)

Since R λ (z, t)Pf  R λ (z, t)f1 ,

(z, t) ∈ H n ,

(8.17)

it follows from (8.16) and (8.17) that P R λ (z, t)  R λ (z, t)P , (z, t) ∈ H n , and hence P is a scalar multiple of the identity operator on L2 (Rn ). Thus, M  L2 (Rn ) or M  {0}. So, we now let A be a bounded linear operator on L2 (Rn ) that commutes with all the operators R λ (z, t), (z, t) ∈ H n , i.e., (R λ (z, t)Af )(x)  (AR λ (z, t)f )(x),

x ∈ Rn ,

(8.18)

for all (z, t) ∈ H n . Therefore, by (8.5) and (8.18), eiλ(q·x+ 2 q·p+ 4 t) (Af )(x + p)  (Aeiλ(q·?+ 2 q·p+ 4 t) f (? + p))(x) 1

1

1

1

(8.19)

for all x in Rn and (z, t) in H n . Let q  0 and t  0 in (8.19). Then x, p ∈ Rn ,

((Tp A)f )(x)  ((ATp )f )(x), n

(8.20) n

where, for any measurable function g on R , Tp g is the function on R defined in Proposition 1.9; i.e., A commutes with translations on Rn . Hence, by (8.20), there exists a function τ in L∞ (Rn ) such that (Af )ˆ  τ fˆ,

f ∈ L2 (Rn ).

(8.21)

Now, let p  0 and t  0 in (8.19). Then ((Mλq A)f )(x)  ((AMλq )f )(x),

x, q ∈ Rn ,

(8.22)

where for any measurable function g on Rn , Mλq g is the function on Rn defined in Proposition 1.9 by (Mλq g)(x)  eiλq·x g(x),

x ∈ Rn ;

(8.23)

i.e., A commutes with modulations on Rn . Thus, by (8.21), (8.22), and (8.23), we get −n/2 ((AMλq )f )(x)  (2π ) eix·ξ τ (ξ )(Mλq f )ˆ(ξ )dξ Rn  (2π)−n/2 eix·ξ τ (ξ )fˆ(ξ − λq)dξ n R −n/2  (2π ) eix·(η+λq) τ (η + λq)fˆ(η)dη Rn ix·λq −n/2 e (2π) eix·η τ (η + λq)fˆ(η)dη, x ∈ Rn , (8.24) Rn

42

8. The Heisenberg Group

for all f in S(Rn ). But by (8.21) and (8.23), ((Mλq A)f )(x)  eix·λq (2π)−n/2 eix·η τ (η)fˆ(η)dη, x ∈ Rn ,

(8.25)

Rn

for all f in S(Rn ). Hence, by (8.24) and (8.25), τ (η + λq)  τ (η)

(8.26)

for almost all η and q in Rn . Therefore, by (8.26), τ is equal to a constant a.e. on Rn . Consequently, by (8.21), A is a scalar multiple of the identity operator on L2 (Rn ), and the proof that R λ is irreducible is complete. Two irreducible unitary representations R1 and R2 of H n on L2 (Rn ) are said to be unitarily equivalent if there is a unitary operator U on L2 (Rn ) such that U R1 (z, t)  R2 (z, t)U, (z, t) ∈ H n .

(8.27)

Proposition 8.6. Two unitary representations R λ and R µ of H n on L2 (Rn ) are unitarily equivalent if and only if λ  µ. Proof. It is enough to prove that if R λ and R µ are unitarily equivalent, then λ  µ. By (8.27), there is a unitary operator on L2 (Rn ) such that U R λ (z, t)  R µ (z, t)U, (z, t) ∈ H n .

(8.28)

Let z  0 in (8.28). Then U R λ (0, t)  R µ (0, t)U, t ∈ R.

(8.29)

So, by (8.5) and (8.29), U e 4 iλt  e 4 iµt U, t ∈ R, 1

1

and hence λ  µ.

✷

Remark 8.7. By a theorem of Stone and von Neumann, we know that up to unitary equivalence, the only irreducible and unitary representations of H n on L2 (Rn ) are given by {R λ : −∞ < λ < ∞} and {Rα,β : α, β ∈ Rn }, where R λ is given by (8.5) for all λ in R and (Rα,β (z, t)f )(x)  ei(α·q+β·p) f (x) for all α and β in Rn , x in Rn , f ∈ L2 (Rn ), and (z, t)  (q + ip, t) in H n . Thus, the only nontrivial irreducible and unitary representations of H n on L2 (Rn ) are given by {R λ : −∞ < λ < ∞}. Remark 8.8. It is interesting to note that the unitary operator ρ(q, p), defined by (2.1) for all q and p in Rn , is equal to R 1 (z, 0), where z  q + ip. This observation will be useful to us in the next chapter.

9 The Twisted Convolution

The aim of this chapter is to express the symbol of the product of two Weyl transforms with symbols in L2 (R2n ) in terms of a twisted convolution, which we now define. Let λ be a fixed real number. Then we define the twisted convolution f ∗λ g of two measurable functions f and g on Cn by f (z − w)g(w)eiλ[z,w] dw, z ∈ Cn , (9.1) (f ∗λ g)(z)  Cn

where [z, w] is the symplectic form of z and w defined by (8.1), dw is the Lebesgue measure on Cn , and provided that the integral exists. Proposition 9.1. Let f and g be measurable functions on Cn such that (f ∗λ g)(z) exists at the point z in Cn . Then (g ∗−λ f )(z) exists, and (f ∗λ g)(z)  (g ∗−λ f )(z). Proof. In (9.1), we change the variable of integration from w to ζ by w  z − ζ . Then we get (f ∗λ g)(z)  g(z − ζ )f (ζ )eiλ[z,z−ζ ] dζ. (9.2) Cn

By (9.1), (9.2), and parts (i) and (iv) of Proposition (8.1), we get g(z − ζ )f (ζ )e−iλ[z,ζ ] dζ (f ∗λ g)(z)  Cn

 (g ∗−λ f )(z).

✷

44

9. The Twisted Convolution

Remark 9.2. It is clear from Proposition 9.1 that the twisted convolution is, in general, noncommutative. We can now give a formula for the product Wσ Wτ of two Weyl transforms Wσ and Wτ in terms of a twisted convolution of σ and τ . We begin with the case when both σ and τ are in S(R2n ). To do this, let ϕ and ψ be in S(Rn ). Then, by (2.3), Theorem 3.1, (3.12), Theorem 4.3, and the adjoint formula in the theory of the Fourier transform, Wσ ϕ, ψ  (2π )−n σˆ (q, p)ρ(q, p)ϕ, ψdq dp. (9.3) Rn

Rn

Therefore, by (9.3) and Fubini’s theorem,   Wσ ϕ, ψ  (2π )−n σˆ (q, p) (ρ(q, p)ϕ)(x)ψ(x)dx dq dp Rn Rn Rn   ψ(x) σˆ (z)(ρ(z)ϕ)(x)dz dx,  (2π)−n Rn

Cn

and hence (Wσ ϕ)(x)  (2π)−n

Cn

σˆ (z)(ρ(z)ϕ)(x)dz, x ∈ Rn ,

(9.4)

for all ϕ in S(Rn ). But by (2.1) and (9.4), (ρ(z)(Wτ ϕ))(x)  ei(q·x+ 2 q·p) (Wτ ϕ)(x + p) 1  ei(q·x+ 2 q·p) (2π)−n τˆ (w)(ρ(w)ϕ)(x + p)dw Cn τˆ (w)(ρ(z)ρ(w)ϕ)(x)dw, x ∈ Rn ,  (2π )−n 1

(9.5)

Cn

for all ϕ in S(Rn ). Thus, by (9.4) and (9.5), −2n (Wσ Wτ ϕ)(x)  (2π ) σˆ (z)τˆ (w)(ρ(z)ρ(w)ϕ)(x)dz dw, x ∈ Rn . Cn

Cn

(9.6) Now, by (2.1), (8.2), (8.5), the fact that R 1 is a unitary representation of H n on L2 (Rn ), and Remark 8.8, we get ρ(z)ρ(w)  R 1 (z, 0)R 1 (w, 0)  R 1 (z + w, [z, w])  ρ(z + w)e 4 i[z,w] , z, w ∈ Cn . 1

So, by (9.6) and (9.7), (Wσ Wτ ϕ)(x)  (2π)−2n



Cn

(9.7)

σˆ (z)τˆ (w)(ρ(z+w)ϕ)(x)e 4 i[z,w] dz dw, x ∈ Rn , 1

Cn

(9.8) for all ϕ in S(Rn ). Now, in (9.8), we change the variable z to ζ by z  ζ − w. Then, by (9.1) and parts (i) and (iv) of Proposition (8.1), we get 1 −2n (Wσ Wτ ϕ)(x)  (2π) σˆ (ζ − w)τˆ (w)(ρ(ζ )ϕ)(x)e 4 i[ζ −w,w] dζ dw Cn

Cn

9. The Twisted Convolution

 (2π )−2n  (2π )−2n  (2π)−2n



Cn

C

n

Cn

45

σˆ (ζ − w)τˆ (w)(ρ(ζ )ϕ)(x)e 4 i[ζ,w] dζ dw   1 σˆ (ζ − w)τˆ (w)e 4 i[ζ,w] dw (ρ(ζ )ϕ)(x)dζ 1

Cn

Cn

(σˆ ∗ 1 τˆ )(ζ )(ρ(ζ )ϕ)(x)dζ, x ∈ Rn . 4

(9.9)

Hence, by (9.4) and (9.9), Wσ Wτ  Wω ,

(9.10)

ωˆ  (2π )−n (σˆ ∗ 1 τˆ ).

(9.11)

where 4

Theorem 9.3. Let σ and τ be in L2 (R2n ). Then Wσ Wτ  Wω , where ωˆ  (2π)−n (σˆ ∗ 1 τˆ ). 4

Proof.

Let

{σk }∞ k1

and

{τk }∞ k1

be sequences of functions in S(R2n ) such that σk → σ

(9.12)

τk → τ

(9.13)

and

in L2 (R2n ) as k → ∞. Then, by (9.10) and (9.11), Wσk Wτk  Wωk ,

(9.14)

ωˆ k  (2π)−n (σˆ k ∗ 1 τˆk )

(9.15)

where 4

for k  1, 2, . . . . Now, by Remark 4.5, (4.9), (9.12), and (9.13), Wσk → Wσ

(9.16)

Wτk → Wτ

(9.17)

and in B(L2 (Rn )) as k → ∞. So, by (9.16) and (9.17), Wωk  Wσk Wτk → Wσ Wτ

(9.18)

n

in B(L (R )) as k → ∞. By (9.1), (9.12), and (9.13), 1 (σˆ k ∗ 1 τˆk )(z)  σˆ k (z − w)τˆk (w)e 4 i[z,w] dw 4 n C 1 → σˆ (z − w)τˆ (w)e 4 i[z,w] dw 2

Cn

 (σˆ ∗ 1 τˆ )(z) 4

(9.19)

46

9. The Twisted Convolution

for almost all z in Cn as k → ∞. On the other hand, by (9.12)–(9.15) and part (v) of Theorem 7.5, we get σˆ k ∗ 1 τˆk − σˆ j ∗ 1 τˆj L2 (R2n ) 4

4

 σˆ k ∗ 1 τˆk − σˆ k ∗ 1 τˆj + σˆ k ∗ 1 τˆj − σˆ j ∗ 1 τˆj L2 (R2n ) 4

4

4

4

≤ σˆ k ∗ 1 (τk − τj )ˆ L2 (R2n ) + (σk − σj )ˆ∗ 1 τˆj L2 (R2n ) 4 3n 2

4

3n 2

 (2π ) Wσk Wτk −τj H S + (2π) Wσk −σj Wτj H S 3n

3n

≤ (2π ) 2 Wσk H S Wτk −τj H S + (2π) 2 Wσk −σj H S Wτj H S n

 (2π ) 2 ( σk L2 (R2n ) τk − τj L2 (R2n ) + σk − σj L2 (R2n ) τj L2 (R2n ) ) →0 as k, j → ∞. Hence, by the Plancherel theorem, there exists a function ω such that σˆ k ∗ 1 τˆk → (2π)n ωˆ 4

(9.20)

in L2 (R2n ) as k → ∞. Therefore, by (9.20), there exists a subsequence {σˆ k ∗ 1 4 τˆk }∞ ˆ k ∗ 1 τˆk }∞ k  1 of {σ k1 such that 4

σˆ k ∗ 1 τˆk → (2π)n ωˆ 4

(9.21)

a.e. on R2n as k  → ∞. Thus, by (9.19) and (9.21), (2π)n ωˆ  σˆ ∗ 1 τˆ 4

(9.22)

a.e. on R2n . By (9.15), (9.20), and the Plancherel theorem, ωk → ω

(9.23)

in L (R ) as k → ∞. Thus, by Remark 4.5, (4.9), and (9.23), 2

2n

Wωk → Wω n

(9.24)

in B(L (R )) as k → ∞. So, by (9.18), (9.22), and (9.24), the proof of the theorem is complete. ✷ 2

10 The Riesz–Thorin Theorem

In Chapters 11–14, we shall define the Weyl transform Wσ corresponding to a tempered distribution σ on R2n and decide whether or not the resulting Weyl transform Wσ , for σ in Lr (R2n ), 1 ≤ r ≤ ∞, is a bounded, or even compact, linear operator from L2 (Rn ) into L2 (Rn ). We need an interpolation theorem of Riesz and Thorin for this task and also for the study of localization operators in Chapter 15. The proof of the interpolation theorem is based on a fact in complex analysis that we now present. Theorem 10.1. (The Three Lines Theorem) Let f be a continuous and bounded function on the strip B  {z ∈ C : α ≤ Rez ≤ β}, where α and β are real numbers such that α < β. Suppose that f is analytic on the interior of B and that there exist constants M1 and M2 such that |f (α + iy)| ≤ M1 ,

−∞ < y < ∞,

(10.1)

|f (β + iy)| ≤ M2 ,

−∞ < y < ∞.

(10.2)

and Then, for α ≤ x ≤ β, we have |f (x + iy)| ≤ M1L(x) M21−L(x) ,

−∞ < y < ∞,

(10.3)

where L(x) 

β −x , β −α

α ≤ x ≤ β.

(10.4)

48

10. The Riesz–Thorin Theorem

Proof.

Let F be the function on B defined by f (z)

F (z) 

M1L(z) M21−L(z)

,

z ∈ B.

(10.5)

We first suppose that f (x + iy) → 0

(10.6)

uniformly with respect to x on [α, β] as |y| → ∞. Note that by (10.4) and (10.5), |F (z)| 

|f (z)|

≤

M1L(x) M21−L(x)

|f (z)| , M

z ∈ B,

(10.7)

where M  min(M1 , M2 ). Hence, by (10.6) and (10.7), F (x + iy) → 0

(10.8)

uniformly with respect to x on [α, β] as |y| → ∞. Thus, by (10.8), there exists a positive number R such that |F (x + iy)| ≤ 1

(10.9)

for α ≤ x ≤ β and |y| ≥ R. On the rectangle defined by α ≤ x ≤ β and |y| ≤ R, (10.9) is valid for |y|  R, and by (10.1), (10.2), and (10.5), |F (α + iy)| ≤ 1,

−∞ < y < ∞,

(10.10)

|F (β + iy)| ≤ 1,

−∞ < y < ∞.

(10.11)

and

Hence, by (10.9), (10.10), (10.11), and the maximum modulus principle, |F (z)| ≤ 1,

z ∈ B.

(10.12)

So, by (10.4), (10.5), and (10.12), we can conclude that (10.3) is true for α ≤ x ≤ β and −∞ < y < ∞. If f (x + iy) does not go to zero uniformly with respect to x on [α, β] as |y| → ∞, then for every positive integer k, we define the function fk on B by z2

fk (z)  e k f (z),

z ∈ B.

(10.13)

Since f is a bounded function on B, and for k  1, 2, . . . , z2

|e k |  e

x 2 −y 2 k

≤e

γ 2 −y 2 k

,

−∞ < y < ∞,

(10.14)

where γ 2  max(α 2 , β 2 ), it follows from (10.13) and (10.14) that for k  1, 2, . . . , |fk (x + iy)| → 0

(10.15)

uniformly with respect to x on [α, β] as |y| → ∞. Furthermore, by (10.1), (10.2), (10.13), and (10.14), α2

|fk (α + iy)| ≤ e k M1 ,

−∞ < y < ∞,

(10.16)

10. The Riesz–Thorin Theorem

49

and β2

|fk (β + iy)| ≤ e k M2 ,

−∞ < y < ∞.

(10.17)

Thus, by (10.15), (10.16), (10.17), and what we have just shown, we get, for k  1, 2, . . . , z2

|e k f (x + iy)| ≤ e

α2 k

L(x)

M1L(x) e

β2 k

(1−L(x))

M21−L(x)

(10.18)

for α ≤ x ≤ β and −∞ < y < ∞. Therefore, by letting k → ∞ in (10.18), the proof of the theorem is complete. ✷ We can now formulate and prove the Riesz–Thorin theorem. Theorem 10.2. (The Riesz–Thorin Theorem) Let (X, µ) be a measure space and (Y, ν) a σ -finite measure space. Let T be a linear transformation with domain D consisting of all µ-simple functions f on X such that µ{s ∈ X : f (s)  0} < ∞ and such that the range of T is contained in the set of all ν-measurable functions on Y . Suppose that α1 , α2 , β1 , and β2 are real numbers in [0, 1] and there exist positive constants M1 and M2 such that Tf

1

L βj (Y )

≤ Mj f

1

L αj (X)

,

f ∈ D,

j  1, 2.

(10.19)

Then, for 0 < θ < 1, α  (1 − θ)α1 + θα2 ,

(10.20)

β  (1 − θ)β1 + θβ2 ,

(10.21)

and

we have Tf

1

L β (Y )

Proof.

≤ M11−θ M2θ f L α1 (X) ,

f ∈ D.

For any complex number z, we let α(z)  (1 − z)α1 + zα2

(10.22)

β(z)  (1 − z)β1 + zβ2 .

(10.23)

and

Let f be a µ-simple function on X and g a ν-simple function on Y such that f L α1 (X)  1

(10.24)

and g

L

( β1 )

(Y )

 1,

(10.25)

50

10. The Riesz–Thorin Theorem

  1 where is the conjugate index of β1 . We first suppose that 1 ≤ β 1 < β1 ≤ ∞. Let

1 α

< ∞ and

f (s)  |f (s)|eiu(s) ,

s ∈ X,

(10.26)

g(t)  |g(t)|eiv(t) ,

t ∈ Y.

(10.27)

and For any z in C, we write



|f (s)| 0,

F (s, z)  and

 G(t, z) 

|g(t)| 0,

for all s in X and t in Y . Let f (s) 

m
j 1

α(z) α

eiu(s) ,

1−β(z) 1−β

eiv(t) ,

f (s)  0, f (s)  0,

(10.28)

g(t)  0, g(t)  0,

(10.29)

cj χSj (s),

s ∈ X,

(10.30)

dk χTk (t),

t ∈ Y,

(10.31)

and g(t) 

n
k1

where c1 , c2 , . . . , cm are distinct nonzero complex numbers; d1 , d2 , . . . , dn are distinct nonzero complex numbers; S1 , S2 , . . . , Sm are pairwise disjoint measurable subsets of X; T1 , T2 , . . . , Tn are pairwise disjoint measurable subsets of Y ; χSj is the characteristic function on Sj , j  1, 2, . . . , m; χTk is the characteristic function on Tk , k  1, 2, . . . , n. Let cj  |cj |eiuj ,

j  1, 2, . . . , m,

(10.32)

dk  |dk |eivk ,

k  1, 2, . . . , n.

(10.33)

and

Then, by (10.26), (10.28), (10.30), and (10.32), m
α(z) |cj | α eiuj χSj (s), F (s, z) 

s ∈ X,

(10.34)

and, by (10.27), (10.29), (10.31), and (10.33), n
1−β(z) G(t, z)  |dk | 1−β eivk χTk (t),

t ∈ Y,

(10.35)

j 1

k1

for all z in C. Let 5 be the function on C defined by 5(z)  (T F (·, z))(t)G(t, z)dν(t), Y

z ∈ C.

(10.36)

10. The Riesz–Thorin Theorem

51

Then, by (10.34), (10.35), and (10.36), 5(z) 

n m



e

i(uj +vk )

|cj |

α(z) α

|dk |

1−β(z) 1−β

Y

j 1 k1

(T χSj )(t)χTk (t)dν(t)

(10.37)

for all z in C. Thus, 5 is entire. For 0 ≤ x ≤ 1 and −∞ < y < ∞, we get, by (10.22), (10.23), and (10.37),   n m

 1−β(x)  α(x) |5(x + iy)| ≤ |cj | α |dk | 1−β  (T χSj )(t)χTk (t)dν(t) . (10.38) Y

j 1 k1

Hence, by (10.38), 5 is a bounded function on the strip {z ∈ C : 0 ≤ Rez ≤ 1}. For Rez  0, we get by (10.20)–(10.23), Reα(z)  α1

(10.39)

Reβ(z)  β1 .

(10.40)

and

Thus, by (10.28) and (10.39),



|F (s, iy)|  and by (10.29) and (10.41),



|G(t, iy)| 

α1

|f (s)| α , 0, 1−β1

|g(t)| 1−β , 0,

f (s)  0, f (s)  0,

(10.41)

g(t)  0, g(t)  0,

(10.42)

for all s in X and t in Y . Therefore, by (10.24) and (10.41),  α1 α1 1 α  |f (s)| dµ(s)  f α 1 F (·, iy) α1 L

1

(X)

L α (X)

X

and by (10.25) and (10.42),  1−β1 1−β1 1 |g(t)| 1−β dν(t)  g 1−β G(·, iy) ( β1 )  ( 1 ) L

1

(Y )

L

Y

β

 1,

(Y )

1

(10.43)

(10.44)

for −∞ < y < ∞. So, by (10.19), (10.36), (10.43), and (10.44),     |5(iy)|   (T F (·, iy))(t)G(t, iy)dν(t) Y

≤ T F (·, iy)

1

L β1 (Y )

G(·, iy)

L

( β1 ) 1 (Y )

≤ M1

(10.45)

for −∞ < y < ∞. Similarly, |5(1 + iy)| ≤ M2 ,

−∞ < y < ∞.

(10.46)

So, by (10.45), (10.46), Theorem 10.1, and the fact that 5 is entire, |5(x + iy)| ≤ M11−x M2x ,

0 ≤ x ≤ 1, −∞ < y < ∞.

(10.47)

52

10. The Riesz–Thorin Theorem

Thus, by (10.47), |5(θ )| ≤ M11−θ M2θ .

(10.48)

Next, note that by (10.20), (10.22), (10.30), (10.32), and (10.34), F (s, θ)  f (s),

s ∈ X,

(10.49)

and, by (10.21), (10.23), (10.31), (10.33), and (10.35), G(t, θ)  g(t),

t ∈ Y.

(10.50)

Therefore, by (10.36) and (10.48)–(10.50),      |5(θ )|   (Tf )(t)g(t)dν(t) ≤ M11−θ M2θ .

(10.51)

Y

So, by (10.51), Tf

1 Lβ

(Y )

     sup  (Tf )(t)g(t)dν(t) ≤ M11−θ M2θ , Y

where the supremum is taken over all ν-simple functions g on Y with g

L

( β1 )

(Y )

 1, and this completes the proof of the theorem for 1 ≤ < ∞ and 1 < ≤ ∞. Now we have to look at case 1: α  0 and β  1; case 2: α  0 and 0 < β < 1; case 3: α  0 and β  0; case 4: α  1 and β  1; and case 5: 0 < α < 1 and β  1. All cases are trivial except cases 2 and 5. For case 2, we take α(z)  1 α for all z in C, and the preceding proof goes through without any change. For case 5, we take 1−β(z)  1 for all z in C; then the preceding proof can be used, until 1−β we get (10.51) for all ν-simple functions g on Y with g L∞ (Y )  1. Thus, for all ν-simple functions g on Y ,      (Tf )(t)g(t)dν(t) ≤ M 1−θ M θ g L∞ (Y ) . (10.52) 2 1   1 α

1 β

Y

Let S  {t ∈ Y : (Tf )(t)  0} and let h be the function on Y defined by  |(Tf )(t)| h(t)  (Tf )(t) , t ∈ S, (10.53) 0, t ∈ S. Now, let {gl }∞ l1 be a sequence of ν-simple functions on Y such that gl L∞ (Y ) ≤ 1

(10.54)

for l  1, 2, . . . , and gl → h a.e. on Y as l → ∞. Therefore, (Tf )gl → (Tf )h

(10.55)

a.e. on Y as l → ∞. By (10.53) and (10.54), |(Tf )(t)gl (t)| ≤ |(Tf )(t)|,

t ∈ Y.

Since β  1, it follows from (10.19) and (10.21) that |(Tf )(t)|dν(t) < ∞. Y

(10.56)

(10.57)

10. The Riesz–Thorin Theorem

53

Thus, using (10.53), (10.55)–(10.57), and the Lebesgue dominated convergence theorem, we get         |(Tf )(t)|dν(t)   (Tf )(t)h(t)dν(t)  lim  (Tf )(t)gl (t)dν(t) . Y

Y

l→∞

Y

(10.58)

So, by (10.52), (10.54), and (10.58),      (Tf )(t)dν(t) ≤ M 1−θ M θ , 2 1   Y

and the proof is therefore complete.

✷

A standard and beautiful application of the Riesz–Thorin theorem is the Hausdorff–Young inequality for the Fourier transform of a function in Lr (Rn ), 1 ≤ r ≤ 2, which we state, prove, and use in Chapter 14. The Hausdorff–Young inequality is also attributed to Titchmarsh. See the paper [30] and the book [31] by Titchmarsh in this connection.

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11 Weyl Transforms with Symbols in Lr (R2n), 1 ≤ r ≤ 2

The following result is an extension of Theorem 4.4 and is essential for the proof that a Weyl transform with symbol in Lr (R2n ), 1 ≤ r ≤ 2, is compact. Theorem 11.1. For 1 ≤ r ≤ 2, there exists a unique bounded linear operator Q : Lr (R2n ) → B(L2 (Rn )) such that (4.8) is valid for all σ in Lr (R2n ) and f and g in S(Rn ). Furthermore, Qσ ∗ ≤ 2

−n

  nr 2 σ Lr (R2n ) , π

σ ∈ Lr (R2n ).

(11.1)

To prove Theorem 11.1, we begin by noting that for r  2, Theorem 11.1 is just Theorem 4.4. For r  1, let σ ∈ S(R2n ). Then, for any f in S(Rn ), we define (Qσ )f by (4.10). Then, for all f and g in S(Rn ), we get, by (4.10) and Theorem 4.3,     −n/2  |(Qσ )f, g|  (2π) σ (x, ξ )W (f, g)(x, ξ )dx dξ   Rn

−n/2

≤ (2π )

Rn

σ L1 (R2n ) W (f, g) L∞ (R2n ) .

(11.2)

But by (3.12),     p  p  −iξ ·p  |W (f, g)(x, ξ )|  (2π) dp f x+ g x−  ne 2 2 R     21     21 p 2 p 2   −n/2 ≤ (2π) f x + g x −  dp  dp 2 2 Rn Rn −n/2

56

11. Weyl Transforms with Symbols in Lr (R2n ), 1 ≤ r ≤ 2 −n/2 n

 (2π ) 

 |f (p)| dp 2

2

Rn

 21 

 n/2 2 f L2 (Rn ) g L2 (Rn ) , π

|g(p)| dp 2

Rn

 21

x, ξ ∈ Rn .

(11.3)

Thus, by (11.2) and (11.3), |(Qσ )f, g| ≤ π −n σ L1 (R2n ) f L2 (Rn ) g L2 (Rn )

(11.4)

for all σ in S(R2n ), and f and g in S(Rn ). Hence, by (11.4), Qσ ∗ ≤ π −n σ L1 (R2n ) ,

σ ∈ S(R2n ).

(11.5)

{σk }∞ k1

Let σ ∈ L1 (R2n ). Then we let be a sequence of functions in S(R2n ) such that σk → σ in L1 (R2n ) as k → ∞. So, by (11.5), Qσk − Qσl ∗ ≤ π −n σk − σl L1 (R2n ) → 0 2 n as k, l → ∞. Therefore, {Qσk }∞ k1 is a Cauchy sequence in B(L (R )), and we can ∞ 2 n define Qσ to be the limit of the sequence {Qσk }k1 in B(L (R )). This definition ∞ 2n is independent of the choice of the sequence {σk }∞ k1 in S(R ). Indeed, let {τk }k1 2n 1 2n be another sequence in S(R ) such that τk → σ in L (R ) as k → ∞. Then, by (11.5),

Qσk − Qτk ∗ ≤ π −n σk − τk L1 (R2n ) → 0 as k → ∞. Therefore, the limits in B(L2 (Rn )) of the sequences {Qσk }∞ k1 and ∞ 1 2n {Qτk }∞ are equal. Let σ ∈ L (R ). Then we let {σ } be a sequence of k k1 k1 functions in S(R2n ) such that σk → σ in L1 (R2n ) as k → ∞. Thus, by (11.5), Qσ ∗  lim Qσk ∗ ≤ lim π −n σk L1 (R2n )  π −n σ L1 (R2n ) . k→∞

k→∞

(11.6)

Furthermore, if f and g are in S(Rn ), then by (4.10) and Theorem 4.3, (Qσ )f, g  lim (Qσk ), g k→∞  lim (2π)−n/2 σk (x, ξ )W (f, g)(x, ξ )dx dξ k→∞ n n R R  (2π )−n/2 σ (x, ξ )W (f, g)(x, ξ )dx dξ. Rn

Rn

To prove the theorem for 1 ≤ r ≤ 2, we fix a function f in S(Rn ). Let T be the linear transformation with domain D consisting of all simple functions σ on R2n with the property that the Lebesgue measure of the set {(x, ξ ) ∈ R2n : σ (x, ξ )  0} is finite, and defined by T σ  (Qσ )f,

σ ∈ D.

(11.7)

Then, by (4.9) and (11.7), T σ L2 (Rn )  (Qσ )f L2 (Rn ) ≤ (2π)−n/2 f L2 (Rn ) σ L2 (R2n ) ,

(11.8)

11. Weyl Transforms with Symbols in Lr (R2n ), 1 ≤ r ≤ 2

57

and by (11.6) and (11.7), T σ L2 (Rn )  (Qσ )f L2 (Rn ) ≤ π −n f L2 (Rn ) σ L1 (R2n )

(11.9)

for all σ in D. So, by (11.8), (11.9), and the Riesz–Thorin theorem, T σ L2 (Rn )  (Qσ )f L2 (Rn ) ≤ {(2π )−n/2 f L2 (Rn ) }2− r {π −n f L2 (Rn ) } r −1 σ Lr (R2n )   nr 2 −n 2 f L2 (Rn ) σ Lr (R2n ) , π 2

and hence Qσ ∗ ≤ 2

−n

2

  nr 2 σ Lr (R2n ) , π

σ ∈ D.

(11.10)

Now, let σ ∈ Lr (R2n ). Then we pick a sequence {sk }∞ k1 of functions in D such that sk → σ in Lr (R2n ) as k → ∞. Then, by (11.10), {Qsk }∞ k1 is a Cauchy sequence in B(L2 (Rn )), and we denote the limit in B(L2 (Rn )) by Qσ . The proof that the limit is independent of the choice of the sequence {sk }∞ k1 and the proof that Qσ satisfies the conclusions of the theorem are the same as before. It is also obvious that Q : Lr (R2n ) → B(L2 (Rn )) is the only bounded linear operator satisfying the conclusions of the theorem. Remark 11.2. It is natural to denote Qσ by Wσ for any σ in Lr (R2n ), 1 ≤ r ≤ 2. Theorem 11.3. Let σ ∈ Lr (R2n ), 1 ≤ r ≤ 2. Then Wσ : L2 (Rn ) → L2 (Rn ) is a compact operator. To prove Theorem 11.3, we use the following lemma. Lemma 11.4. Let Sh be the Hilbert–Schmidt operator on L2 (Rn ) corresponding to the kernel h in L2 (R2n ). Then Sh is compact. Proof. By Proposition 6.2, we can get a sequence {tk }∞ k1 of finite linear combinations of tensor products of functions in L2 (Rn ) such that tk → h

(11.11)

in L2 (R2n ) as k → ∞. By (5.1), (5.2), and (11.11), Stk − Sh ∗ ≤ tk − h L2 (R2n ) → 0

(11.12)

as k → ∞. Hence, by (11.12), Sh is compact if we can prove that each Stk is compact. To prove that each Stk is compact, it is enough to prove that Sa⊗b is compact for all a and b in L2 (Rn ). But if a and b are in L2 (Rn ), then (Sa⊗b f )(x)  a(x)b(y)f (y)dy Rn

 f, ba(x),

x ∈ Rn ,

for all f in L2 (Rn ). So, Sa⊗b is of finite rank and hence compact.

✷

58

11. Weyl Transforms with Symbols in Lr (R2n ), 1 ≤ r ≤ 2

Proof of Theorem 11.3. Let σ ∈ Lr (R2n ), 1 ≤ r ≤ 2. Then we pick a sequence 2n r 2n {σk }∞ k1 of functions in S(R ) such that σk → σ in L (R ) as k → ∞. By 2 n 2 n Theorem 6.8, Wσk : L (R ) → L (R ) is a Hilbert–Schmidt operator, and hence, by Lemma 11.4, compact for k  1, 2, . . . . By (11.1) and Remark 11.2, Wσ is the 2 n 2 n limit in B(L2 (Rn )) of the sequence {Wσk }∞ k1 . Thus, Wσ : L (R ) → L (R ) is compact. ✷

12 Weyl Transforms with Symbols in L∞(R2n)

We begin this chapter by defining the notion of a Weyl transform with symbol in S  (R2n ). To this end, let σ be in S  (R2n ). Then, for all f in S(Rn ), we define the linear functional Wσ f on S(Rn ) by (Wσ f )(g)  (2π)−n/2 σ (W (f, g)), ¯

g ∈ S(Rn ),

(12.1)

where W (f, g) ¯ is the Wigner transform of f and g¯ defined by (3.12). Theorem 12.1. For all σ in S  (R2n ) and f in S(Rn ), Wσ f is a tempered distribution on Rn . To prove Theorem 12.1, we need a lemma. n Lemma 12.2. Let f be in S(Rn ) and {gk }∞ k1 a sequence of functions in S(R ) n such that gk → 0 in S(R ) as k → ∞. Then, for all multi-indices α, β, γ , and δ,   p  δ p  |x||α| |p||β| (∂ γ f ) x +  |(∂ gk )(x − )|dp → 0 2 2 Rn

uniformly with respect to x on Rn as k → ∞. For k  1, 2, . . . , we define the function Ik on Rn by    p   δ p   Ik (x)  |x||α| |p||β| (∂ γ f ) x +  (∂ gk ) x −  dp, x ∈ Rn . 2 2 Rn (12.2)

Proof.

12. Weyl Transforms with Symbols in L∞ (R2n )

60

Then, by (12.2), Ik (x) ≤ 2

|β|



    p |α|  p |β|  p |α| p |β|   x +  + x −  x +  + x −  2 2 2 2 Rn       p p   δ  × (∂ γ f ) x + (12.3)  (∂ gk ) x −  dp 2 2

for all x in Rn . Now,   p |α+β|  γ  p   δ p   x ±  (∂ f ) x +  (∂ gk ) x −  dp → 0 2 2 2 Rn

(12.4)

and

   p |α|  p |β|  p   δ p   x ±  x ∓  (∂ γ f ) x +  (∂ gk ) x −  dp → 0 (12.5) 2 2 2 2 Rn

uniformly with respect to x on Rn as k → ∞. Thus, by (12.3), (12.4), and (12.5), ✷ Ik → 0 uniformly on Rn as k → ∞. n Proof of Theorem 12.1. Let {gk }∞ k1 be a sequence of functions in S(R ) such n that gk → 0 in S(R ) as k → ∞. Let α, β, γ , and δ be multi-indices. Then, for all x and ξ in Rn , we get, by (3.12),

x α ξ β (∂xγ ∂ξδ W (f, g k ))(x, ξ )  p p  α β γ δ −n/2 gk x − dp e−ip·ξ f x +  x ξ ∂x ∂ξ (2π) 2 2 Rn   
γ p γ −γ  γ  x α ξ β (2π)−n/2 (∂ (−ip)δ e−ip·ξ f ) x + gk ) (∂ γ 2 Rn γ  ≤γ  p dp × x− 2   
γ 1 p −n/2 |δ| β −ip·ξ α δ γ   (2π ) (−i) (∂ e )x p (∂ f ) x + p |β| γ 2 Rn (−i) γ  ≤γ  p  dp × (∂ γ −γ gk ) x − 2    
γ p γ −γ  −n/2 |δ+β| −ip·ξ α β δ γ  (2π ) p (∂ (−i) e x ∂ (∂ f ) x + gk ) p γ 2 Rn γ  ≤γ  p  dp × x− 2   

 γ  β  1 |β−β | −n/2 |δ+β|  (2π ) − (−i) γ  β 2 γ  ≤γ β  ≤β     p p     (∂ β−β +γ −γ gk ) x − dp × e−ip·ξ x α ∂pβ p δ (∂ γ f ) x + 2 2 Rn

12. Weyl Transforms with Symbols in L∞ (R2n )

61

      


 γ  β  β   1 |β−β | 1 |β −β | −n/2 |δ+β| (2π) − (−i) γ  β  β  2 2 γ  ≤γ β  ≤β β  ≤β    p β−β  +γ −γ  p     (∂ dp × e−ip·ξ x α (∂pβ p δ )(∂ β −β +γ f ) x + gk ) x − 2 2 Rn       


 γ  β  β   1 |β−β | 1 |β −β | −n/2 |δ+β| (2π) −  (−i) γ  β  β  2 2 γ  ≤γ β  ≤β β  ≤β    δ × β  !  β   p β−β  +γ −γ  p     × e−ip·ξ x α p δ−β (∂ β −β +γ f ) x + (∂ dp, gk ) x − 2 2 Rn



and hence, by Lemma 12.2, |x α ξ β (∂xγ ∂ξδ W (f, g k ))(x, ξ )|


 γ  β  β    δ  ≤ β  !     γ β β β γ  ≤γ β  ≤β β  ≤β     p   β−β  +γ −γ  p       × |x||α| |p||δ−β | (∂ β −β +γ f ) x + gk ) x −  (∂  dp 2 2 Rn →0 (12.6) uniformly with respect to (x, ξ ) on R2n as k → ∞. Since σ is in S  (R2n ), it follows ✷ from (12.6) that σ (W (f, g¯ k )) → 0 as k → ∞. Remark 12.3. Let σ ∈ Lr (R2n ), 1 ≤ r ≤ 2. Then, for all f and g in S(Rn ), we get, by Theorem 12.1, (Wσ f )(g)  (2π)−n/2 σ (W (f, g)) ¯  (2π )−n/2 σ (x, ξ )W (f, g)(x, ¯ ξ )dx dξ. Rn

(12.7)

Rn

Thus, for all σ in Lr (R2n ), 1 ≤ r ≤ 2, we get, by (12.7), (Wσ f )(g)  Wσ f, g, ¯

f, g ∈ S(Rn ),

(12.8)

where Wσ on the right-hand side of (12.8) is the bounded linear operator Qσ on L2 (Rn ) provided by Theorem 11.1. By Theorem 11.3, Wσ : L2 (Rn ) → L2 (Rn ) is compact if σ is in Lr (R2n ), 1 ≤ r ≤ 2. Now, let σ be in Lr (R2n ), 2 < r ≤ ∞, and let f be in S(Rn ). Is Wσ f , which by Theorem 12.1 is a tempered distribution on Rn , always a function in L2 (Rn )? If so, can Wσ be extended to a bounded or even compact operator from L2 (Rn ) into L2 (Rn )? We attempt to answer these questions in this and the next chapter. Theorem 12.4. There exists a function σ in L∞ (R2n ) for

which Wσ f is not a function in L2 (Rn ) for all functions f in S(Rn ) such that Rn f (x)dx  0.

62

12. Weyl Transforms with Symbols in L∞ (R2n )

Proof.

Let σ be the function on R2n defined by x, ξ ∈ Rn .

σ (x, ξ )  e2ix·ξ ,

(12.9)

Then, for all f and g in S(Rn ), we get, by (3.12), (12.1), (12.9), and the Fourier inversion formula, (Wσ f )(g)  (2π)−n/2 σ (W (f, g)) ¯  (2π )−n/2 σ (x, ξ )W (f, g)(x, ¯ ξ )dx dξ n n    R R  p  p −n 2ix·ξ −ip·ξ  (2π) e e f x+ g x− dp dξ dx 2 2 Rn Rn Rn    f (2x)g(0)dx  2−n f (x)dx g(0), Rn

Rn

and hence Wσ f  2−n

 Rn

 f (x)dx δ,

n

where δ : S(R ) → C is the Dirac delta given by δ(ϕ)  ϕ(0),

ϕ ∈ S(Rn ).

✷

13 Weyl Transforms with Symbols in Lr (R2n), 2 < r < ∞

The following result can be found in the paper [25] by Simon. Theorem 13.1. For 2 < r < ∞, there exists a function σ in Lr (R2n ) such that the Weyl transform Wσ , defined by (12.1), is not a bounded linear operator on L2 (Rn ). To prove Theorem 13.1, we need some preparations. Lemma 13.2. Suppose that for all σ in Lr (R2n ), 2 < r < ∞, the Weyl transform Wσ , defined by (12.1), is a bounded linear operator on L2 (Rn ). Then there exists a positive constant C such that Wσ ∗ ≤ C σ Lr (R2n ) , Proof. that

σ ∈ Lr (R2n ).

(13.1)

Suppose that for all σ in Lr (R2n ), there exists a positive constant Cσ such Wσ f L2 (Rn ) ≤ Cσ f L2 (Rn ) ,

f ∈ L2 (Rn ).

(13.2)

Let f and g be functions in S(Rn ) such that f L2 (Rn )  g L2 (Rn )  1. Consider the bounded linear functional Qf,g : Lr (R2n ) → C defined by Qf,g σ  Wσ f, g,

σ ∈ Lr (R2n ).

(13.3)

Then, by (13.2) and (13.3), sup |Qf,g σ | ≤ Cσ ,

σ ∈ Lr (R2n ),

(13.4) n

where the supremum is taken over all functions f and g in S(R ) such that f L2 (Rn )  g L2 (Rn )  1. Thus, by (13.4) and the uniform boundedness

64

13. Weyl Transforms with Symbols in Lr (R2n ), 2 < r < ∞

principle, there exists a positive constant C such that Qf,g ≤ C

(13.5)

for all f and g in S(Rn ) with f L2 (Rn )  g L2 (Rn )  1, where Qf,g is the norm of the bounded linear functional Qf,g : Lr (R2n ) → C. Therefore, by (13.3) and (13.5), sup

σ Lr (R2n ) 1

|Wσ f, g| ≤ C

(13.6)

for all f and g in S(Rn ) with f L2 (Rn )  g L2 (Rn )  1. So, by (13.6), |Wσ f, g| ≤ C σ Lr (R2n ) f L2 (Rn ) g L2 (Rn )

(13.7)

for all σ in Lr (R2n ), and f and g in S(Rn ). Thus, (13.1) follows from (13.7).

✷

Lemma 13.3. Let α ∈ (0, 1). Then

π 2

lim

ρ→∞ 0

Proof.

Let θ 

π 2

e−ρ cos θ ρ α dθ  0.

− ϕ. Then

π 2

e

−ρ cos θ



α

ρ dθ 

0

π 2

e−ρ sin ϕ ρ α dϕ.

(13.8)

0

Since the sine function is concave down on [0, π2 ], it follows that sin ϕ ≥

 π ϕ ∈ 0, . 2

2 ϕ, π

(13.9)

Thus, by (13.8) and (13.9),

π 2

0

e

−ρ cos θ

α



ρ dθ ≤

π 2

e− π ϕ ρ α dϕ 2ρ

0

  π 2ρ π  2  ρ α e− π ϕ − 2ρ 0 π  ρ α−1 (1 − e−ρ ) → 0 2

as ρ → ∞. Lemma 13.4. Let α ∈ (0, 1). Then ∞  πα L(α). t α−1 cos tdt  cos 2 0

✷

13. Weyl Transforms with Symbols in Lr (R2n ), 2 < r < ∞

65

Proof.

Fig. 2

Let Lρ,r be the simple closed curve in Fig. 2 traversed once in the positive direction. Let f be the function on the cut plane C − (−∞, 0] defined by f (z)  e−z zα−1  e−z e(α−1)Log−π z ,

z ∈ C − (−∞, 0],

(13.10)

where Log−π z  ln |z| + iArg−π z and −π < Arg−π z < π . Then, by Cauchy’s theorem, 0 f (z)dz  f (z)dz+ f (z)dz+ f (z)dz+ f (z)dz (13.11) Lρ,r

1 γρ,r

γρ

2 γρ,r

γr

for all real numbers ρ and r satisfying 0 < r < ρ. Of course, ρ f (z)dz  e−x x α−1 dx, 0 < r < ρ. Since

γρ

(13.12)

r

1 γρ,r

f (z)dz 

γρ



π 2

e−z e(α−1)Log−π z dz iθ

e−ρe e(α−1){ln ρ+iθ} iρeiθ dθ

0



π 2

iθ

e−ρe ρ α e(α−1)iθ ieiθ dθ,

0

it follows from Lemma 13.3 that   π   2   e−ρ cos θ ρ α dθ → 0  f (z)dz ≤  γρ  0

(13.13)

66

13. Weyl Transforms with Symbols in Lr (R2n ), 2 < r < ∞

as ρ → ∞. Next,





2 γρ,r

f (z)dz 

r

iπ

e−it e(α−1){ln t+ 2 } idt

ρ r

iπα

iπ

e−it t α−1 e 2 e− 2 idt ρ r iπα e 2 e−it t α−1 dt ρ r iπα e 2 (cos t − i sin t)t α−1 dt 

(13.14)

ρ

for all real numbers ρ and r satisfying 0 < r < ρ. Finally, for r > 0, f (z)dz  e−z e(α−1)Log−π z dz γr

γr



π 2

−

iθ

e−re e(α−1){ln r+iθ} ireiθ dθ

0 π 2

−

iθ

e−re r α e(α−1)iθ ieiθ dθ.

(13.15)

0

Thus, by (13.15),

     f (z)dz ≤   γr

π 2

e−r cos θ r α dθ → 0

(13.16)

0

as r → 0. So, letting ρ → ∞ and r → 0, we get, by (13.10)–(13.14) and (13.16), ∞ ∞ iπα −x α−1 2 e x dx − e (cos t − i sin t)t α−1 dt  0. (13.17) 0

0

Therefore, by (13.17), ∞ iπα α−1 2 e (cos t − i sin t)t dt  0

∞

e−x x α−1 dx  L(α),

0

✷

and the lemma is proved. Theorem 13.1 is a consequence of Lemma 13.2 and the following lemma.

Lemma 13.5. For 2 < r < ∞, there is no positive constant C such that (13.1) is valid. Proof. Suppose that there exists a positive constant C such that (13.1) is valid. Then, by (12.1) and (13.1),     sup  σ (x, ξ )W (f, g)(x, ξ )dx dξ  W (f, g) Lr  (R2n )  σ Lr (R2n ) 1

≤

sup

Rn

Rn

n 2

(2π) |Wσ f, g|

σ Lr (R2n ) 1

13. Weyl Transforms with Symbols in Lr (R2n ), 2 < r < ∞

≤

sup

67

n

(2π) 2 Wσ f L2 (Rn ) g L2 (Rn )

σ Lr (R2n ) 1 n

≤ (2π) 2 C f L2 (Rn ) g L2 (Rn ) ,

f, g ∈ S(Rn ).

(13.18)

∞ Let f and g be in L2 (Rn ). Then, we let {fk }∞ k1 and {gk }k1 be sequences of n 2 n functions in S(R ) such that fk → f and gk → g in L (R ) as k → ∞. Thus, using the bilinearity of W ,

W (fk , gk ) − W (fl , gl )  W (fk , gk ) − W (fk , gl ) + W (fk , gl ) − W (fl , gl )  W (fk , gk − gl ) − W (fk − fl , gl ), k, l  1, 2, . . . , and hence by (13.18), W (fk , gk ) − W (fl , gl ) Lr  (R2n ) ≤ W (fk , gk − gl ) Lr  (R2n ) + W (fk − fl , gl ) Lr  (R2n ) n

≤ (2π) 2 C{ fk L2 (Rn ) gk − gl L2 (Rn ) + fk − fl L2 (Rn ) gl L2 (Rn ) } 

r 2n → 0 as k, l → ∞. Therefore, {W (fk , gk )}∞ k1 is a Cauchy sequence in L (R ), r 2n and it is easy to see that its limit in L (R ) is equal to W (f, g). Thus, by (13.18), n

W (f, g) Lr  (R2n ) ≤ (2π) 2 C f L2 (Rn ) g L2 (Rn )

(13.19)

for all f and g in L2 (Rn ). Now, let f be in L2 (Rn ) such that supp(f ) ⊆ {x ∈ Rn : |x| ≤ 1},

(13.20) n

where supp(f ) is the support of the function f . Let x and ξ be in R . Then, by (3.12), W (f )(x, ξ )  0 only if there is a point p in Rn such that |x ± p2 | ≤ 1, and hence W (f )(x, ξ )  0 only if 1  p  p  |x| ≤  x + + x−  ≤ 1. 2 2 2 Thus, |x| > 1 implies that W (f )(x, ξ )  0 for all ξ in Rn . Now, by (13.19),  |W (f )(x, ξ )|r dx dξ < ∞. (13.21) Rn

Rn

Then, for any real-valued function θ on R2n , we get, by (13.20) and (13.21),   r   r1    W (f )(x, ξ )eiθ(x,ξ ) dx  dξ  n R



≤

|x|≤1

 Rn

|x|≤1

 ≤ Thus, by (13.22),



|x|≤1

dx

|W (f )(x, ξ )|r dξ

 1r 

|x|≤1

Rn

 1 r

dx 

Rn

|W (f )(x, ξ )|r dx dξ



W (f )(x, ξ )eiθ(x,ξ ) dx ∈ Lr (Rn )

 1 r

< ∞.

(13.22)

(13.23)

68

13. Weyl Transforms with Symbols in Lr (R2n ), 2 < r < ∞

as a function of ξ on Rn . Now, let x, ξ ∈ Rn .

θ (x, ξ )  2x · ξ,

(13.24)

Then, by (3.12) and (13.24), W (f )(x, ξ )eiθ(x,ξ ) dx |x|≤1



  p  p e−iξ ·p f x + dp eiθ (x,ξ ) dx f x− 2 2 Rn Rn   p p p e2iξ ·(x− 2 ) f x + f x− dx dp (13.25)  (2π)−n/2 2 2 Rn Rn

 (2π )−n/2



for all ξ in Rn . Let u  x + p2 and v  x − p2 in (13.25). Then, using the Fourier inversion formula, W (f )(x, ξ )eiθ(x,ξ ) dx n R e2iξ ·v f (u)f (v)du dv  (2π)−n/2 Rn

 (2π )

−n/2



Rn

Rn



f (u)

 Rn

 (2π)n/2 fˆ(0)fˆ(2ξ ),

e−2iξ ·v f (v)dv

du

ξ ∈ Rn .

(13.26)

Now, let Q  {x ∈ Rn : −a ≤ xj ≤ a, j  1, 2, . . . , n} be a cube lying inside {x ∈ Rn : |x| ≤ 1}. Let α ∈ (0, 21 ) and let f be the function on Rn defined by  n −α j 1 |xj | , x ∈ Q, xj  0, j  1, 2, . . . , n, f (x)  0, otherwise. Then fˆ(ξ )  (2π)−n/2  (2π)−n/2 Now, for ξj > 0, a −a

e

−ixj ξj

−α

|xj |

Q

e−ix·ξ

n 

n 

|xj |−α dx

j 1 a

j 1 −a

e−ixj ξj |xj |−α dxj ,

dxj 

a

−a



ξ ∈ Rn .

cos(xj ξj )|xj |−α dxj a

cos(xj ξj )xj−α dxj  aξj  −α 2 t cos t dt ξj−1+α . 2

(13.27)

0

0

(13.28)

13. Weyl Transforms with Symbols in Lr (R2n ), 2 < r < ∞

69

By Lemma 13.4, there is a positive constant A such that   aξj   −α  t cos t dt  ≥ A 

(13.29)

for ξj large enough. Thus, by (13.27), (13.28), and (13.29),   n  −1+α −n/2 |fˆ(ξ )| ≥ (2π) 2 n An ξj

(13.30)

0

j 1

whenever ξ1 , ξ2 , . . . , ξn are all larger than some positive number R. Hence, by (13.30), n ∞   nr     |fˆ(ξ )|r dξ ≥ (2π)− 2 2nr Anr ξj(−1+α)r dξj  ∞ (13.31) Rn

j 1 R

if (1 − α)r  < 1. Therefore, n

Wf Lr  (R2n ) ≤ (2π) 2 C f 2L2 (Rn )

(13.32)

is impossible if (1 − α)r  < 1, and (13.33) is obtained if we pick α to be some number in

(13.33) ( 1r , 21 ).

✷

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14 Compact Weyl Transforms

In view of the positive result, i.e., Theorem 11.1, and the negative results, i.e., Theorem 12.4 and Simon’s Theorem 13.1, it is of genuine interest to have a sufficient condition on a function σ in Lr (R2n ), 2 ≤ r ≤ ∞, such that the Weyl transform Wσ , defined by (12.1), is a compact operator from L2 (Rn ) into L2 (Rn ). To do this, we introduce the space Lr∗ (R2n ), 1 ≤ r ≤ ∞, defined by 

Lr∗ (R2n )  {σ ∈ Lr (R2n ) : σˆ ∈ Lr (R2n )}.

(14.1)

To gain some insight into the space Lr∗ (R2n ), 1 ≤ r ≤ ∞, we use the following theorem. Theorem 14.1. (The Hausdorff–Young Inequality) The Fourier transform is a  bounded linear operator from Lr (Rn ) into Lr (Rn ), 1 ≤ r ≤ 2. In fact, n

fˆ Lr  (Rn ) ≤ (2π)− 2 (1− r  ) f Lr (Rn ) , Proof.

2

f ∈ Lr (Rn ).

(14.2)

By the Plancherel theorem, we get fˆ L2 (Rn )  f L2 (Rn ) ,

f ∈ L2 (Rn ),

(14.3)

and using the definition of the Fourier transform, we get fˆ L∞ (Rn ) ≤ (2π)−n/2 f L1 (Rn ) ,

f ∈ L1 (Rn ).

(14.4)

Let β1  21 , α1  21 , β2  0, and α2  1. Let M1  1 and M2  (2π )−n/2 . Let θ  1 − r2 . Then, by (14.3), (14.4), and the Riesz–Thorin theorem, n 2 fˆ Lr  (Rn ) ≤ (2π)− 2 (1− r  ) f Lr (Rn ) ,

f ∈ D,

(14.5)

72

14. Compact Weyl Transforms

where D is the set of all simple functions f on Rn such that the Lebesgue measure of the set {x ∈ Rn : f (x)  0} is finite. Since D is dense in Lr (Rn ), a limiting argument can be used to complete the proof that the Fourier transform is a bounded linear operator from Lr (Rn ) into  Lr (Rn ) and (14.2) is valid. ✷ As an immediate consequence of the Hausdorff–Young inequality, we give the following corollary. Corollary 14.2. Lr∗ (R2n )  Lr (R2n ), 1 ≤ r ≤ 2. We can now give a result supplementing Theorems 11.1 and 11.3. Theorem 14.3. Let σ ∈ Lr∗ (R2n ), 2 ≤ r ≤ ∞. Then Wσ : L2 (Rn ) → L2 (Rn ) is a bounded linear operator. In fact, n

Wσ ∗ ≤ (2π)− r  σˆ Lr  (R2n ) ,

σ ∈ Lr∗ (R2n ).

(14.6)

Furthermore, Wσ : L2 (Rn ) → L2 (Rn ) is a compact operator. To prove Theorem 14.3, we need some lemmas. Lemma 14.4. Let f and g be in S(Rn ). Then W (f, g)(x, ξ )  2n V (f, g)(−2ξ, ˜ 2x), Proof.

Let q 

x, ξ ∈ Rn .

p 2

in (3.12). Then, by Proposition 2.3,  p  p W (f, g)(x, ξ )  (2π)−n/2 e−iξ ·p f x + g x− dp 2 2 Rn  2n (2π)−n/2 e−2iξ ·q f (x + q)g(x − q)dq Rn  2n (2π)−n/2 e−2iξ ·q f (q + x)g(q ˜ − x)dq Rn

˜ 2x),  2n V (f, g)(−2ξ,

x, ξ ∈ Rn , ✷

and the proof is complete. Lemma 14.5. Let σ ∈ by

Lr∗ (R2n ),

1 ≤ r ≤ ∞. Then the function τ on R

2n

defined

τ (x, ξ )  σ (ξ, x),

x, ξ ∈ Rn ,

(14.7)

τˆ (q, p)  σˆ (p, q),

q, p ∈ Rn .

(14.8)

is also in Lr∗ (R2n ), and To prove Lemma 14.5, we need another lemma. Lemma 14.6. Let f and g be in Lr (Rn ), 1 ≤ r ≤ 2. Then fˆ(x)g(x)dx. f (x)g(x)dx ˆ  Rn

Rn

14. Compact Weyl Transforms

73

∞ Proof. Let f and g be in Lr (Rn ), 1 ≤ r ≤ 2. Then we let {fk }∞ k1 and {gk }k1 be sequences of functions in S(Rn ) such that fk → f and gk → g in Lr (Rn ) as k → ∞. Then, by the Hausdorff–Young inequality, fˆk → fˆ and gˆ k → gˆ in  Lr (Rn ) as k → ∞. Hence, by Proposition 1.10, f (x)g(x)dx ˆ  lim fk (x)gˆ k (x)dx k→∞ Rn Rn ✷  lim fˆk (x)gk (x)dx  fˆ(x)g(x)dx. k→∞ Rn

Rn

Proof of Lemma 14.5. For any ϕ in S(R2n ), we get, by Lemma 14.6, τˆ (ϕ)  τ (ϕ) ˆ  τ (x, ξ )ϕ(x, ˆ ξ )dx dξ n n R R ˆ  τ (x, ξ )ψ(ξ, x)dx dξ, Rn

(14.9)

Rn

where ψ(q, p)  ϕ(p, q),

q, p ∈ Rn .

Thus, by (14.7), (14.9), (14.10), and Lemma 14.6, ˆ σ (ξ, x)ψ(ξ, x)dξ dx τˆ (ϕ)  Rn Rn  σˆ (p, q)ψ(p, q)dq dp n n R R  σˆ (p, q)ϕ(q, p)dq dp, ϕ ∈ S(R2n ). Rn

(14.10)

(14.11)

Rn



Thus, by (14.11), τˆ ∈ Lr (R2n ), and (14.8) is valid.

✷

Proof of Theorem 14.3. Let f and g be in S(Rn ). Then, by (12.1), Theorem 3.1, (3.12), and Lemmas 14.4–14.6, ¯  (2π)−n/2 σ (W (f, g)) (Wσ f )(g)  (2π)−n/2 σ (x, ξ )W (f, g)(x, ξ )dx dξ n n R R  (2π)−n/2 σ (x, ξ )V (f, g)ˆ(x, ξ )dx dξ Rn Rn  (2π)−n/2 σˆ (q, p)V (f, g)(q, p)dq dp Rn Rn p q  2−n (2π)−n/2 σˆ (q, p)W (f, g) ˜ ,− dq dp n n 2 2 R R  2n (2π)−n/2 σˆ (−2q, 2p)W (f, g)(p, ˜ q)dq dp Rn

Rn

74

14. Compact Weyl Transforms

 2n (2π)−n/2  2n (2π)−n/2  2n (2π)−n/2





Rn

σˆ (−2p, 2q)W (f, g)(q, ˜ p)dq dp

Rn





n

n

Rn

Rn

R R

τˆ (2q, −2p)W (f, g)(q, ˜ p)dq dp (δ τˆ )(q, p)W (f, g)(q, ˜ p)dq dp,

(14.12)

q, p ∈ Rn .

(14.13)

where (δ τˆ )(q, p)  τˆ (2q, −2p), r

Since δ τˆ ∈ L (R ), it follows from Theorem 11.1, (14.12), and (14.13) that 2n

(Wσ f )(g) ¯  2n Wδτˆ f, g, ˜ and hence |(Wσ f )(g)| ¯ ≤

f, g ∈ S(Rn ),

  n 2 r δ τˆ L2 (R2n ) f L2 (Rn ) g L2 (Rn ) π

(14.14)

for all f and g in S(Rn ). Therefore, by (14.14), Wσ : L2 (Rn ) → L2 (Rn ) is a bounded linear operator and   n 2 r Wσ ∗ ≤ δ τˆ Lr  (R2n ) . (14.15) π But, by (14.13),  δ τˆ Lr  (R2n ) 



Rn

  2

Rn − 2n r





Rn

|τˆ (2q, −2p)|r dq dp 

Rn

 1

|σˆ (−2p, 2q)|r dq dp

r

 1

σˆ Lr  (R2n ) ,

r

(14.16)

and hence (14.6) follows from (14.15) and (14.16). To prove that Wσ : L2 (Rn ) → 2n L2 (Rn ) is compact, let {τk }∞ k1 be a sequence of functions in S(R ) such that r 2n 2n τk → σˆ in L (R ) as k → ∞. For k  1, 2, . . . , let σk ∈ S(R ) be such that σˆ k  τk . Then, by (14.6), n

Wσ − Wσk ∗ ≤ (2π )− r  σˆ − σˆ k Lr  (R2n ) n

 (2π)− r  σˆ − τk Lr  (R2n ) , and hence Wσk → Wσ in B(L2 (Rn )) as k → ∞. By Theorem 6.8, Wσk is a Hilbert–Schmidt operator on L2 (Rn ), and hence, by Lemma 11.4, compact, for k  1, 2, . . . . Thus, Wσ : L2 (Rn ) → L2 (Rn ) is compact. ✷

15 Localization Operators

Our aim for Chapters 15–17 is to show that the localization operators introduced by Daubechies in the paper [3] as filters in signal analysis are examples of Weyl transforms that enjoy good mapping properties as compact operators from L2 (Rn ) into L2 (Rn ). In this chapter, we define the notion of a localization operator with symbol in Lr (R2n ), 1 ≤ r ≤ ∞, and prove that a localization operator with symbol in Lr (R2n ), 1 ≤ r ≤ ∞, is a bounded linear operator from L2 (Rn ) into L2 (Rn ). Let Z be the set of all integers. Then Cn × R/2πZ is a locally compact and Hausdorff group in which the group law is given by (q1 , p1 , t1 ) · (q2 , p2 , t2 )  (q1 + q2 , p1 + p2 , t1 + t2 + q1 · p2 ) for all (q1 , p1 , t1 ) and (q2 , p2 , t2 ) in Cn × R/2πZ, where q1 · p2 is the Euclidean inner product of q1 and p2 in Rn ; t1 +t2 and t1 +t2 +q1 ·p2 are cosets in the quotient group R/2π Z in which the group law is addition modulo 2π . On Cn × R/2π Z, the left Haar measure coincides with the right Haar measure and can be identified with the Lebesgue measure dq dp dt on the measurable space Cn × R/2π Z. The locally compact Hausdorff space Cn × R/2π Z endowed with the left (and right) Haar measure dq dp dt is hence unimodular. It is called the Weyl–Heisenberg group, and we denote it by (W H )n . Let π : (W H )n → B(L2 (Rn )) be the mapping defined by (π (q, p, t)f )(x)  ei(p·x+ 2 q·p+t) f (x − q), 1

x ∈ Rn ,

for all (q, p, t) in (W H )n and f in L2 (Rn ). That π : (W H )n → B(L2 (Rn )) is an irreducible unitary representation is left as an exercise.

76

15. Localization Operators

Let ϕ be the function on Rn defined by n

ϕ(x)  π − 4 e−

|x|2 2

,

x ∈ Rn .

(15.1)

Then ϕ L2 (Rn )  1,

(15.2)

and it is also an easy exercise to prove that the number cϕ defined by 2π |ϕ, π(q, p, t)ϕ|2 dq dp dt cϕ  Rn

0

Rn

is finite, and in fact cϕ  (2π)n+1 .

(15.3)

The function ϕ is called an admissible wavelet for the irreducible unitary representation π : (W H )n → B(L2 (Rn )), and the representation π : (W H )n → B(L2 (Rn )) is called square integrable. It is left as an exercise that we have the resolution of the identity formula, i.e., 1 2π f, π(z, t)ϕπ (z, t)ϕ, gdz dt (15.4) f, g  cϕ 0 Cn for all f and g in L2 (Rn ). The theory of the representation π : (W H )n → B(L2 (Rn )) hitherto described is an important, albeit special, case of the theory of square-integrable representations studied in the papers [11,12] by Grossmann, Morlet, and Paul, and the book [17] by Holschneider, among others. For q and p in Rn , we define the function ϕq,p on Rn by ϕq,p (x)  eip·x ϕ(x − q),

x ∈ Rn .

(15.5)

Let F be a measurable function on Cn . Then we define σ : (W H )n → C by σ (z, t)  F (z),

z ∈ Cn , t ∈ [0, 2π ].

(15.6)

Let F be in L1 (Cn ) or L∞ (Cn ). Then, for any f in L2 (Rn ), we define the function LF f on Rn by 1 2π σ (z, t)f, π(z, t)ϕπ(z, t)ϕ, gdz dt LF f, g  cϕ 0 Cn or, in view of (15.3) and (15.6), F (z)f, ϕz ϕz , gdz, g ∈ L2 (Rn ). LF f, g  (2π)−n

(15.7)

Cn

We have the following result. Proposition 15.1. Let F ∈ L1 (Cn ). Then LF : L2 (Rn ) → L2 (Rn ) is a bounded linear operator, and LF ∗ ≤ (2π)−n F L1 (Cn ) .

(15.8)

15. Localization Operators

Proof.

77

Since by (15.2) and (15.5), z ∈ Cn ,

ϕz L2 (Rn )  1,

(15.9)

it follows from (15.9) that for all z in Cn , |f, ϕz ϕz , g| ≤ f L2 (Rn ) g L2 (Rn ) ,

f, g ∈ L2 (Rn ).

(15.10)

Since F ∈ L1 (Cn ), it follows from (15.7) and (15.10) that |LF f, g| ≤ (2π)−n F L1 (Cn ) f L2 (Rn ) g L2 (Rn ) for all f and g in L2 (Rn ) and hence LF : L2 (Rn ) → L2 (Rn ) is a bounded linear operator and (15.8) is valid. ✷ We also have the following. Proposition 15.2. Let F ∈ L∞ (Cn ). Then LF : L2 (Rn ) → L2 (Rn ) is a bounded linear operator, and LF ∗ ≤ F L∞ (Cn ) .

(15.11)

Let f and g be in L2 (Rn ). Then, by (15.7),   21  −n 2 |LF f, g| ≤ (2π ) F L∞ (Cn ) |f, ϕz| dz

Proof.

Cn

|ϕz , g| dz 2

Cn

 21

.

(15.12)

But using the resolution of the identity formula, i.e., (15.4), 2 −n |f, ϕz |2 dz f L2 (Rn )  (2π)

(15.13)

Cn

and g 2L2 (Rn )

−n

 (2π)

Cn

|g, ϕz |2 dz.

(15.14)

So, by (15.12), (15.13), and (15.14), |LF f, g| ≤ F L∞ (Cn ) f L2 (Rn ) g L2 (Rn ) , ✷

and the proof is complete. n

n

Remark 15.3. The bounded linear operators LF : L (R ) → L (R ) already introduced, and to be introduced, in this chapter are called localization operators. This terminology appears to be first used in the papers [3, 4] by Daubechies and [5] by Daubechies and Paul. To understand the terminology better, let us note that when F is identically equal to 1 on Cn , the resolution of the identity formula, i.e., (15.4), implies that the corresponding operator LF is equal to the identity. Thus, the role of the “symbol” F is to assign different weights to different parts of the phase space Cn , i.e., localize on Cn , in order to produce a mathematically interesting operator with applications in various disciplines in science and engineering. To wit, applications to signal analysis can be found in the above-mentioned papers, the paper [14] by He, and the paper [16] by He and Wong. 2

2

78

15. Localization Operators

We can now associate a localization operator LF : L2 (Rn ) → L2 (Rn ) to every F in Lr (Cn ), 1 < r < ∞. The main result is the following theorem. Theorem 15.4. Let F ∈ Lr (Cn ), 1 < r < ∞. Then there exists a unique bounded linear operator LF : L2 (Rn ) → L2 (Rn ) such that n

LF ∗ ≤ (2π)− r F Lr (Cn ) ,

(15.15)

and LF f, g, for all f and g in L2 (Rn ), is given by (15.7) for all simple functions F on Cn for which the Lebesgue measure of the set {z ∈ Cn : F (z)  0} is finite. Proof. Let D be the set of all simple functions F on Cn such that the Lebesgue measure of the set {z ∈ Cn : F (z)  0} is finite. Let f ∈ L2 (Rn ) and let T be the linear transformation from D into the set of all Lebesgue-measurable functions on Rn defined by T F  LF f,

F ∈ D.

(15.16)

Then, by (15.8), (15.11), (15.16), and the Riesz–Thorin theorem, n

LF f L2 (Rn ) ≤ (2π)− r F Lr (Cn ) f L2 (Rn )

(15.17)

for all F in D. Hence (15.15) follows immediately from (15.17). Now, let F ∈ Lr (Cn ). Then there exists a sequence {Fk }∞ k1 of functions in D such that Fk → F in Lr (Cn ) as k → ∞. Thus, by (15.15), n

LFk − LFj ∗ ≤ (2π)− r Fk − Fj Lr (Cn ) → 0 ∞. Therefore, by (15.18), {LFk }∞ k1

(15.18)

as k, j → is a Cauchy sequence in B(L2 (Rn )). Thus, there is a bounded linear operator LF : L2 (Rn ) → L2 (Rn ) such that LFk → LF in B(L2 (Rn )) as k → ∞. That the limit is independent of the choice of the 2 n 2 n sequence {Fk }∞ k1 in D and that LF : L (R ) → L (R ) so defined is the unique bounded linear operator satisfying the conclusions of the theorem should, by now, be obvious, or if not, it follows from a standard argument, which we leave as an exercise. ✷

16 A Fourier Transform

In order to study localization operators in some detail, we first compute the Fourier transform of a function on Cn . Theorem 16.1. Let ϕ be the function on Rn defined by (15.1), and for any x in Rn and f in S(Rn ), let fx be the function on Cn defined by fx (z)  f, ϕz ϕz (x),

z ∈ Cn .

(16.1)

Then fˆx (ζ )  e−

|ζ |2 4

(ρ(−ζ )f )(x),

ζ ∈ Cn ,

(16.2)

where ρ(ζ ), for any ζ in Cn , is given by (2.1). Proof.

Let ε be any positive number. Then we define the function Iε on Cn by ε 2 |p|2 e−i(q·ξ +p·η) e− 2 f, ϕz ϕz (x)dz (16.3) Iε (ζ )  (2π)−n Rn

Rn

for all ζ  ξ + iη in C , where z  q + ip is a point in Cn . Then, by (15.1), (16.3), and the fact that the Fourier transform of the function ψ on Rn given by n

ψ(x)  e−

|x|2 2

,

x ∈ Rn ,

(16.4)

is equal to itself, we get (2π)n Iε (ζ )   2 2 −i(q·ξ +p·η) − ε |p| −ip·y 2 e e f (y)e ϕ(y − q)dy eip·x ϕ(x − q)dq dp  Rn Rn

Rn

80

16. A Fourier Transform





−iq·ξ

2 |p|2

 dp dq

f (y)ϕ(y − q)ϕ(x − q) e e Rn |η+y−x|2  (2π)n/2 e−iq·ξ f (y)ϕ(y − q)ϕ(x − q)ε −n e− 2ε2 dq n n   R R |η+y−x|2  (2π)n/2 e−iq·ξ ϕ(x − q) ε −n e− 2ε2 f (y)ϕ(y − q)dy dq Rn Rn

e

−ip·(η+y−x) − ε

Rn

2

(16.5)

Rn

for all ζ  ξ + iη in Cn . Now, for each q in Rn , we define Fq : Rn → C by Fq (y)  f (y)ϕ(y − q),

y ∈ Rn .

Then, by (16.4), (16.5), and (16.6), Iε (ζ )  (2π)−n/2 e−iq·ξ ϕ(x − q)(Fq ∗ ψε )(x − η)dq

(16.6)

(16.7)

Rn

for all ζ  ξ + iη in Cn , where ψε (x)  ε−n ψ

x

(16.8) , x ∈ Rn , ε and ψ is the function on Rn given by (16.4). Now, for each fixed q in Rn , we get, by (16.4), (16.6), and (16.8),   n (16.9) ψ(x)dx Fq  (2π) 2 Fq F q ∗ ψε → Rn

uniformly on compact subsets of Rn as ε → 0. Furthermore, there exists a positive constant C such that |(Fq ∗ ψε )(w)| ≤ Fq L∞ (Rn ) ψε L1 (Rn ) ≤ sup (|f (y) ϕ(y − q)|) ψ L1 (Rn ) y∈Rn

≤ C, w, q ∈ Rn .

(16.10)

So, by (16.6), (16.7), (16.9), (16.10), and the Lebesgue dominated convergence theorem,   lim Iε (ζ )  e−iq·ξ ϕ(x − q)ϕ(x − η − q)dq f (x − η) (16.11) ε→0

Rn

for all ζ  ξ + iη in Cn . But, using (16.1), (16.3), and the Lebesgue dominated convergence theorem, lim Iε (ζ )  (2π)−n e−i(q·ξ +p·η) f, ϕz ϕz (x)dz  fˆx (ζ ) (16.12) ε→0

Rn

n

Rn

for all ζ  ξ + iη in C . Thus, by (16.11), (16.12), (2.1), and Proposition 2.3,   −iq·ξ ˆ e ϕ(x − q)ϕ(x − η − q)dq f (x − η) fx (ζ )  Rn   −i(x−q)·ξ  e ϕ(q)ϕ(q − η)dq f (x − η) Rn

16. A Fourier Transform

 e−ix·ξ f (x − η)



81

 η η η  ϕ q− dq ei(q+ 2 )·ξ ϕ q + 2 2 Rn n

 e−ix·ξ e 2 iξ ·η f (x − η)(2π ) 2 V (ϕ, ϕ)(ξ, η) n  (ρ(−ξ, −η)f )(x)(2π) 2 V (ϕ, ϕ)(ξ, η). 1

(16.13)

But by Proposition 2.3, (15.1), and the fact that the Fourier transform of the function ψ on Rn defined by (16.4) is equal to itself, we get  η  η V (ϕ, ϕ)(ξ, η)  (2π)−n/2 e−iy·ξ ϕ y + ϕ y− dy 2 2 Rn η 2 η 2 n 1 1  (2π)−n/2 π − 2 eiy·ξ e− 2 |y+ 2 | e− 2 |y− 2 | dy n R 1 2 |η|2 −n/2 − n2  (2π) π eiy·ξ e− 2 {2|y| + 2 } dy Rn |η|2 n 2 eiy·ξ e−|y| dy  π − 2 e− 4 (2π)−n/2 Rn 2 |y|2 i √y ·ξ − n2 − |η|4 − n2 −n/2 π e 2 (2π) e 2 e− 2 dy Rn

 (2π)−n/2 e n

2 − |η|4

e

2 − |ξ4|

 (2π)−n/2 e−

|ζ |2 4

for all ζ  ξ + iη in C . Thus, (16.2) follows from (16.13) and (16.14).

(16.14) ✷

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17 Compact Localization Operators

The aim of this chapter is to use the Fourier transform computed in Chapter 16 to prove that every localization operator is a Weyl transform and the fact that a localization operator with symbol in Lr (R2n ), 1 ≤ r < ∞, is compact. Theorem 17.1. Let M be the function on Cn defined by M(z)  π −n e−|z| , 2

z ∈ Cn .

(17.1)

Then, for all F in Lr (Cn ), 1 ≤ r ≤ ∞, the Weyl transform WF ∗M , initially defined on S(Rn ), can be extended to a bounded linear operator from L2 (Rn ) into L2 (Rn ) that is equal to LF . Proof. We begin with the case when F is in S(Cn ). Then, for all f in S(Rn ), we get, by (15.7) and (16.1), (LF f )(x)  (2π)−n F (z)fx (z)dz Cn  (2π)−n Fˇ (ζ )fˆx (ζ ) dζ, x ∈ Rn , (17.2) Cn

where Fˇ is the inverse Fourier transform of F . Thus, by (17.2), 1 2 (LF f )(x)  (2π )−n Fˇ (ζ )e− 4 |ζ | (ρ(−ζ )f )(x)dζ n C 1 2  (2π)−n Fˆ (ζ )e− 4 |ζ | (ρ(ζ )f )(x)dζ, x ∈ Rn . Cn

(17.3)

84

17. Compact Localization Operators

But, by (9.4), (Wσ f )(x)  (2π)−n

Cn

σˆ (ζ )(ρ(ζ )f )(x)dζ, x ∈ Rn ,

(17.4)

for any σ in S(Cn ). Then, by (17.1), (17.3), (17.4), and Proposition 1.7, LF  WF ∗M .

(17.5)

Now, let F ∈ Lr (Cn ), 1 ≤ r < ∞. Then there exists a sequence {Fk }∞ k1 of functions in S(Cn ) such that Fk → F in Lr (Cn ) as k → ∞. Thus, by Proposition 15.1, Theorem 15.4, and (17.5), WFk ∗M  LFk → LF n

(17.6) n

in B(L (R )) as k → ∞. But for all f and g in S(R ), we get, by (12.1) and (17.6), 2

(WFk ∗M f )(g)  (2π)−n/2 (Fk ∗ M)(W (f, g)) ¯ ¯  (WF ∗M f )(g) → (2π)−n/2 (F ∗ M)(W (f, g))

(17.7)

as k → ∞. Thus, for all f in S(Rn ), we get, by (17.7), WFk ∗M f → WF ∗M f

(17.8)

in S  (Rn ) as k → ∞. But of course, by (17.6), we get, for all f in S(Rn ), WFk ∗M f → LF f 

n

(17.9)

n

in S (R ) as k → ∞. Thus, for all f in S(R ), we get, by (17.8) and (17.9), WF ∗M f  LF f

(17.10)

in the sense of distributions. Thus, by (17.10), the Weyl transform WF ∗M , initially defined on S(Rn ), can be extended to a bounded linear operator from L2 (Rn ) into L2 (Rn ) that is equal to LF . Now, let F ∈ L∞ (Cn ). Then we can find a sequence n {Fk }∞ k1 of simple functions on C such that the Lebesgue measure of the set {z ∈ Cn : Fk (z)  0} is finite for k  1, 2, . . . , and Fk → F a.e. on Cn as k → ∞. Now, for all f and g in S(Rn ), we get, by (12.1), (17.5), and the Lebesgue dominated convergence theorem, (WF ∗M f )(g)  (2π )−n/2 (F ∗ M)(z)W (f, g)(z)dz ¯ Cn (Fk ∗ M)(z)W (f, g)(z)dz ¯  lim (2π)−n/2 k→∞

Cn

 lim (WFk ∗M f )(g) k→∞

 lim (LFk f )(g)  (LF f )(g).

(17.11)

f ∈ S(Rn ).

(17.12)

k→∞

So, by (17.11), WF ∗M f  LF f,

17. Compact Localization Operators

85

Therefore, by (17.12), the Weyl transform WF ∗M , initially defined on S(Rn ), can be extended to a bounded linear operator from L2 (Rn ) into L2 (Rn ) that is equal to LF . ✷ Theorem 17.2. Let F ∈ Lr (Cn ), 1 ≤ r < ∞. Then the localization operator LF : L2 (Rn ) → L2 (Rn ) is compact. n Proof. Let {Fk }∞ k1 be a sequence of functions in S(C ) such that Fk → F in r n L (C ) as k → ∞. Then, by Theorem 6.8 and Lemma 11.4, LFk  WFk ∗M is a Hilbert–Schmidt operator and hence compact because Fk ∗ M is a function in S(Cn ) for k  1, 2, . . . . But by Proposition 15.1 and Theorem 15.4, LFk → LF in B(L2 (Rn )) as k → ∞. Therefore, LF is compact. ✷

Remark 17.3. That Theorem 17.2 is false for r  ∞ can be seen easily by taking the function F on Cn to be such that F (z)  1,

z ∈ Cn .

For then, by the resolution of the identity formula, i.e., (15.4), LF is equal to the identity operator on L2 (Rn ) and hence cannot be compact. Remark 17.4. A more precise and more general result than Theorem 17.2 has been proved. See Theorem 6.1 in the paper [15] by He and Wong in this connection.

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18 Hermite Polynomials

We are now interested in criteria for the boundedness and/or compactness of Weyl transforms on L2 (R) using an orthonormal basis for L2 (R2 ) consisting of Hermite functions on C ( R2 ), i.e., Wigner transforms of Hermite functions on R. The inverse Fourier transforms of functions in this orthonormal basis, i.e., the Fourier– Wigner transforms of Hermite functions on R, are shown in Chapter 22 to be eigenfunctions of a partial differential operator on C. In this and the next two chapters we lay out the basic properties of Hermite polynomials, Hermite functions, and Laguerre polynomials, which we shall use to study Weyl transforms. A good reference for these topics is Chapter 6 of the book [7] by Folland. The properties of Hermite functions on C to be used in this book are given in Chapters 21, 22, and 23. Let n  0, 1, 2, . . . . Then we define the function Hn on R by  n d 2 n x2 (e−x ), x ∈ R. (18.1) Hn (x)  (−1) e dx We call Hn the Hermite polynomial of degree n. It is easy to see that H0 (x)  1, H1 (x)  2x, and so on. Proposition 18.1. For n  1, 2, . . . ,  Hn (x)  2xHn−1 (x) − Hn−1 (x),

Proof. e

−x 2

x ∈ R.

By (18.1), n

Hn (x)  (−1)



d dx

n (e

−x 2

n

)  (−1)



d dx



d dx

n−1

(e−x ) 2

88

18. Hermite Polynomials



d − dx



n−1



(−1)

d dx



n−1 (e

−x 2

) −

d −x 2 (e Hn−1 (x)) dx

 (x))  −(−2xe−x Hn−1 (x) + e−x Hn−1 2

2

 (x),  2xe−x Hn−1 (x) − e−x Hn−1 2

2

x ∈ R, ✷

and Proposition 18.1 follows.

Remark 18.2. Using Proposition 18.1 repeatedly, we see that the highest power in Hn (x) is equal to (2x)n multiplied by the highest power in H0 (x), which is then equal to 2n x n . Let w be the function on R defined by w(x)  e−x , 2

x ∈ R.

(18.2)

Then we define L2w (R) to be the set of all complex-valued functions on R such that ∞ |f (x)|2 w(x)dx < ∞. −∞

Then L2w (R) is a Hilbert space in which the inner product  , w and norm w are, respectively, given by ∞ f, gw  f (x)g(x)w(x)dx (18.3) −∞

and  f w 

∞

−∞

|f (x)| w(x)dx 2

 21 (18.4)

for all f and g in L2w (R). Proposition 18.3. {Hn : n  0, 1, 2, . . .} is an orthogonal set in L2w (R). Moreover, √ Hn 2w  2n n! π, n  0, 1, 2, . . . . Proof. Let m and n be nonnegative integers such that m ≤ n. Then, by (18.1), (18.2), and (18.3), we get ∞ 2 Hm , Hn w  Hm (x)Hn (x)e−x dx −∞  n ∞ d 2 2 n x2  Hm (x)(−1) e (e−x )e−x dx dx −∞  n ∞ d 2 n  (−1) Hm (x) (e−x )dx dx −∞ ∞ 2 Hm(n) (x)e−x dx. (18.5)  −∞

18. Hermite Polynomials

89

Thus, by (18.5), Hm , Hn w  0 if m  n. Moreover, for n  0, 1, 2, . . . , by Remark 18.2, (18.3), (18.4), and (18.5), we get ∞ ∞ √ 2 2 Hn(n) (x)e−x dx  2n n! e−x dx  2n n! π Hn 2w  −∞

−∞

✷

and hence complete the proof. We can strengthen Proposition 18.3 and get the following result.

Theorem 18.4. { H1n w Hn : n  0, 1, 2, . . .} is an orthonormal basis for L2w (R). To prove Theorem 18.4, we use two lemmas. Lemma 18.5. Let {pn : n  0, 1, 2, . . .} be a sequence of nonzero polynomials such that the degree of pn is equal to n. Let P be any polynomial of degree k. Then there exist constants c0 , c1 , c2 , . . . , ck such that P 

k


cn p n .

n0

Proof. Let P be a polynomial of degree zero. Then P (x)  α for all x in R, where α is a constant. Suppose that p0 (x)  β for all x in R, where β is a nonzero constant. Then α α P (x)  β  p0 (x), x ∈ R. β β Suppose that Lemma 18.5 is true for all polynomials of degree at most k. Let P be a polynomial of degree k + 1. Then we choose the constant ck+1 such that P and ck+1 pk+1 have the same highest power. Thus, P − ck+1 pk+1 is a polynomial of degree at most k. Hence P − ck+1 pk+1 

k


c n pn

n0

for some constants c0 , c1 , c2 , . . . , ck , and the proof is complete. Lemma 18.6. Let f be a measurable function on R such that ∞ 2 |f (x)|e|xξ | e−x dx < ∞

✷

(18.6)

−∞

for all ξ in R. If



∞

−∞

f (x)P (x)e−x dx  0 2

(18.7)

for all polynomials P , then f  0 a.e. on R. Proof. We begin by noting that ∞ ∞
∞ (ixξ )n 2 2 eixξ f (x)e−x dx  f (x)e−x dx, ξ ∈ R. n! −∞ −∞ n0

(18.8)

90

18. Hermite Polynomials

Now,

  ∞ N 

(ixξ )n |xξ |n 2  −x 2  f (x)e  ≤ |f (x)|e−x   n0 n!  n0 n! ≤ e|xξ | |f (x)|e−x , 2

x, ξ ∈ R.

(18.9)

Hence, using (18.6)–(18.9) and the Lebesgue dominated convergence theorem, ∞ ∞
(iξ )n ∞ n 2 ixξ −x 2 e f (x)e dx  x f (x)e−x dx  0. (18.10) n! −∞ −∞ n0 Thus, by (18.10), f (x)e−x complete.

2

 0 for almost all x in R, and the proof is ✷

Proof of Theorem 18.4. In view of Proposition 18.3, we need only prove that if f ∈ L2w (R) is such that f, Hn w  0,

n  0, 1, 2, . . . ,

then f  0 a.e. on R. But by the Schwarz inequality,  ∞  21  ∞ |tx| −x 2 2 −x 2 |f (x)|e e dx ≤ |f (x)| e dx −∞

−∞

∞ −∞

(18.11)

e

2|tx| −x 2

e

dx

 21

<∞

(18.12) for all t in R. Let P be any polynomial of degree k. Then, by Lemma 18.5, we can find constants c0 , c1 , c2 , . . . , ck such that P (x) 

k


cn Hn (x),

x ∈ R.

(18.13)

n0

Thus, by (18.11) and (18.13), ∞ k
2 f (x)P (x)e−x dx  cn −∞

n0

∞ −∞

f (x)Hn (x)e−x dx  0. 2

(18.14)

Hence, by (18.12) and (18.14), Lemma 18.6 can be used to conclude that f  0 a.e. on R. ✷ The following properties of Hermite polynomials will be useful to us. Proposition 18.7. For all x in R, (i) H0 (x)  0; (ii) Hn (x)  2nHn−1 (x), n  1, 2, . . . ; (iii) Hn (x) − 2xHn (x) + 2nHn (x)  0, n  0, 1, 2, . . . . Proof. To prove part (iii), note that the equation is trivially true for n  0. Now, suppose that   Hn−1 (x) − 2xHn−1 (x) + 2(n − 1)Hn−1 (x)  0

(18.15)

18. Hermite Polynomials

91

for some positive integer n. By Proposition 18.1 and (18.15), we get   Hn (x)  2Hn−1 (x) + 2xHn−1 (x) − Hn−1 (x)

   2Hn−1 (x) + 2xHn−1 (x) − 2xHn−1 (x) + 2(n − 1)Hn−1 (x)

 2nHn−1 (x),

x ∈ R.

(18.16)

So, by (18.16),  Hn (x)  2nHn−1 (x),

x ∈ R.

(18.17)

Thus, by Proposition 18.1, (18.16) and (18.17), Hn (x) − 2xHn (x) + 2nHn (x)

   2nHn−1 (x) − 4nxHn−1 (x) + 4nxHn−1 (x) − 2nHn−1 (x)  0

for all x in R. Part (i) is obvious. To prove part (ii), note that by part (iii) with n replaced by n − 1 and Proposition 18.1, we get   Hn (x)  2xHn−1 (x) + 2Hn−1 (x) − 2xHn−1 (x) + 2(n − 1)Hn−1 (x)

 2nHn−1 (x),

x ∈ R.

✷

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19 Hermite Functions

For n  0, 1, 2, . . . , we define the function hn on R by x2

hn (x)  e− 2 Hn (x),

x ∈ R.

(19.1)

We call hn the Hermite function of order n. Theorem 19.1. For all x in R, (i) xhn (x) + hn (x)  2nhn−1 (x), n  1, 2, . . . ; (ii) xhn (x) − hn (x)  hn+1 (x), n  0, 1, 2, . . . ; (iii) hn (x) − x 2 hn (x) + (2n + 1)hn (x)  0, n  0, 1, 2, . . . . d d d 2 , x − dx , and − dx Remark 19.2. The differential operators x + dx 2 +x in Theorem 19.1 are, respectively, called the annihilation operator, the creation operator, and the harmonic oscillator. The Hermite function hn of order n is an eigenfunction of the harmonic oscillator corresponding to the eigenvalue 2n + 1. 2

Proof of Theorem 19.1. For n  1, 2, . . . , by (19.1) and part (ii) of Proposition 18.7, x2

2ne 2 hn−1 (x)  2nHn−1 (x)  Hn (x) d x2 x2 x2  (e 2 hn (x))  xe 2 hn (x) + e 2 hn (x) dx for all x in R, and part (i) is proved. For part (ii), using (19.1) and Proposition 18.1, we get, for n  1, 2, . . . , x2

x2

e 2 hn+1 (x)  2xe 2 hn (x) −

d x2 (e 2 hn (x)) dx

94

19. Hermite Functions x2

x2

x2

 2xe 2 hn (x) − xe 2 hn (x) − e 2 hn (x) x2

x2

 xe 2 hn (x) − e 2 hn (x),

x ∈ R,

and part (ii) follows. For part (iii), we use parts (i) and (ii) to get hn (x)  xhn−1 (x) − hn−1 (x)     1 1  1 1 1  x xhn (x) + hn (x) + hn (x) − xhn (x) + hn (x) 2n 2n 2n 2n 2n 1 2 1 1  (19.2)  − hn (x) + x hn (x) − hn (x), x ∈ R, 2n 2n 2n for n  1, 2, . . . . That part (iii) is also true for n  0 is obvious. ✷ For n  1, 2, . . . , we define the function en on R by en (x) 

1 √

(2n n!

1

π) 2

hn (x),

x ∈ R.

(19.3)

Then we have the following important theorem. Theorem 19.3. {en : n  0, 1, 2, . . .} is an orthonormal basis for L2 (R). Proof. Let m and n be nonnegative integers. Then, by (19.1), (19.3), and Proposition 18.3, ∞ ∞ 1 em (x)en (x)dx  hm (x)hn (x)dx 1 −∞ (2m+n m!n!π) 2 −∞ ∞ 1 2 Hm (x)Hn (x)e−x dx  1 m+n (2 m!n!π) 2 −∞  0, m  n,  1, m  n. So, it remains to prove that if f ∈ L2 (R) is such that f, hn   0,

n  0, 1, 2, . . . ,

(19.4)

then f  0 a.e. on R. To this end, let F be the function on R defined by x2

F (x)  f (x)e 2 , Then F ∈

L2w (R),

x ∈ R.

(19.5)

and for n  0, 1, 2, . . . , we get, by (19.4), F, Hn w  f, hn   0.

(19.6)

So, by Proposition 18.4 and (19.6), F  0 a.e. on R, and hence, by (19.4), f  0 a.e. on R. ✷

20 Laguerre Polynomials

Let α > −1. Then, for n  0, 1, 2, . . . , we define the function Lαn on R by Lαn (x) 

x −α ex n!



d dx

n

(e−x x α+n ),

x > 0.

We call Lαn the Laguerre polynomial of degree n and order α. If we write out Lαn (x), x > 0, in detail, then we get Lαn (x) 

 n−k n   n d x −α ex
(−1)k e−x (x α+n ), n! k0 k dx

x > 0.

(20.1)

Thus, Lαn (x) 

n−1 (α + n)(α + n − 1) · · · (α + k + 1) (−1)n n
x + (−x)k , x > 0. (20.2) n! (n − k)!k! k0

For the highest power in Lαn (x), x > 0, we let k  n in (20.1) or use (20.2) to get Lαn (x) 

(−1)n n x + ···, n!

x > 0.

Let u be the function on (0, ∞) defined by u(x)  x α e−x ,

x ∈ (0, ∞).

(20.3)

96

20. Laguerre Polynomials

Then we define L2u (0, ∞) to be the set of all complex-valued functions f on (0, ∞) such that ∞ |f (x)|2 u(x)dx < ∞. 0

L2u (0, ∞)

is a Hilbert space in which the inner product  , u and norm u Then are, respectively, given by ∞ f, gu  f (x)g(x)u(x)dx 0

and



∞

f u 

|f (x)| u(x)dx 2

 21

0

for all f and g in L2u (0, ∞). Proposition 20.1. {Lαn : n  0, 1, 2, . . .} is an orthogonal set in L2u (0, ∞). Moreover, L(α + n + 1) , n  0, 1, 2, . . . . n! Proof. Let m and n be nonnegative integers such that m < n. Then ∞ α α Lm , Ln u  Lαm (x)Lαn (x)x α e−x dx 0  n 1 ∞ α d  Lm (x) (e−x x α+n )dx n! 0 dx   (−1)n ∞ d n α {Lm (x)}e−x x α+n dx  0.  n! dx 0 Lαn 2u 

Next, by (20.3) and the same computations as before, 1 ∞ −x α+n L(α + n + 1) α 2 α α Ln u  Ln , Ln u  e x dx  n! 0 n! for n  1, 2, . . . , and that the same formula is valid for n  0 follows from the definition of Lα0 . ✷ Theorem 20.2. { L1α u Lαn : n  0, 1, 2, . . .} is an orthonormal basis for L2u (0, ∞). n

Proof. In view of Proposition 20.1, we only need to prove that if g ∈ L2u (0, ∞) is such that g, Lαn u  0,

n  0, 1, 2, . . . ,

(20.4)

then g  0 a.e. on (0, ∞). Now, for n  0, 1, 2, . . . , we get, by Lemma 18.5, xn 

n
k0

ck Lαk (x),

x > 0,

(20.5)

20. Laguerre Polynomials

97

where c0 , c1 , c2 , . . . , cn are constants. Thus, for n  0, 1, 2, . . . , we get, by (20.4) and (20.5), ∞ ∞ n
n α −x g(x)x x e dx  ck g(x)Lαk (x)x α e−x dx  0. (20.6) 0

0

k0

Let x  y . Then, by (20.6), we get, for n  0, 1, 2, . . . , ∞ ∞ 2 2 g(y 2 )y 2n y 2α+1 e−y dy  0 ⇒ 2 g(y 2 )y 2n |y|2α+1 e−y dy  0 2 0 0 ∞ 2 g(y 2 )y 2n |y|2α+1 e−y dy  0. (20.7) ⇒ 2

−∞

Of course,



∞

−∞

g(y 2 )y 2n+1 |y|2α+1 e−y dy  0, 2

So, by (20.7) and (20.8), ∞ 2 g(y 2 )y n |y|2α+1 e−y dy  0,

n  0, 1, 2, . . . .

n  0, 1, 2, . . . .

−∞

(20.8)

(20.9)

Let P be any polynomial of degree k. Then P (y) 

k


an y n ,

y ∈ R,

n0

where a0 , a1 , a2 , . . . , ak are constants. So, by (20.9), ∞ 2 g(y 2 )P (y)|y|2α+1 e−y dy  0.

(20.10)

−∞

Also, for all ξ in R, ∞ 2 |g(y 2 )||y|2α+1 e|yξ | e−y dy −∞

≤



∞

−∞

 

∞

|g(y )| |y| 2 2

2α+1 −y 2

2 α −x

|g(x)| x e

e

dx

dy

 21 

0

 21 

∞ −∞

∞ −∞

|y|

|y|

2α+1 2|yξ | −y 2

e

2α+1 −2|yξ | −y 2

e

e

e

dy

dy

 21

 21

< ∞. (20.11)

Thus, by Lemma 18.6, (20.10), and (20.11), g(y 2 )|y|2α+1  0 for almost all y in R. Therefore, g  0 a.e. on (0, ∞). ✷ Theorem 20.3. For each fixed positive number x, ∞
n0

xz

Lαn (x)zn

e− 1−z  , (1 − z)α+1

|z| < 1,

where the series is uniformly and absolutely convergent on every compact subset of {z ∈ C : |z| < 1}.

98

20. Laguerre Polynomials xz

e− 1−z the generating function of the Laguerre (1 − z)α+1 α polynomials Ln , n  0, 1, 2, . . . . Remark 20.4. We call

Proof of Theorem 20.3.

γ ✙

r ✒

|

|

|

|

|

|

|

|

|

|

|

|

|

|

x

Fig. 3 Let γ be a circle with center at x and lying inside the right half plane (Fig. 3). Now,   ∞ ∞

x −α ex d n −x α+n n Lαn (x)zn  (e x )z n! dx n0 n0 ∞ e−ζ ζ α+n x −α ex
 zn dζ, (20.12) n+1 2πi n0 γ (ζ − x) where the principal branch of ζ α+n is taken, i.e., ζ α+n  e(α+n)Log−π ζ and Log−π ζ  ln |ζ | + iArg−π ζ,

−π < Arg−π ζ < π.

Next, for n  1, 2, . . . ,  −ζ α+n   e−Reζ e(α+n) ln |ζ |  e ζ    (ζ − x)n+1   r n+1 e−(x−r) (x + r)α+n r n+1  α  x+r n −(x−r) (x + r) . e r r ≤

20. Laguerre Polynomials

99

∞
zn e−ζ ζ α+n is uniformly and (ζ − x)n+1 n0 absolutely convergent with respect to z on {z ∈ C : |z| < rx } and ζ on γ , where r rx is any number in (0, x+r ). Therefore, by (20.12),  n −ζ α
∞
zζ e ζ ∞ x −α ex α n Ln (x)z  dζ (20.13) 2πi γ ζ − x n0 ζ − x n0

Then, for all z in C with |z| <

for |z| < rx . But

r , x+r

the series

   zζ  |z|(x + r)   < 1, ζ − x  ≤ r

and hence, by (20.13), ∞
n0

Lαn (x)zn  

x −α ex 2πi x −α ex 2πi −α x



γ

e−ζ ζ α 1 dζ ζ − x 1 − ζzζ −x

γ

e−ζ ζ α dζ ζ − x − zζ

e−ζ ζ α dζ γ (1 − z)ζ − x x −α ex 1 e−ζ ζ α  x dζ 1 − z 2πi γ ζ − 1−z



x e 2πi

for |z| < rx . But for sufficiently small z, ∞
n0

Lαn (x)zn

x 1−z

x −α ex − x  e 1−z 1−z



(20.14)

is inside γ . So, by (20.14), x 1−z

α

− xz

e 1−z  (1 − z)α+1

− xz

e 1−z is an analytic function on {z ∈ C : |z| < (1 − z)α+1 1}. Thus, by the principle of analytic continuation,

for sufficiently small z. Now, ∞
n0

−xz

Lαn (x)zn 

and Theorem 20.3 is proved.

e 1−z , (1 − z)α+1

|z| < 1, ✷

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21 Hermite Functions on C

We can now give in this and the next two chapters a self-contained treatment of Hermite functions on C, i.e., the Fourier–Wigner transforms of Hermite functions on R, which we need in this book. A good reference for these topics is Chapter 1 of the book [29] by Thangavelu. For j, k  0, 1, 2, . . . , we define the function ej,k on C by ej,k (z)  V (ej , ek )(q, p)

(21.1)

for all z  q + ip in C. Proposition 21.1. {ej,k : j, k  0, 1, 2, . . .} is an orthonormal set in L2 (R2 ). Proof. By (21.1), Theorem 19.3, the Moyal identity for the Fourier–Wigner transform, and the Plancherel theorem, we get, for all nonnegative integers j1 , j2 , k1 , and k2 , ej1 ,k1 , ej2 ,k2   V (ej1 , ek1 ), V (ej2 , ek2 )  ej1 , ej2 ek1 , ek2   0 unless j1  j2 and k1  k2 ; and if j1  j2 and k1  k2 , then ej1 ,k1 , ej2 ,k2   1.

✷

Theorem 21.2. {ej,k : j, k  0, 1, 2, . . .} is an orthonormal basis for L2 (R2 ). Proof. In view of Proposition 21.1, we only need to prove that if f ∈ L2 (R2 ) is such that f, ej,k   0,

j, k  0, 1, 2, · · · ,

(21.2)

102

21. Hermite Functions on C

then f  0 a.e. on R2 . To this end, we let g ∈ L2 (R2 ) be such that gˆ  f . Then, by Theorem 4.4, Lemma 14.6, and (21.1), ∞ ∞ 1 g(x, ξ )W (ej , ek )(x, ξ )dx dξ Wg ej , ek   (2π)− 2 −∞ −∞ ∞ ∞ 1 f (q, p)V (ej , ek )(q, p)dq dp  (2π)− 2 −∞ −∞ ∞ ∞ 1 f (q, p)ej,k (q, p)dq dp (21.3)  (2π )− 2 −∞

−∞

for j, k  0, 1, 2, . . . . By Theorem 19.3, (21.2), and (21.3), Wg ej  0,

j  0, 1, 2, . . . .

(21.4)

Now, let h ∈ L2 (R) and ε be any
positive number. Then, by Theorem 19.3, we can find a finite linear combination ajk ejk of the ej ’s such that 
   ajk ejk − h 2 < ε. (21.5)  L (R)

So, by (21.4) and (21.5),   
  a j k ej k  Wg h L2 (R) ≤ Wg h −

L2 (R)

 
   + Wg ajk ejk 

L2 (R)

≤ ε Wg ∗ . Since ε is arbitrary, it follows that Wg h  0,

h ∈ L2 (R).

Therefore, Wg  0. But by part (v) of Theorem 7.5, (2π )− 2 g L2 (R2 )  Wg H S  0. 1

Thus, g  0 a.e. on R2 . Consequently, f  0 a.e. on R2 .

✷

22 Vector Fields on C

In this chapter, we introduce the analogues of the annihilation operator, the creation operator, and the harmonic oscillator for the Hermite functions ej,k , j, k  0, 1, 2, . . . , on R2 . To do this, we define the vector fields Z and Z on C, respectively, by Z

∂ 1 + z¯ ∂z 2

and ∂ 1 Z¯  − z, ∂ z¯ 2 where ∂ ∂ ∂  −i ∂z ∂q ∂p and ∂ ∂ ∂  +i . ∂ z¯ ∂q ∂p We also define the vector field L on C by 1 ¯ L  − (Z Z¯ + ZZ). 2 Theorem 22.1. For all z in C, (i) (ii)

1

(Zej,k )(z)  i(2k) 2 ej,k−1 (z), j  0, 1, 2, . . . , k  1, 2, . . . ; ¯ j,k )(z)  i(2k + 2) 21 ej,k+1 (z), j, k  0, 1, 2, . . . . (Ze

22. Vector Fields on C

104

Proof.

By (21.1), we get, for all q and p in R, ∞  ∂ej,k p  p 1 (q, p)  i(2π)− 2 yeiqy ej y + ek y − dy. ∂q 2 2 −∞

(22.1)

Now, using (22.1) and the formula  p p  + y− , 2y  y + 2 2 we get, for all q and p in R, ∂ej,k (q, p)  J (+) (q, p) + J (−) (q, p), ∂q where J (±) (q, p) 

i 1 (2π)− 2 2



(22.2)

 p  p  p eiqy y ± ej y + ek y − dy. (22.3) 2 2 2 −∞ ∞

Next, for all q and p in R, i

∂ej,k (q, p)  K (1) (q, p) − K (2) (q, p), ∂p

where i 1 K (q, p)  (2π)− 2 2



(1)

and K (2) (q, p) 

i 1 (2π)− 2 2



(22.4)

 p  p ek y − dy eiqy ej y + 2 2 −∞

(22.5)

 p   p eiqy ej y + ek y − dy. 2 2 −∞

(22.6)

∞

∞

Now, by Theorem 19.1 and (19.3), we get, for k  0, 1, 2, . . . ,    d 1 ek (x)  (2k + 2) 2 ek+1 (x), x ∈ R, x− dx and, for k  1, 2, . . . ,    d 1 ek (x)  (2k) 2 ek−1 (x), x+ dx

x ∈ R.

(22.7)

(22.8)

So, by (22.2)–(22.8), we get, for j  0, 1, 2, . . . and k  1, 2, . . . , ∂ej,k (z)  (J (+) (q, p) − K (1) (q, p)) + (J (−) (q, p) + K (2) (q, p)) ∂z  ∞  p  p i 1 1 eiqy (2j + 2) 2 ej +1 y + ek y − dy  (2π)− 2 2 2 2 −∞  ∞   p p 1 eiqy (2k) 2 ej y + ek−1 y − dy + 2 2 −∞ i 1 1 (22.9)  {(2j + 2) 2 ej +1,k (z) + (2k) 2 ej,k−1 (z)}, z ∈ C. 2

22. Vector Fields on C

105

Also, by (22.2)–(22.8), we get, for j  1, 2, . . . and k  0, 1, 2, . . . , ∂ej,k (z)  (J (+) (q, p) + K (1) (q, p)) + (J (−) (q, p) − K (2) (q, p)) ∂ z¯  ∞  p  p i 1 1 eiqy (2j ) 2 ej −1 y + ek y − dy  (2π)− 2 2 2 2 −∞  ∞   p p 1 + eiqy (2k + 2) 2 ej y + ek+1 y − dy 2 2 −∞ i 1 1 (22.10)  {(2j ) 2 ej −1,k (z) + (2k + 2) 2 ej,k+1 (z)}, z ∈ C. 2 Now, by (21.1), (22.5), (22.6), and an integration by parts,   ∞ 1 ∂ iqy i p  p 1 qej,k (q, p)  − (2π) 2 ej y + e ek y − dy 2 2 2 2 −∞ ∂y  K (1) (q, p) + K (2) (q, p),

q, p ∈ R.

(22.11)

Using (21.1), (22.3), and the formula  p  p p y+ − y− , 2 2 we get i pej,k (q, p)  J (+) (q, p) + J (−) (q, p), 2

q, p ∈ R.

(22.12)

So, by (22.1), (22.11), and (22.12), we get, for j  1, 2, . . . and k  0, 1, 2, . . . ,  ∞  1 i p  p 1 − 21 eiqy (2j ) 2 ej −1 y + zej,k (z)  (2π) ek y − dy 2 2 2 2 −∞  ∞   p p 1 iqy 2 − e (2k + 2) ej y + ek+1 y − dy 2 2 −∞ i 1 1 (22.13)  {(2j ) 2 ej −1,k (z) − (2k + 2) 2 ej,k+1 (z)} 2 for all z in C. By (22.1), (22.11), and (22.12), we also get, for j  0, 1, 2, . . . and k  1, 2, . . . ,  ∞  1 p  p i 1 1 z¯ ej,k (z)  (2π)− 2 − eiqy (2j + 2) 2 ej +1 y + ek y − dy 2 2 2 2 −∞  ∞   p p 1 eiqy (2k) 2 ej y + ek−1 y − dy + 2 2 −∞ i 1 1 (22.14)  {(2k) 2 ej,k−1 (z) − (2j + 2) 2 ej +1,k (z)} 2 for all z in C. Therefore, by (22.9) and (22.14), 1

Zej,k  i(2k) 2 ej,k−1 ,

j  0, 1, 2, . . . , k  1, 2, . . . ,

106

22. Vector Fields on C

and by (22.10) and (22.13), ¯ j,k  i(2j + 2) 21 ej,k+1 , Ze

j  1, 2, . . . , k  0, 1, 2, . . . .

That the preceding formula is also true for e0,k , k  0, 1, 2, . . . , should by now be obvious and is left as an exercise. ✷ From Theorem 22.1, we get the following theorem. Theorem 22.2. Lej,k  (2k + 1)ej,k , Proof.

j, k  0, 1, 2, . . . .

By Theorem 22.1, we get, for j  0, 1, 2, . . . and k  1, 2, . . . , ¯ j,k  i(2k + 2) 2 Zej,k+1  −(2k + 2)ej,k Z Ze 1

and ¯ j,k  i(2k) 21 Zej,k−1  −2kej,k . ZZe Thus, for j  0, 1, 2, . . . and k  1, 2, . . . , 1 ¯ Lej,k  − (Z Z¯ + ZZ)e j,k  (2k + 1)ej,k . 2 That the preceding formula is also true for ej,0 , j  0, 1, 2, . . . , follows from the fact that Zej,0  0, and is left as an exercise.

j  0, 1, 2, . . . , ✷

Remark 22.3. In view of Theorem 22.1, we call Z and Z the annihilation operator and the creation operator, respectively, for the Hermite functions ej,k , j, k  0, 1, 2, . . . , on C. Theorem 22.2 says that for k  0, 1, 2, . . . , 2k + 1 is an eigenvalue of the “harmonic oscillator” L and the Hermite functions ej,k , j, k  0, 1, 2, . . . , on C are eigenfunctions of L corresponding to the eigenvalue 2k + 1. The partial differential operator L is in fact a “Laplacian” with variable coefficients on C.

23 Laguerre Formulas for Hermite Functions on C

We are now interested in expressing some classes of Hermite functions on C in terms of Laguerre polynomials. We begin with a formula. Theorem 23.1. (Mehler’s Formula) For all x and y in R and all w in C with |w| < 1, ∞
hk (x)hk (y) k0

2k k!

w k  (1 − w 2 )− 2 e 1

− 21

1+w2 1−w2

(x 2 +y 2 )+

2w 1−w2

xy

,

where the series is uniformly and absolutely convergent on {w ∈ C : |w| < 1}. Remark 23.2. To be specific, we use the principal branch of (1 − w2 )− 2 , i.e., 1

(1 − w 2 )− 2  e− 2 Log−π (1−w ) , 1

1

2

where Log−π ζ  ln |ζ | + iArg−π ζ,

−π < Arg−π ζ < π.

− 21

Thus, (1 − w2 ) is analytic on the cut plane C − {x ∈ R : x ≤ −1 or x ≥ 1} and, for any w in R with |w| < 1, we get (1 − w2 )− 2  e− 2 ln(1−w ) > 0. 1

1

2

Proof of Theorem 23.1. We begin with the formula ∞ 1 2 −x 2 √ e−u +2ixu du, x ∈ R. e π −∞

(23.1)

108

23. Laguerre Formulas for Hermite Functions on C

So, for k  0, 1, 2, . . . ,



 ∞ d k 1 2 e−u +2ixu du √ dx π −∞ ∞ x2 2 1 2 (2iu)k e−u +2ixu du  e− 2 (−1)k ex √ π −∞ ∞ 1 x2 2 uk e−u +2ixu du, x ∈ R.  √ (−2i)k e 2 π −∞ x2

hk (x)  e− 2 (−1)k ex

2

(23.2)

Hence, for any x and y in R and any w in (−1, 1), we have, by (23.1) and (23.2), ∞
hk (x)hk (y)

2k k!

k0

      

wk

∞ 1 1 (x 2 +y 2 )
(−2i)2k ∞ ∞ −u2 −v2 +2ixu+2iyv k k k e2 e u v w du dv π 2k k! −∞ −∞ k0 ∞ ∞ ∞ (−2uvw)k −u2 −v2 +2ixu+2iyv 1 1 (x 2 +y 2 )
e2 du dv e π k! k0 −∞ −∞ 1 1 (x 2 +y 2 ) ∞ ∞ −2uvw −u2 −v2 +2ixu+2iyv e2 e e du dv π −∞ −∞  ∞  1 1 (x 2 +y 2 ) ∞ −u2 +2ixu 2 e e−v +2iyv−2uvw dv du e2 π −∞ −∞  ∞  1 1 (x 2 +y 2 ) ∞ −u2 +2ixu 2 2 2 2 2 2iyv e2 e e−(v +2uvw+u w )+u w +e dv du π −∞ −∞  ∞  ∞ 1 1 (x 2 +y 2 ) 2 2 2 2 e−u +2ixu+u w e−v e2iy(v−uw) dv du e2 π −∞ −∞ 1 1 (x 2 +y 2 ) −y 2 ∞ −u2 +2ixu+u2 w2 −2iyuw e e du. (23.3) √ e2 π −∞

Therefore, for any x and y in R and any w in (−1, 1), we have, by (23.1) and (23.3), ∞
1 1 (x 2 −y 2 ) ∞ −(1−w2 )u2 +2i(x−yw)u hk (x)hk (y) k 2 e  w e du. (23.4) √ 2k k! π −∞ k0 1

Let (1 − w2 ) 2 u  t. Then, for any x and y in R and any w in (−1, 1), we have, by (23.1) and (23.4), ∞
x−yw hk (x)hk (y) k 1 1 (x 2 −y 2 ) ∞ −t 2 +2i (1−w 1 2 )1/2 t 2 e  e dt(1 − w 2 )− 2 w √ k k! 2 π −∞ k0 1

 e 2 (x

2

(x−yw) −y 2 ) − 1−w2

e

 (1 − w 2 )− 2 e 1

− 21

2

(1 − w 2 )− 2

1+w2 1−w2

1

(x 2 +y 2 )+

2w 1−w2

xy

.

(23.5)

23. Laguerre Formulas for Hermite Functions on C

109

Now, the last term in (23.5) is analytic on the cut plane C−{x ∈ R : x ≤ −1 or x ≥ 1}, and the first term in (23.5) is analytic on the open disk {w ∈ C : |w| < 1}. The two terms are equal on (−1, 1). Hence, by the principle of analytic continuation, the proof of the theorem is complete. ✷ Here is a formula expressing the Hermite functions ej,j , j  0, 1, 2, . . . , on C in terms of Laguerre polynomials. Theorem 23.3. For j  0, 1, 2, . . . and any z in C,   1 2 − 21 0 1 2 ej,j (z)  (2π ) Lj |z| e− 4 |z| . 2 Proof. For j  0, 1, 2, . . . , y, p in R, and all r on R with |r| < 1, we get, by Mehler’s formula in Theorem 23.1,   ∞ ∞ 
hk (y + p2 )hk y − p2 k p  p k
r ek y + ek y − r  √ 2 2 2k k! π k0 k0 2 2 2 1 1 − 1 1+r (2y 2 + p )+ 2r (y 2 − p ) 2 4 1−r 2  √ (1 − r 2 )− 2 e 2 1−r 2 π 2 2 2 2 1 1 − 1+r y 2 − 1+r p + 2r y 2 − 2r p 1−r 2 4 1−r 2 1−r 2 4  √ (1 − r 2 )− 2 e 1−r 2 π 2 2 2 1 1 − 1−2r+r y 2 − 1+2r+r p 4 1−r 2  √ (1 − r 2 )− 2 e 1−r 2 π 1 1 1−r 2 1+r p 2  √ (1 − r 2 )− 2 e− 1+r y − 1−r 4 . (23.6) π

Taking the inverse Fourier transform of the first and last terms in (23.6) with respect to y, we get, for all z  q + ip in C and all r in R with |r| < 1, ∞ ∞
1 1 1 1+r p 2 1−r 2 ek,k (z)r k  √ (1 − r 2 )− 2 e− 1−r 4 √ eiqy e− 1+r y dy π 2π −∞ k0 1 1 − 1 |z|2 r − 1 |z|2 √ e 2 1−r e 4 . (23.7) 2π 1 − r So, by Theorem 20.3 and (23.7),   ∞ 1 2 k − 1 |z|2 1
ek,k (z)r k  √ L0k |z| r e 4 , 2 2π k0 k0

∞


and hence ek,k (z)  (2π )−1/2 L0k for k  0, 1, 2, . . . .



 1 2 − 1 |z|2 |z| e 4 , 2

z ∈ C, ✷

Now, we can give another set of formulas expressing Hermite functions on C in terms of Laguerre polynomials.

23. Laguerre Formulas for Hermite Functions on C

110

Theorem 23.4. For j, k  0, 1, 2, . . . and any z in C, j! } 2 ( √i 2 )k (¯z)k Lkj ( 21 |z|2 )e− 4 |z| , ej +k,j (z)  (2π )−1/2 { (j +k)! 1

(i) (ii)

1

2

j! ej,j +k (z)  (2π)−1/2 { (j +k)! } 2 ( √i 2 )k zk Lkj ( 21 |z|2 )e− 4 |z| . 1

1

2

Remark 23.5. Let k  0. Then Theorem 23.4 becomes Theorem 23.3. Remark 23.6. We note that, for j, k  0, 1, 2, . . . , ej,j +k (z)  V (ej , ej +k )(z)  W (ej , ej +k )ˇ(z)  (W (ej +k , ej ))ˇ(z)  (W (ej +k , ej ))ˆ(z)  (W (ej +k , ej ))ˇ(−z)  (V (ej +k , ej ))(−z)  ej +k,j (−z),

z ∈ C.

(23.8)

Thus, if part (i) of Theorem 23.4 is true, then, by (23.8),   21     i k j! 1 2 − 1 |z|2 −1/2 k k ej,j +k (z)  (2π ) −√ (−z) Lj |z| e 4 (j + k)! 2 2     21   j! i k k k 1 2 − 1 |z|2 −1/2 |z| e 4 , z ∈ C.  (2π ) z Lj √ (j + k)! 2 2 Thus, to prove Theorem 23.4, we only need to prove part (i). The following lemma will be used in the proof of Theorem 23.4. Lemma 23.7. For α > −1 and k  1, 2, . . . , d α (L (x))  −Lα+1 k−1 (x), dx k Proof.

x > 0.

By (20.2), k L(k + α + 1) d
(−x)j d α (Lk (x))  dx dx j 0 L(k − j + 1)L(j + α + 1) j !

−

k
j 1

− 

k−1


(−x)j −1 L(k + α + 1) L(k − j + 1)L(j + α + 1) (j − 1)! L(k − 1 + α + 1 + 1) (−x)l L(k − 1 − l + 1)L(l + α + 1 + 1) l!

l0 −Lα+1 k−1 (x),

x > 0.

✷

Proof of Theorem 23.4. In view of Remark 23.6, it is enough to prove part (i). By Theorem 23.3, the formula is true if k  0. Suppose that the formula is true for all nonnegative integers j and all nonnegative integers k with k ≤ l, say. Then, by part (i) of Theorem 22.1, ej +k+1,j  −i(2j + 2)− 2 Zej +k+1,j +1 . 1

(23.9)

23. Laguerre Formulas for Hermite Functions on C

111

Now, by the induction hypothesis, we have, for all z ∈ C,     21   (j + 1)! i k k k 1 2 − 1 |z|2 (¯z) Lj +1 |z| e 4 . ej +k+1,j +1 (z)  (2π )−1/2 √ (j + k + 1)! 2 2 (23.10) Let fj be the function on C defined by   1 2 − 1 |z|2 k k fj (z)  (¯z) Lj +1 |z| e 4 , z ∈ C. (23.11) 2 Then, for k ≥ 1,

       ∂fj 1 2 1 1 2 1 1 2 2 − q e− 4 |z| (z)  (¯z)k (∂Lkj +1 ) |z| qe− 4 |z| + Lkj +1 |z| ∂q 2 2 2   1 1 2 + k(¯z)k−1 Lkj +1 (23.12) |z|2 e− 4 |z| , z ∈ C, 2

and        ∂fj i 1 2 1 2 1 1 2 2 i − p e− 4 |z| (z)  (¯z)k (∂Lkj +1 ) |z| ipe− 4 |z| + Lkj +1 |z| ∂p 2 2 2   1 2 − 1 |z|2 + k(¯z)k−1 Lkj +1 (23.13) |z| e 4 , z ∈ C. 2 So, by (23.11), (23.12), and (23.13), (Zfj )(z)  (¯z)k+1 (∂Lkj +1 )



 1 2 − 1 |z|2 |z| e 4 , 2

z ∈ C.

(23.14)

It is easy to see that (23.14) is also true for k  0. Thus, by (23.9)–(23.11) and (23.14), ej +k+1,j (z)



 21 (j + 1)!  (2π) (−i)(2j + 2) (j + k + 1)!    k i 1 2 − 1 |z|2 × √ (¯z)k+1 (∂Lkj +1 ) |z| e 4 2 2 −1/2

− 21

(23.15) for all z in C. But, by (23.15) and Lemma 23.7, ej +k+1,j (z)

   21   (j + 1)! i k k+1 k+1 1 2 − 1 |z|2 (¯z) Lj |z| e 4 √ (j + k + 1)! 2 2     21  k+1 j! i 1 2 − 1 |z|2  (2π )−1/2 (¯z)k+1 Lk+1 |z| e 4 , z ∈ C, √ j (j + k + 1)! 2 2 and the proof is complete. ✷  (2π)−1/2 i(2j + 2)− 2

1



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24 Weyl Transforms on L2(R) with Radial Symbols

For Weyl transforms on L2 (R) with radial symbols, we can give a sufficient and necessary condition for boundedness. A sufficient and necessary condition for compactness can also be given. In order to obtain these conditions, we need the Wigner transforms of Hermite functions on R. For j, k  0, 1, 2, . . . , we define the function ψj,k on R2 by ψj,k (x, ξ )  W (ej , ek )(x, ξ ),

x, ξ ∈ R.

Theorem 24.1. For j, k  0, 1, 2, . . . , we get, for any ζ  x + iξ ,   21 √ 1 2 j! (i) ψj +k,j (ζ )  2(−1)j (2π )− 2 (j +k)! ( 2)k (ζ¯ )k Lkj (2|ζ |2 )e−|ζ | ,   21 √ 1 2 j! ( 2)k ζ k Lkj (2|ζ |2 )e−|ζ | . (ii) ψj,j +k (ζ )  2(−1)j (2π)− 2 (j +k)! Proof. It is easy to see from Proposition 18.1 that ek is even or odd if k is, respectively, even or odd. Thus, for k  0, 1, 2, . . . , e˜k (x)  ek (−x)  (−1)k ek (x), So, by (24.1), we get V (ej , e˜k )(q, p)  (2π)−1/2



x ∈ R.

(24.1)

 p  p e˜k y − dy eiqy ej y + 2 2 −∞ ∞

 (−1)k V (ej , ek )(q, p),

q, p ∈ R.

(24.2)

Thus, by Lemma 14.4 and (24.2), W (ej , ek )(x, ξ )  (−1)k 2V (ej , ek )(−2ξ, 2x)  (−1)k 2ej,k (−2ξ, 2x), x, ξ ∈ R.

(24.3)

114

24. Weyl Transforms on L2 (R) with Radial Symbols

For all ζ  x + iξ in C, we get, by (24.3), ψj,k (ζ )  (−1)k 2ej,k (2iζ ).

(24.4)

Hence, by Theorem 23.4 and (24.4), ψj +k,j (ζ )  2(−1)j ej +k,j (2iζ )   21   j! i k k 2 2 (−i)k (ζ¯ )k Lkj (2|ζ |2 )e−|ζ |  2(−1)j (2π)−1/2 √ (j + k)! 2   21 √ j! 2  2(−1)j (2π)−1/2 ( 2)k (ζ¯ )k Lkj (2|ζ |2 )e−|ζ | , ζ ∈ C, (j + k)! and ψj,j +k (ζ )  2(−1)j +k ej,j +k (2iζ )   21   j! i k k k k k 2 j +k −1/2 2 i ζ Lj (2|ζ |2 )e−|ζ |  2(−1) (2π) √ (j + k)! 2   21 √ j! 2 ✷  2(−1)j (2π)−1/2 ( 2)k ζ k Lkj (2|ζ |2 )e−|ζ | , ζ ∈ C. (j + k)! To see the role of the function ψj,k , j, k  0, 1, 2, . . . , in the study of the Weyl transform, we use the following result in the paper [24] by Simon. We skip the proof. Theorem 24.2. Let f be any function in S(R). Then f 

∞


f, ek ek ,

k0

where the convergence is in S(R), i.e., for all nonnegative integers α and β,   N  
 α β  α β f, ek x (∂ ek )(x) → 0 sup x (∂ f )(x) −   x∈R k0 as N → ∞. Let σ be a tempered function on R2 . Suppose that σ is radial, i.e., σ (x, ξ )  σ (r), x, ξ ∈ R,  where r  x 2 + ξ 2 . Now, by Theorem 24.1, for j, k  0, 1, 2, . . . and j ≥ k,   21 √ k! 1 2 j −k ψj,k (x, ξ )  2(−1)k (2π)− 2 ( 2)j −k (ζ¯ )j −k Lk (2|ζ |2 )e−|ζ | (24.5) j! for all ζ  x + iξ in C. Now, for all f and g in S(R), we can use (12.1), Theorem 24.2, the bilinearity of W , the proof of Theorem 12.1, and (24.5) to get (Wσ f )(g) ¯  (2π)−1/2 σ (W (f, g))

24. Weyl Transforms on L2 (R) with Radial Symbols





 (2π)−1/2 σ W f,  (2π)

−1/2

σ

 ∞


∞


115

 g, ek ek

k0



g, ¯ ek W (f, ek )

k0

 (2π)−1/2  (2π )−1/2

∞


g, ¯ ek σ (W (f, ek ))

k0 ∞ ∞



g, ¯ ek f, ej σ (ψj,k ).

(24.6)

k0 j 0

Remark 24.3. We have shown only that (24.6) is valid in the sense that we sum with respect to j first and then with respect to k. Now, for j, k  0, 1, 2, . . . and j ≥ k, we get, by (24.5), ∞ ∞ σ (ψj,k )  σ (x, ξ )ψj,k (x, ξ )dx dξ −∞



2π

0

−∞



∞

k

σ (ρ)2(−1) (2π)

− 21



0

k! j!

 21

√ j −k ( 2)j −k ρ j −k e−i(j −k)θ Lk (2ρ 2 )

× e−ρ ρdρdθ   21 2π ∞ k! 1 1 −i(j −k)θ  e dθ σ (ρ) 2 2 (j −k)+1 (−1)k (2π )− 2 ρ j −k+1 j! 0 0 2

j −k

× Lk

(2ρ 2 )e−ρ dρ. 2

(24.7)

Thus, by (24.7), σ (ψj,k )  0,

j  k.

(24.8)

So, by (24.6), (24.7), and (24.8), we get (Wσ f )(g) ¯  (2π)−1/2

∞


f, ek g, ¯ ek σ (ψk,k ),

(24.9)

k0

where σ (ψk,k )  (2π ) 2 (−1)k 2 1

0

∞

σ (ρ)L0k (2ρ 2 )e−ρ ρdρ, 2

k  0, 1, 2, . . . , (24.10)

for all f and g in S(R). Remark 24.4. The convergence in (24.9) is valid in the sense that the sequence of partial sums of the series is convergent. Theorem 24.5. Let σ be a tempered function on R2 . Suppose that σ is radial, i.e., σ (x, ξ )  σ (ρ),

x, ξ ∈ R,

116

24. Weyl Transforms on L2 (R) with Radial Symbols

where ρ 

 x 2 + ξ 2 . For k  0, 1, 2, . . . , let ∞ 2 ak  σ (ρ)L0k (2ρ 2 )e−ρ ρdρ. 0

Then Wσ is a bounded linear operator from L2 (R) into L2 (R) if and only if the sequence {ak }∞ k0 is bounded, (ii) Wσ is a compact operator from L2 (R) into L2 (R) if and only if ak → 0 as k → ∞.

(i)

Proof.

Suppose that there is a positive constant M such that |ak | ≤ M, k  0, 1, 2, . . . .

(24.11)

Then, for all f and g in S(R), we get, by (24.9) and (24.10), (Wσ f )(g)  (2π)−1/2

∞


(2π) 2 (−1)k 2ak f, ek g, ek , 1

(24.12)

k0

and hence, by the Schwarz inequality and (24.11),  21   21  ∞ ∞

|(Wσ f )(g)| ≤ 2M |f, ek |2 |g, ek |2 k0

k0

 2M f L2 (R) g L2 (R) . Thus, Wσ f L2 (R) ≤ 2M f L2 (R) ,

f ∈ S(R).

Therefore, by a limiting argument, Wσ is a bounded linear operator from L2 (R) into L2 (R). Conversely, suppose that Wσ is a bounded linear operator from L2 (R) into L2 (R). Then, by (24.12), (Wσ ej )(ej )  (−1)j 2aj ,

j  0, 1, 2, . . . .

(24.13)

Therefore, by (24.13) and the Schwarz inequality, |aj |  |(Wσ ej )(ej )|  |Wσ ej , ej | ≤ Wσ ej L2 (R) ej L2 (R) ≤ Wσ ∗ ,

j  0, 1, 2, . . . ,

and part (i) is proved. To prove part (ii), suppose that ak → 0 as k → ∞. For N  0, 1, 2, . . . , we define the bounded linear operator WN : L2 (R) → L2 (R) by WN f, g 

N


(−1)k 2ak f, ek g, ¯ ek 

k0

for all f and g in L2 (R). Then WN f 

N
k0

(−1)k 2ak f, ek ek ,

f ∈ L2 (R),

(24.14)

24. Weyl Transforms on L2 (R) with Radial Symbols

117

for N  0, 1, 2, . . . . Therefore, WN : L2 (R) → L2 (R) is a finite rank operator, and hence compact for N  0, 1, 2, . . . . Now, for all f and g in S(R), by (24.12) and (24.14), |(Wσ − WN )f, g| ≤ 2( sup |ak |) f L2 (R) g L2 (R) k≥N +1

(24.15)

for N  0, 1, 2, . . . . Then, by (24.15), Wσ − WN ∗ ≤ 2( sup |ak |) → 0 k≥N +1

as N → ∞. Therefore, Wσ is the limit in B(L2 (R)) of a sequence of compact operators from L2 (R) into L2 (R). So, Wσ is a compact operator from L2 (R) into L2 (R). Conversely, suppose that Wσ is a compact operator from L2 (R) into L2 (R). Since ej → 0 weakly in L2 (R) as j → ∞, it follows that Wσ ej → 0 in L2 (R) as j → ∞. But by (24.13) and the Schwarz inequality, 2|aj | ≤ Wσ ej L2 (R) ej L2 (R)  Wσ ej L2 (R) → 0 as j → ∞.

✷

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25 Another Fourier Transform

We compute in this chapter the Fourier transform of a function related to the Laguerre polynomial of degree k and order 0, k  0, 1, 2, . . . . This Fourier transform will be used in the next chapter to obtain a criterion for the compactness of the Weyl transform on L2 (R). We begin with some complex analysis. Let f be a continuous function on [0, ∞) such that we can find a positive constant A and a constant c for which |f (t)| ≤ Aect ,

t ≥ 0.

Then we define the function F on the region {z ∈ C : Rez > c} by ∞ F (z)  e−zt f (t)dt, Rez > c.

(25.1)

(25.2)

0

The function F is in fact the Laplace transform of f . Theorem 25.1. F is analytic on the region {z ∈ C : Rez > c}. To prove Theorem 25.1, we need some preparations. Lemma 25.2. Let ϕ be an entire function. Let g be a continuous function on a closed and bounded interval [a, b] and let G be the function on C defined by b G(z)  g(t)ϕ(zt)dt, z ∈ C. (25.3) a

Then G is continuous on C. Proof. Let z0 ∈ C. Let {zk }∞ k1 be a sequence of complex numbers such that zk → z0 as k → ∞. Let D be a fixed disk centered at z0 such that z0 t ∈ D for all t

120

25. Another Fourier Transform

in [a, b]. Then ϕ is uniformly continuous on D. So, for any given positive number ε, there exists a positive number δ such that |ϕ(z) − ϕ(w)| < ε for all z and w in D with |z − w| < δ. Thus, there exists a positive integer K such that k ≥ K ⇒ |ϕ(zk t) − ϕ(z0 t)| < ε for all t in [a, b]. So, g(t)ϕ(zk t) → g(t)ϕ(z0 t) uniformly with respect to t on [a, b] as k → ∞. Therefore, G(zk ) → G(z0 ) as k → ∞. ✷ Lemma 25.3. The function G on C defined by (25.3) is entire. Proof. Let C be a simple closed curve in C. Then   b G(z)dz  g(t)ϕ(zt)dt dz a C C    b b  g(t)ϕ(zt)dz dt  g(t) ϕ(zt)dz dt  a

 0.

a

C

So, by Lemma 25.2 and Morera’s theorem, G is entire.

C

✷

Proof of Theorem 25.1. We see that by (25.1) and (25.2), F (z) exists for all z in the region {z ∈ C : Rez > c}. Indeed, ∞ ∞ −zt |e f (t)|dt  e−(Rez)t |f (t)|dt 0 0 ∞ A e−(Rez−c)t dt  < ∞. ≤A Rez −c 0 Next, for j  1, 2, . . . , we define the function Fj on C by j Fj (z)  e−zt f (t)dt, z ∈ C. (25.4) 0

By Lemma 25.3, Fj is entire for j  1, 2, . . . . Now, let c1 > c. Then, for Rez > c1 , we get, by (25.1) and (25.4), j j e−(Rez−c)t dt ≤ A e−(c1 −c)t dt |Fj (z) − Fl (z)| ≤ l l  −(c1 −c)l  e − e−(c1 −c)j A →0 c1 − c as j, l → ∞. Thus, Fj → F uniformly on the region {z ∈ C : Rez > c1 } as j → ∞, and hence F is analytic on {z ∈ C : Rez > c1 }. But c1 is an arbitrary number larger than c. So, F is analytic on the region {z ∈ C : Rez > c}. ✷

25. Another Fourier Transform

121

Corollary 25.4. For k  0, 1, 2, . . . , the function Fk defined on the region {z ∈ C : Rez > 0} by ∞ e−zt t k dt, Rez > 0, Fk (z)  0

is analytic. Proof. Let ε be any positive number. Then we can find a positive constant Aε such that |t k | ≤ Aε eεt ,

t ≥ 0.

So, by Theorem 25.1, Fk is analytic on the region {z ∈ C : Rez > ε}. Since ε is an arbitrary positive number, it follows that Fk is analytic on {z ∈ C : Rez > 0}. ✷ For k  0, 1, 2, . . . , we define the function lk on R by  x −2 0 lk (x)  e Lk (x), x > 0, 0, x ≤ 0.

(25.5)

The aim of this chapter is to compute the Fourier transform of the function lk , k  0, 1, 2, . . . . Theorem 25.5. For k  0, 1, 2, . . . ,   k 1 ˆlk (ξ )  2 2iξ − 1 , π 2iξ + 1 2iξ + 1 Proof.

ξ ∈ R.

By (25.5) and the definition of Lk (x), x > 0, ∞ 1 e−ixξ lk (x)dx lˆk (ξ )  (2π )− 2 0

  ex d k −x k e e (e x )dx  (2π) k! dx 0 ∞  k d (−1)k 1 x (2π)− 2 (e 2 (1−2iξ ) )e−x x k dx  k! dx 0  k ∞ k (−1) x − 21 (1−2iξ ) 1 − 2iξ 2 (2π) e e−x x k dx  k! 2 0   ∞ (−1)k 1 − 2iξ k 1 x (2π)− 2 e− 2 (1+2iξ ) x k dx  k! 2 0 − 21



∞

−ixξ − x2

(25.6)

for all ξ in R. Now, for z > 0,  k ∞ ∞ s 1 ds k! −zt k −s  k+1 L(k + 1)  k+1 . e t dt  e (25.7) z z z z 0 0

∞ By Corollary 25.4, 0 e−zt t k dt is analytic on {z ∈ C : Rez > 0}. So, by (25.7) and the principle of analytic continuation, ∞ k! e−zt t k dt  k+1 , Rez > 0. (25.8) z 0

122

25. Another Fourier Transform

So, by (25.8), we get ∞

e−x(

1+2iξ 2

0

) k

x dx 

k! ( 1+2iξ )k+1 2

,

ξ ∈ R.

(25.9)

Therefore, by (25.6) and (25.9),

  2k+1 1 − 2iξ k 1 lˆk (ξ )  (2π)− 2 (−1)k 2 (1 + 2iξ )k+1   k 2 2iξ − 1 1  , ξ ∈ R. π 2iξ + 1 2iξ + 1

✷

26 A Class of Compact Weyl Transforms on L2(R)

The space Lr∗ (R), 1 ≤ r < ∞, defined by (14.1) will be used again in this chapter to give a criterion for the compactness of the Weyl transform on L2 (R). Let g ∈ Lr∗ (R), 1 ≤ r < ∞, and let σ be the function on R2 defined by σ (x, ξ )  g(ρ ˆ 2 ), x, ξ ∈ R,   where ρ  x 2 + ξ 2 . Then σ ∈ Lr (R2 ). Indeed, 2π ∞ ∞ ∞  r |σ (x, ξ )| dx dξ  |g(ρ ˆ 2 )|r ρdρdθ −∞ −∞ 0 0 ∞  r |g(ρ)| ˆ dρ ≤ π g ˆ rLr  (R2 ) < ∞. π 0

Theorem 26.1. Let g ∈ Lr∗ (R), 1 ≤ r < ∞, and let σ be the function on R2 defined by σ (x, ξ )  g(ρ ˆ 2 ), x, ξ ∈ R,  where ρ  x 2 + ξ 2 . Then Wσ : L2 (R) → L2 (R) is a compact operator. Proof. By part (ii) of Theorem 24.5, we only need to prove that the sequence {ak }∞ k1 of complex numbers defined by ∞ 2 ak  σ (ρ)L0k (2ρ 2 )e−ρ ρdρ, k  0, 1, 2, . . . , 0

where σ (ρ)  g(ρ ˆ ), is such that limk→∞ ak  0. To do this, let lk and Ik be functions on R defined, respectively, by (25.5) and 2

Ik (x)  lk (2x),

x ∈ R,

(26.1)

26. A Class of Compact Weyl Transforms on L2 (R)

124

for k  0, 1, 2, . . . . Then, by Theorem 25.5 and (26.1),     iξ − 1 k 1 1ˆ ξ 1 ˆ √ , Ik (ξ )  lk 2 2 2π iξ + 1 iξ + 1

ξ ∈ R,

for k  0, 1, 2, . . . . Now, for 1 < s < ∞, we get, by (26.2),  ∞  1s 1 1 dξ Iˆk Ls (R)  √ s 2π −∞ |iξ + 1|

(26.2)

(26.3)

and 1 Iˆk L∞ (R) ≤ √ 2π

(26.4)

for k  0, 1, 2, . . . . Thus, for 1 < s ≤ ∞, by (26.3) and (26.4) we can get a positive constant Cs such that Iˆk Ls (R) ≤ Cs ,

k  0, 1, 2, . . . .

Note that for k  0, 1, 2, . . . , ∞ ∞ 2 2 σ (ρ)L0k (2ρ 2 )e−ρ ρdρ  g(ρ ˆ 2 )L0k (2ρ 2 )e−ρ ρdρ ak  0 0 1 ∞ 1 ∞ 0 −t  g(t)L ˆ (2t)e dt  g(t)I ˆ k (t)dt. k 2 0 2 −∞ Now, if



∞ −∞

g(t)I ˆ k (t)dt 

∞

−∞

g(t)Iˆk (t)dt, k  0, 1, 2, . . . ,

then, by (26.2), (26.6), and (26.7),   ∞ 1 iξ − 1 k 1 ak  √ g(ξ ) dξ, k  0, 1, 2, . . . . iξ + 1 iξ + 1 2 2π −∞ Let ξ  tan θ , − π2 < θ < 1 ak  √ 2 2π



π 2

− π2

(26.5)

(26.6)

(26.7)

(26.8)

π . 2

Then, by (26.8), we get, for k  0, 1, 2, . . . ,   1 i tan θ − 1 k g(tan θ) (26.9) sec2 θdθ. i tan θ + 1 i tan θ + 1

But for k  0, 1, 2, . . . and − π2 < θ < π2 ,   (−1)k (cos θ − i sin θ )k i tan θ − 1 k   (−1)k e−2ikθ . i tan θ + 1 (cos θ + i sin θ )k Thus, for k  0, 1, 2, . . . , we have, by (26.9) and (26.10), (−1)k π ak  √ f (θ)e−2ikθ dθ, 2 2π −π

(26.10)

(26.11)

26. A Class of Compact Weyl Transforms on L2 (R)

where f (θ ) 

  0, 

g(tan θ) sec θ , i tan θ+1

0,

2

125

−π < θ < − π2 ,

− π2 < θ < π2 , π < θ < π. 2

Note that f ∈ L1 [−π, π]

(26.12)

because by H¨older’s inequality there exists a positive constant Cr such that   ∞ π   g(ξ )  2  g(tan θ) sec2 θ     dθ   dξ  i tan θ + 1    −∞ iξ + 1 − π2  ∞  1 r 1 dξ < ∞. ≤ g Lr (R)  r −∞ |iξ + 1| Thus, by (26.11), (26.12), and the Riemann–Lebesgue lemma, ak → 0 as k → ∞. Thus, the proof is complete, provided that we can prove (26.7). To do this, we first note that for 1 ≤ r ≤ 2, (26.7) follows from Lemma 14.6. For 2 < r < ∞, by ˆ ˆ (26.5) we can find a sequence {ψj }∞ j 1 of functions in S(R) such that ψj → Ik in ∞ r L (R) as j → ∞. Now, let {ϕl }l1 be a sequence of functions in S(R) such that ϕl → g in Lr (R) as l → ∞. Then ϕl → g in S  (R) as l → ∞, so that ϕˆl → gˆ in S  (R) as l → ∞. Therefore, ∞ ∞ g(t)Iˆk (t)dt  lim ϕl (t)Iˆk (t)dt l→∞ −∞ −∞ ∞  lim lim ϕl (t)ψˆ j (t)dt l→∞ j →∞ −∞ ∞  lim lim ϕˆl (t)ψj (t)dt. (26.13) l→∞ j →∞ −∞

Now, if we can prove that ∞ lim lim ϕˆl (t)ψj (t)dt  lim lim l→∞ j →∞ −∞

∞

j →∞ l→∞ −∞

then, by (26.13) and (26.14), ∞ g(t)Iˆk (t)dt  lim −∞

∞

j →∞ −∞

ϕˆl (t)ψj (t)dt,

g(t)ψ ˆ j (t)dt.

(26.14)

(26.15)

But by the Fourier inversion formula and the Hausdorff–Young inequality, ψj → Ik in Lr (R) as j → ∞. So, by (26.15), ∞ ∞ g(t)Iˆk (t)dt  g(t)I ˆ k (t)dt. −∞

−∞

So, it remains to prove (26.14), or equivalently, ∞ ˆ lim lim ϕl (t)ψj (t)dt  lim lim l→∞ j →∞ −∞

∞

j →∞ l→∞ −∞

ϕl (t)ψˆ j (t)dt.

(26.16)

126

26. A Class of Compact Weyl Transforms on L2 (R)

To prove (26.16), let ε be any positive number. Then we pick N1 to be any positive integer such that ε j ≥ N1 ⇒ ψˆ j − Iˆk Lr  (R) < , (26.17) 2M1 where M1  sup{ ϕl Lr (R) : l  0, 1, 2, . . .}.

(26.18)

Let N2 be another positive integer such that ε . ˆ 2 Ik Lr  (R)

l ≥ N2 ⇒ ϕl − g Lr (R) <

(26.19)

Thus, for j, l ≥ max(N1 , N2 ), we get, by (26.17), (26.18), and (26.19),  ∞  ∞    ˆ ˆ ϕl (t)ψj (t)dt − g(t)Ik (t)dt   −∞ −∞  ∞   ∞         ˆ ˆ ˆ ≤ ϕl (t)(ψj (t) − Ik (t))dt  +  (ϕl (t) − g(t))Ik (t)dt  −∞

−∞

≤ ϕl Lr (R) ψˆ j − Iˆk Lr  (R) + ϕl − g Lr (R) Iˆk Lr  (R) ε ε ε ε < M1 + Iˆk Lr  (R) < +  ε. ˆ 2M1 2 2  2 Ik Lr (R) Therefore,



∞

−∞

ϕl (t)ψˆ j (t)dt →



∞

−∞

g(t)Iˆk (t)dt

as l, j → ∞, and so the left-hand side and the right-hand side of (26.16) exist. So, we can conclude that (26.16) holds, and this completes the proof of the theorem. ✷

27 A Class of Bounded Weyl Transforms on L2(R)

In order to give another sufficient condition for the Weyl transform to be a bounded linear operator from L2 (R) into L2 (R), we begin with two basic facts in Fourier series. Lemma 27.1. Let f be a C 1 function on [−π, π] such that f (−π )  f (π ). For k  0, ±1, ±2, . . . , let ck and ck be numbers defined by π 1 ck  f (θ)e−ikθ dθ 2π −π and ck 

1 2π



π

−π

f  (θ)e−ikθ dθ.

Then ck  ikck ,

0, ±1, ±2, . . . .

Proof.

For k  0, ±1, ±2, . . . , π 1 f  (θ)e−ikθ dθ ck  2π −π ik π 1 f (θ)e−ikθ dθ  ikck . f (θ)e−ikθ |π−π +  2π 2π −π Lemma 27.2. Let f and ck be as in Lemma 27.1. Then ∞
k−∞

|ck | < ∞.

✷

128

27. A Class of Bounded Weyl Transforms on L2 (R)

Proof. By Lemma 27.1, the Schwarz inequality, and the Parseval identity in the L2 theory of Fourier series, ∞
k−∞

|ck |  |c0 | +




|ck |  |c0 | +

k0

k0

|ck |

1 k

1
1 2  |c0 | + k2 k0 k0  1
1 2  |c0 | + f  L2 [−π,π] < ∞. k2 k0 




 21 




|ck |2

✷

We can now give a class of bounded Weyl transforms on L2 (R). Theorem 27.3. Let σ be a function on R2 defined by σ (x, ξ )  R(r)P(θ ),

x, ξ ∈ R2 − {(0, 0)},

where P is a C 1 function on [−π, π] such that P(−π )  P(π ), and R is a tempered function on (0, ∞), i.e., ∞ |R(ρ)| dρ < ∞ (1 + ρ)N 0 for some nonnegative integer N . For j, k  0, 1, 2, . . . , let aj k be the number defined by 

∞ 1 1 √ 2 j −k  2(−1)k (2π)− 2 ( jk!! ) 2 ( 2)j −k 0 R(ρ)ρ j −k+1 Lk (2ρ 2 )e−ρ dρ,   j ≥ k, √ (27.1)

aj k  − 21 j ! 21 j k−j ∞ k−j +1 k−j 2 −ρ 2  2(−1) (2π) ( ) ( 2) R(ρ)ρ L (2ρ )e dρ,  j 0 k!  j < k. (27.2) Suppose that there exists a positive constant C such that |aj k | ≤ C,

j, k  0, 1, 2, . . . .

Then Wσ : L2 (R) → L2 (R) is a bounded linear operator. Proof.

By Theorem 24.1,  1 1 √ 2 j −k  2(−1)k (2π)− 2 ( jk!! ) 2 ( 2)j −k (ζ¯ )j −k Lk (2|ζ |2 )e−|ζ | ,   j ≥ k, √ ψj,k (ζ )  1 1 2 k−j  2(−1)j (2π)− 2 ( jk!! ) 2 ( 2)k−j ζ k−j Lj (2|ζ |2 )e−|ζ | ,   k > j. So, by (27.1)–(27.4),



σ (ψj,k )  aj k

π −π

(27.3) (27.4)

P(θ )e−i(j −k)θ dθ

ˆ − k),  aj k (2π )P(j

j, k  0, ±1, ±2, . . . ,

(27.5)

27. A Class of Bounded Weyl Transforms on L2 (R)

where ˆ P(l) 



1 2π

π

−π

P(θ )e−ilθ dθ, l  0, ±1, ±2, . . . .

129

(27.6)

Let f and g be in S(R). Then, by (24.6) and (27.5), there exists a positive constant C  such that   ∞ ∞

  −1/2  |(Wσ f )(g)|  (2π) g, ek f, ej σ (ψj,k )   k0 j 0  ≤ C

∞ ∞



ˆ − k) f, ej  g, ek | |P(j

k0 j 0

 C

∞


|g, ek |

k0

 ≤ C

∞


ˆ − j ) f, ej | |5(k

j 0

∞


 21   2  21 ∞ ∞

ˆ − j ) f, ej |  |g, ek |2  |5(k

k0

k0

  C  g L2 (R) 

 ∞ ∞

k0

j 0

2  21 ˆ − j ) f, ej |  , |5(k

(27.7)

−π ≤ θ ≤ π.

(27.8)

j 0

where ˜ 5(θ )  P(θ), By Lemma 27.2, (27.6), and (27.8), ∞


ˆ |5(k)| < ∞.

(27.9)

k−∞

So, by (27.7), (27.8), (27.9), and Young’s inequality,   21  ∞ ∞

 2 ˆ |5(k)| |f, ek | |(Wσ f )(g)| ≤ C g L2 (R)   C

k−∞ ∞




k0

ˆ |5(k)| f L2 (R) g L2 (R)

(27.10)

k−∞

for all f and g in S(R). Therefore, by (27.10),   ∞
 ˆ |5(k)| f L2 (R) , Wσ f L2 (R) ≤ C

f ∈ S(R),

k−∞

and the proof is complete.

✷

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28 A Weyl Transform with Symbol in S (R2)

We can compute in this chapter the eigenvalues and eigenfunctions of a specific Weyl transform with symbol in S  (R2 ) as a compact and self-adjoint operator on L2 (R). Let ρ be a positive real number. Then we define the linear functional δρ : S(R2 ) → C by π δρ (ϕ)  ϕ(ρeiθ )ρdθ, ϕ ∈ S(R2 ). −π

Then it is easy to check that δρ is a tempered distribution on R2 . For k  0, 1, 2, . . . , we define the number λk by λk  2(−1)k ρL0k (2ρ 2 )e−ρ . 2

(28.1)

Theorem 28.1. Wδρ : L2 (R) → L2 (R) is a compact and self-adjoint operator. Furthermore, the nonzero eigenvalues of Wδρ : L2 (R) → L2 (R) are precisely equal to the numbers defined by (28.1), and ∞


|λk |r < ∞,

r > 4.

k0

Proof.

Let f and g be in S(R). Then, by (24.6), (Wδρ f )(g)  (2π )−1/2

∞ ∞

k0 j 0

g, ek f, ej δρ (ψj,k ).

(28.2)

132

28. A Weyl Transform with Symbol in S  (R2 )

For j, k  0, 1, 2, . . . , by (27.3) and (27.4), we get δρ (ψj,k )    1 √ j −k π j −k+1 −i(j −k)θ j −k 2 − 1 k! 2 k   e Lk (2ρ 2 )e−ρ dθ,  2(−1) (2π) 2 j ! ( 2) −π ρ  j ≥ k, 

π 1 √ 1 2 k−j j   2(−1) (2π)− 2 jk!! 2 ( 2)k−j −π ρ k−j +1 e−i(j −k)θ Lj (2ρ 2 )e−ρ dθ,   k ≥ j. (28.3) Thus, by (28.3), δρ (ψj,k ) 



0, j  k, 1 2 2(−1)k (2π) 2 ρL0k (2ρ 2 )e−ρ , j  k.

(28.4) (28.5)

So, for all f and g in S(R), by (28.1), (28.2), (28.4), and (28.5), (Wδρ f )(g) 

∞


λk f, ek g, ek .

(28.6)

k0

We now use an asymptotic formula for the Laguerre polynomials in the book [28] by Szeg˝o, which states that for arbitrary but fixed positive numbers ε and ω we get  π 1 x 1 1 1 3 (28.7) + O(k − 4 ), x > 0, L0k (x)  π − 2 e 2 x − 4 k − 4 cos 2(kx) 2 − 4 as k → ∞, where the O-term is uniform with respect to x on [ε, ω]. Thus, by (28.1) and (28.7), there exists a positive constant C such that |λk | ≤ Ck − 4

1

(28.8)

for k large enough, say, for k ≥ k0 . Therefore, for r > 4, by (28.8), we get ∞


|λk |r ≤ C r

kk0

∞


r

k − 4 < ∞.

(28.9)

kk0

Now, for every positive integer N , the linear operator WN : L2 (R) → L2 (R) defined by WN f 

N


λk f, ek ek , f ∈ L2 (R),

(28.10)

k0

is of finite rank and hence compact. Moreover, for all f and g in S(R), by (28.6) and (28.10),   ∞ 
   λk f, ek g, ek  |(Wδρ f )(g) − (WN f )(g)| ≤  kN +1  ≤ ( sup |λk |) f L2 (R) g L2 (R) . k≥N +1

Thus, by (28.9) and (28.11), Wδρ − WN ∗ ≤ sup |λk | → 0 k≥N +1

(28.11)

28. A Weyl Transform with Symbol in S  (R2 )

133

as N → ∞. Therefore, Wδρ : L2 (R) → L2 (R) is compact. To see that Wδρ : L2 (R) → L2 (R) is self-adjoint, we note that for all f and g in S(R), by (28.6), Wδρ f, g  (Wδρ f )(g) ¯ 

∞


λk f, ek g, ¯ ek 

k0

and f, Wδρ g  Wδρ g, f   (Wδρ g)(f¯) 

∞


λk f, ek g, ¯ ek .

k0

For k  0, 1, 2, . . . , λk is an eigenvalue of Wδρ : L2 (R) → L2 (R) because for any g in S(R), by (28.6), we get (Wδρ ej )(g) 

∞


λk ej , ek g, ek   (λj ej )(g),

j  0, 1, 2, . . . .

k0

Therefore, λj is an eigenvalue of Wδρ : L2 (R) → L2 (R) for j  0, 1, 2, . . . . Finally, let λ be an eigenvalue of Wδρ : L2 (R) → L2 (R) and let f ∈ L2 (R) be a corresponding eigenfunction. Then   ∞ ∞

Wδρ f, ej ej  λ f, ej ej j 0

⇒

∞


j 0

f, ej Wδρ ej 

j 0

⇒

∞
j 0

∞


f, ej λej

j 0

f, ej λj ej 

∞


f, ej λej

j 0

⇒ f, ej λj  f, ej λ, j  0, 1, 2, . . . .

(28.12)

Since {ej : j  0, 1, 2, . . .} is an orthonormal basis for L (R) and f  0, it follows that f, ek   0 for some k  0, 1, 2, . . . . Thus, by (28.12), λ  λk . So, every eigenvalue of Wδρ : L2 (R) → L2 (R) is equal to λk for some k  0, 1, 2, . . . . ✷ 2

To put Theorem 28.1 in a proper perspective of quantum mechanics, it is imperative to note that due to quantization described at the end of Chapter 4, the numerical values of measurements of an observable in classical mechanics with phase space R2n are replaced by the spectrum of the self-adjoint Weyl transform representing the observable. This explains why the computation of the spectrum of a self-adjoint Weyl transform on L2 (Rn ) is an important chapter in applied mathematics.

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29 The Symplectic Group

It is shown in this chapter that the Weyl transform is invariant with respect to the symplectic group, i.e., the group of all symplectic linear transformations from Cn into Cn . A linear transformation a : Cn → Cn satisfying [a(z), a(z )]  [z, z ],

z, z ∈ Cn ,

(29.1)

where [ , ] is the symplectic form on Cn defined by (8.1), is said to be symplectic. We provide a sufficiently self-contained treatment of symplectic linear transformations from Cn into Cn and the symplectic group in this chapter. Related matters can be found in the books [9], [13], and [34] by Greub, Halmos, and Weyl, respectively. Remark 29.1. Let a : Cn → Cn be a linear transformation. We call the matrix of a with respect to the standard basis for Cn ( R2n ) the standard matrix of a. We n shall q identify a with its standard matrix. A point (q, p) in C will also be denoted by p . a linear transformation such that the Proposition 29.2. Let a : Cn → Cn be  A B , where A, B, C, and D are n × n standard matrix of a is equal to C D matrices with real entries. Then a is symplectic if and only if At D − C t B  I, At C  C t A,

(29.2) (29.3)

B t D  D t B,

(29.4)

and where ( )t is the transpose of the matrix ( ) and I is the n × n identity matrix.

136

29. The Symplectic Group

Proof. For j  1, 2, . . . , n, we let εj be the point in Rn such that all coordinates except the j th coordinate are equal to zero, and the j th coordinate is equal to 1. Then, for j, k  1, 2, . . . , n,      εj Aεj A B , (29.5)  a(εj , 0)  C D 0 Cεj  a(0, εk ) 

A C

B D

    0 Bεk  , εk Dεk

(29.6)

and hence, by (8.1), (29.5), and (29.6), [a(εj , 0), a(0, εk )]  2(εkt B t Cεj − εkt D t Aεj ).

(29.7)

Since by (8.1), [(εj , 0), (0, εk )]  −2εkt εj ,

j, k  1, 2, . . . , n,

(29.8)

it follows from (29.1), (29.7), and (29.8) that D t A−B t C  I , and (29.2) is proved. Next, for j, k  1, 2, . . . , n, we get, by (8.1) and (29.5), [a(εj , 0), a(εk , 0)]  2(εkt At Cεj − εkt C t Aεj )

(29.9)

[(εj , 0), (εk , 0)]  0.

(29.10)

and So, by (29.1), (29.9), and (29.10), (29.3) is proved. Finally, for j, k  1, 2, . . . , n, we get, by (8.1) and (29.6), [a(0, εj ), a(0, εk )]  2(εkt B t Dεj − εkt D t Bεj )

(29.11)

[(0, εj ), (0, εk )]  0.

(29.12)

and So, by (29.1), (29.11), and (29.12), (29.4) is proved. Conversely, for z  x + iξ and z  x  + iξ  in Cn , we get      x Ax + Bξ A B  (29.13) a(z)  C D ξ Cx + Dξ and a(z ) 



A C

B D

      x Ax + Bξ   . ξ Cx  + Dξ 

(29.14)

Hence, by (8.1), (29.13), and (29.14), [a(z), a(z )]  2(x  )t (At C − C t A)x + 2(ξ  )t (B t D − D t B)ξ + 2(x  )t (At D − C t B)ξ − 2x t (At D − C t B)ξ  . (29.15) So, by (8.1), (29.2), (29.3), (29.4), and (29.15), [a(z), a(z )]  2(x  · ξ − x · ξ  )  [z, z ], and hence a is symplectic.

✷

29. The Symplectic Group

137

Proposition 29.3. Let Sp(n, R) be the set of all symplectic linear transformations from Cn into Cn . Then Sp(n, R) is a group with respect to the usual composition of mappings.     A2 B2 A1 B1 Proof. Let and be in Sp(n, R). Then C1 D1 C2 D2      A1 B1 A 2 B2 A B ,  C D C1 D1 C2 D2 where A  A1 A2 + B1 C2 , B  A1 B2 + B1 D2 , C  C1 A2 + D1 C2 , and D  C1 B2 + D1 D2 . So, some easy computations and Proposition 29.2 give At D − C t B  I, At C  C t A, and B t D  D t B, and hence, by Proposition 29.2 again,   A1 B1 A2 C1 D1 C2

B2 D2

 ∈ Sp(n, R).

The associative law follows  fromthe usual association law for the compositions I 0 of mappings. The matrix in Sp(n, R) is obviously the identity element.  0 I A B Finally, let a  ∈ Sp(n, R). Let C D   t −B t D . b −C t At Then, by Proposition 29.2,     t  I 0 A B D −B t .  0 I C D −C t At   A B So, is invertible, and C D  −1  t  A B D −B t .  C D −C t At

138

29. The Symplectic Group

Therefore, it remains to prove that b ∈ Sp(n, R). To do this, we note that for all z and z in Cn , we get, by (29.1), [z, z ]  [(ab)(z), (ab)(z )]  [a(b(z)), a(b(z ))]  [b(z), b(z )], i.e., b is symplectic. Corollary 29.4. Let Proof.



A C

B D

✷



∈ Sp(n, R). Then AB t  BAt and CD t  DC t .

From the proof of Proposition 29.3,   −1  t D −B t A B ∈ Sp(n, R).  −C t At C D

Hence, by Proposition 29.2, the proof is complete.

✷

λ

Proposition 29.5. Let R be the irreducible and unitary representation of the Heisenberg group H n on L2 (Rn ) defined by (8.5). Let a ∈ Sp(n, R). Then the mapping Raλ : H n → G defined by Raλ (z, t)  R λ (a(z), t),

(z, t) ∈ H n ,

(29.16)

is also an irreducible and unitary representation of H n on L2 (Rn ). Proof.

Let (z, t) and (z , t  ) be in H n . Then, by (8.2) and (29.1), Raλ ((z, t) · (z , t  ))  Raλ (z + z , t + t  + [z, z ])  R λ (a(z + z ), t + t  + [z, z ])  R λ (a(z) + a(z ), t + t  + [a(z), a(z )])  R λ ((a(z), t) · (a(z ), t  ))  R λ (a(z), t)R λ (a(z ), t  )  Raλ (z, t)Raλ (z , t  ).

Next, it is easy to see that for all f in L2 (Rn ), we get, by (29.16), Raλ (z, t)f  R λ (a(z), t)f → f in L2 (Rn ) as (z, t) → (0, 0). Thus, Raλ : H n → G is a unitary representation of H n on L2 (Rn ). Finally, let M be a closed subspace of L2 (Rn ) such that M is invariant under all the operators Raλ (z, t), (z, t) ∈ H n . Let (z, t) ∈ H n and let w ∈ Cn be such that z  a(w). Then M is invariant under the unitary operator Raλ (w, t). But by (29.16), Raλ (w, t)  R λ (a(w), t)  R λ (z, t). So, M is invariant under all the operators R λ (z, t), (z, t) ∈ H n , and using the irreducibility of the unitary representation R λ of H n on L2 (Rn ), we conclude that M  L2 (Rn ) or M  {0}. Therefore, Raλ is irreducible. ✷ Proposition 29.6. The irreducible and unitary representations Raλ and R λ of H n on L2 (Rn ) are equivalent for all a in Sp(n, R).

29. The Symplectic Group

Proof.

139

We first note that by (8.5), Raλ (0, t)  R λ (a(0), t)  R λ (0, t)  e 4 iλt , 1

t ∈ R. ✷

So, by Proposition 8.6 and Remark 8.7, the proof is complete.

Remark 29.7. Let λ  1. Then, for all a in Sp(n, R), we get, by Proposition 29.6, a unitary operator Va on L2 (Rn ) such that R 1 (a(z), t)  Ra1 (z, t)  Va R 1 (z, t)Va−1 ,

(z, t) ∈ H n ,

and hence, by Remark 8.8, ρ(a(z))  R 1 (a(z), 0)  Ra1 (z, 0)  Va ρ(z)Va−1 ,

z ∈ Cn .

Proposition 29.8. Let a ∈ Sp(n, R). Then there exists a unitary operator Ua on L2 (Rn ) such that ρ(a(z)) ˜  Ua ρ(z)Ua−1 ,

z ∈ Cn ,

where a˜  (a t )−1 . Proof. In view of Remark 29.7, we only need to prove that a˜ ∈ Sp(n, R). In view of Proposition 29.3, we only need to prove thatthe transpose of any a in  A B Sp(n, R) is in Sp(n, R). To this end, let a  ∈ Sp(n, R) and let C D   0 I J  . (29.17) −I 0 Then, by Proposition 29.2, it is easy to see that J ∈ Sp(n, R). It has been shown in the proof of Proposition 29.3 that   0 −I (29.18) J −1  I 0 and a −1 



Dt −C t

So, by (29.17), (29.18), and (29.19),  t    t D 0 −I A Ct t  a  I 0 B t Dt −C t

−B t At

 .

−B t At



(29.19)

0 −I

and hence a t ∈ Sp(n, R).

I 0



 J −1 a −1 J, ✷

Remark 29.9. For each a in Sp(n, R), the unitary operator Ua on L2 (Rn ), generated by Proposition 29.8, is uniquely determined up to a constant multiple in the sense that if Ua and Va are unitary operators on L2 (Rn ) such that ρ(a(z)) ˜  Ua ρ(z)Ua−1 ,

z ∈ Cn ,

(29.20)

ρ(a(z)) ˜  Va ρ(z)Va−1 ,

z ∈ Cn ,

(29.21)

and

140

29. The Symplectic Group

then Va  ca Ua for some constant ca . Indeed, by (29.20) and (29.21), we get Ua ρ(z)Ua−1  Va ρ(z)Va−1 ,

z ∈ Cn ,

ρ(z)Ua−1 Va  Ua−1 Va ρ(z),

z ∈ Cn .

and hence (29.22)

So, by (8.5) and (29.22), R 1 (z, t)Ua−1 Va  Ua−1 Va R 1 (z, t), (z, t) ∈ H n . n

(29.23) n

Since R is an irreducible and unitary representation of H on L (R ), it follows that the only bounded linear operators on L2 (Rn ) that commute with all operators R 1 (z, t), (z, t) ∈ H n , are constant multiples of the identity I on L2 (Rn ). Hence, by (29.23), we get a constant ca such that 1

2

Ua−1 Va  ca I, or Va  ca Ua . Proposition 29.10.   0 I (i) Let a  . Then Ua  F.  −I 0 I 0 , where C is a symmetric matrix. Then (ii) Let a  C I (Ua f )(x)  e 2 i(Cx)·x f (x),

x ∈ Rn ,

1

for all f inL2 (Rn ).  A 0 (iii) Let a  , where A is an invertible matrix. Then 0 (At )−1 (Ua f )(x)  f (A−1 x)| det A|− 2 , 1

x ∈ Rn ,

for all f in L2 (Rn ). Proof.

For all z in Cn and all f in S(Rn ), we get, by (2.1),

(ρ(z)fˇ)(x)  eiq·x+ 2 iq·p fˇ(x + p) 1 ei(x+p)·ξ f (ξ )dξ  eiq·x+ 2 iq·p (2π)−n/2 Rn iq·x+ 21 iq·p −n/2 (2π) ei(x+p)·(η−q) f (η − q)dη e Rn 1 eix·η−ix·q+ip·η−iq·p f (η − q)dη  eiq·x+ 2 iq·p (2π)−n/2 Rn 1 −n/2 eix·η eip·η− 2 iq·p f (η − q)dη  (2π) 1

Rn

 (ρ(p, −q)f )ˇ(x),

x ∈ Rn .

(29.24)

29. The Symplectic Group

Also,

 a(q, ˜ p) 

0 −I

I 0

    q p  , p −q

q, p ∈ Rn .

141

(29.25)

So, by (29.24) and (29.25), ρ(a(z)) ˜  Fρ(z)F −1 ,

z ∈ Rn ,

and hence Ua  F. To prove part (ii), we first note that −1    I C I −C t −1 , a˜  (a )   0 I 0 I and hence, for all z  q + ip in Cn ,      q q − Cp I −C  . a(z) ˜  0 I p p

(29.26)

Now, we only need to prove that the Ua given by the equation in part (ii) satisfies ˜ )(x), (Ua ρ(z)Ua−1 f )(x)  (ρ(a(z))f

x ∈ Rn ,

(29.27)

for all z in Cn and all f in L2 (Rn ). But, by (29.26), (Ua ρ(z)Ua−1 f )(x)  e 2 i(Cx)·x (ρ(z)Ua−1 f )(x) 1

 e 2 i(Cx)·x eiq·x e 2 iq·p e− 2 i(C(x+p))·(x+p) f (x + p) 1

1

1

 e 2 i(Cx)·x+iq·x+ 2 iq·p− 2 i(Cx)·x−i(Cx)·p− 2 i(Cp)·p f (x + p) 1

1

1

1

 ei(q−Cp)·x+ 2 i(q−Cp)·p f (x + p) 1

 (ρ(q − Cp, p)f )(x)  (ρ(a(z))f ˜ )(x),

x ∈ Rn ,

for all z in Cn and all f in L2 (Rn ). Thus, (29.27) follows. To prove part (iii), we note that   t −1  t −1 (A ) 0 0 A , a˜   0 A−1 0 A and hence, for all z  q + ip in Cn ,  t −1     t −1  q (A ) q (A ) 0 a(z) ˜   . 0 A p Ap

(29.28)

We only need to prove that the Ua given by the equation in part (iii) satisfies ˜ )(x), (Ua ρ(z)Ua−1 f )(x)  (ρ(a(z))f

x ∈ Rn ,

(29.29)

for all z in Cn and all f in L2 (Rn ). But (Ua−1 f )(x)  f (Ax)| det A| 2 , 1

x ∈ Rn ,

(29.30)

142

29. The Symplectic Group

and hence, by (29.28) and (29.30), (Ua ρ(z)Ua−1 f )(x)  (ρ(z)Ua−1 f )(A−1 x)| det A|− 2

1

 eiq·(A

−1

x)

t −1

 ei((A ) e

e 2 iq·p (Ua−1 f )(A−1 x + p)| det A|− 2 1

1

q)·x+ 21 iq·p

f (x + Ap)

i((At )−1 q)·x+ 21 i((At )−1 q)·(Ap)

f (x + Ap)

t −1

 (ρ((A ) q, Ap)f )(x)  (ρ(a(z))f ˜ )(x),

x ∈ Rn ,

for all z in Cn and all f in L2 (Rn ). Thus, (29.29) is proved.

✷

Remark 29.11. It can be proved that the group Sp(n, R) is finitely generated by matrices of the three types given in Proposition 29.10. So, we are able to compute Ua , up to a constant multiple, for all a in Sp(n, R). It can also be proved that the constants can be chosen such that Ua Ub  ±Uab ,

a, b ∈ Sp(n, R).

Thus, the mapping from Sp(n, R) into G given by a → Ua is a “metaplectic representation” of Sp(n, R) on L2 (Rn ). See Chapter 4 of the book [6] by Folland for a discussion of the metaplectic representation and related matters. Remark 29.12. The unitary operators Ua provided by Proposition 29.10 are obviously homeomorphisms from S(Rn ) onto S(Rn ). It is also easy to see that the unitary operators Ua of the types given by parts (i) and (ii) of Proposition 29.10 are homeomorphisms from S  (Rn ) onto S  (Rn ). Let Ua be a unitary operator of the type given by part (iii) of Proposition 29.10. Then, for all f in S  (Rn ), we define Ua f : S(Rn ) → C by (Ua f )(ϕ)  f (Ua−1 ϕ),

ϕ ∈ S(Rn ).

It is then easy to check that Ua is a homeomorphism from S  (Rn ) onto S  (Rn ). Let σ ∈ S  (R2n ) and a ∈ Sp(n, R). Then we define the mapping σ ◦ a : S(R2n ) → C by (σ ◦ a)(ϕ)  σ (ϕ ◦ a −1 ),

ϕ ∈ S(R2n ).

(29.31)

We leave it as an exercise to prove that σ ◦ a ∈ S  (R2n ). We can now give the main result in this chapter. Theorem 29.13. Let a ∈ Sp(n, R). Then, for all σ in S  (R2n ), Wσ ◦a  Ua−1 Wσ Ua , where Ua is any unitary operator guaranteed by Proposition 29.8. Proof. Let σ ∈ S(R2n ). Then, for all ϕ and ψ in S(Rn ), we get, by (2.3), (3.12), Remark 29.12, Theorem 3.1, Theorem 4.3, and the adjoint formula for the Fourier

29. The Symplectic Group

143

transform, Ua−1 Wσ Ua ϕ, ψ  Wσ Ua ϕ, Ua ψ σ (x, ξ )W (Ua ϕ, Ua ψ)(x, ξ )dx dξ  (2π )−n/2 n C  (2π )−n/2 σˆ (z)V (Ua ϕ, Ua ψ)(z)dz n C  (2π)−n σˆ (z)ρ(z)Ua ϕ, Ua ψdz n C σˆ (z)Ua−1 ρ(z)Ua ϕ, ψdz. (29.32)  (2π )−n Cn

So, by (29.32), Proposition 29.8, and Remark 29.11, Ua−1 Wσ Ua ϕ, ψ  (2π)−n σˆ (z)ρ(a t (z))ϕ, ψdz Cn −n  (2π) σˆ (a(z))ρ(z)ϕ, ˜ ψdz n C  (2π)−n (σ ◦ a)ˆ(z)ρ(z)ϕ, ψdz Cn

 Wσ ◦a ϕ, ψ for all ϕ and ψ in S(Rn ). So, Wσ ◦a  Ua−1 Wσ Ua ,

σ ∈ S(R2n ).

(29.33)

Let σ ∈ S  (R2n ). Then, by Proposition 1.18, we can find a sequence {σj }∞ j 1 of functions in S(R2n ) such that σj → σ in S  (R2n ) as j → ∞. Then, by (29.31), (29.33), and Remark 29.12, we get, for all ϕ in S(Rn ), Wσj ◦a ϕ → Wσ ◦a ϕ and Ua−1 Wσj Ua ϕ → Ua−1 Wσ Ua ϕ in S  (Rn ) as j → ∞, and hence Wσ ◦a  Ua−1 Wσ Ua .

✷

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30 Symplectic Invariance of Weyl Transforms

We make precise in this chapter the fact that the symplectic invariance property in Theorem 29.13 characterizes the quantization σ → σ (x, D), described at the end of Chapter 4, as the Weyl transform Wσ . The main result, i.e., Theorem 30.1, obtained by Shale in [23], is another testimonial to the vision of Hermann Weyl that Wσ is the correct quantization of the classical observable σ . We begin with a proposition. Proposition 30.1. Let σ be the function on R2n defined by σ (x, ξ )  ei

n

j 1

xj

x, ξ ∈ Rn .

,

(30.1) n

Then, for all q  (q1 , q2 , . . . , qn ) and p  (p1 , p2 , . . . , pn ) in R with qj  0, j  1, 2, . . . , n, there exists an element a in Sp(n, R) such that (σ ◦ a)(x, ξ )  σ (a(x, ξ ))  ei(q·x+p·ξ ) , Proof.

x, ξ ∈ Rn .

By (30.1), σ (x, ξ )  ei(u,0)·(x,ξ ) ,

x, ξ ∈ Rn ,

where u is the point in Rn in which all coordinates are equal to 1. Let   Q P a , 0 Q−1 where

(30.2)



q1 0 Q

0 q2

0

0

 ··· 0 ··· 0   ··· · · · qn

(30.3)

(30.4)

146

30. Symplectic Invariance of Weyl Transforms

and



p1  0 P 

0 p2

0

0

 ··· 0 ··· 0  . ··· · · · pn

By Proposition 29.2, a ∈ Sp(n, R). Now, by (30.4),      x Qx + P ξ Q P  , a(x, ξ )  0 Q−1 ξ Q−1 ξ

x, ξ ∈ Rn .

(30.5)

Thus, by (30.3) and (30.5), (σ ◦ a)(x, ξ )  ei(u,0)·(Qx+P ξ )  ei(q·x+p·ξ ) ,

x, ξ ∈ Rn , ✷

and the proposition is proved.

Let L(S(Rn ), S  (Rn )) be the set of all continuous linear mappings from S(Rn ) n  n into S  (Rn ). Let {Ak }∞ k1 be a sequence of elements in L(S(R ), S (R )) and let A n  n n  n be in L(S(R ), S (R )). We say that Ak → A in L(S(R ), S (R )) as k → ∞ if Ak ϕ → Aϕ 

n

in S (R ) as k → ∞ for all ϕ in S(Rn ). We can now state and prove the main result in the last chapter of the book. Theorem 30.2. Let A : S  (R2n ) → L(S(Rn ), S  (Rn )) be a linear mapping. We suppose that it is continuous in the sense that σk → σ in S  (R2n ) ⇒ Aσk → Aσ in L(S(Rn ), S  (Rn )) as k → ∞. Moreover, we suppose that ((Aσ )ϕ)(x)  σ (x)ϕ(x), n

∞

x ∈ Rn ,

(30.6)

n

for all ϕ in S(R ) and all σ in L (R ), and A(σ ◦ a)  Ua−1 (Aσ )Ua

(30.7)

for all σ in S  (R2n ) and all a in Sp(n, R), where Ua is a unitary operator on L2 (Rn ) given by Proposition 29.8. Then Aσ  Wσ ,

σ ∈ S  (R2n ).

To prove Theorem 30.2, we need a lemma. Lemma 30.3. For any q and p in Rn , let eq,p be the function on R2n such that eq,p (x, ξ )  ei(q·x+p·ξ ) ,

x, ξ ∈ Rn .

Then Weq,p  ρ(q, p), where ρ(q, p) is given by (2.1).

q, p ∈ Rn ,

30. Symplectic Invariance of Weyl Transforms

147

Let q and p be in Rn . Then, by (2.3), (3.12), Theorem 3.1, and (12.1),

Proof. we get

¯  (2π)−n/2 (Weq,p ϕ)(ψ)



Rn

Rn

ei(q·x+p·ξ ) W (ϕ, ψ)(x, ξ )dx dξ

 (2π)n/2 W (ϕ, ψ)ˇ(q, p)  (2π)n/2 V (ϕ, ψ)(q, p)  ρ(q, p)ϕ, ψ,

ϕ, ψ ∈ S(Rn ), ✷

and hence the proof is complete.

Proof of Theorem 30.2. We begin with the observation that by Theorem 29.13, (30.1), (30.2), (30.6), (30.7), and Lemma 30.3, Aeq,p  Weq,p  ρ(q, p)

(30.8)

for all q and p in Rn such that every coordinate in q is nonzero. Let {Qk }∞ k1 be a sequence of concentric cubes in Cn with centers at the origin and edges parallel to the coordinate axes, and lim k→∞ |Qk |  ∞, where |Qk | is the volume of Qk , k  1, 2, . . . . Then, using (9.4) and the Lebesgue dominated convergence theorem, we get, for all σ in S(R2n ) and all ϕ in S(Rn ), −n (Wσ ϕ)(x)  lim (2π) (ρ(q, p)ϕ)(x)σˆ (q, p)dq dp, x ∈ Rn . (30.9) k→∞

Qk

Let σ ∈ S(R2n ). Then, by (30.9), Fubini’s theorem, and the Lebesgue dominated convergence theorem, −n (2π) (ρ(q, p)ϕ)(·)σˆ (q, p)dq dp → Wσ ϕ (30.10) Qk



n

in S (R ) as k → ∞ for all ϕ in S(Rn ). Now, for k  1, 2, . . . , we let σk be the function on R2n defined by σk (x, ξ )  (2π )−n eq,p (x, ξ )σˆ (q, p)dq dp, x, ξ ∈ Rn . Qk

Then σk → σ in S  (R2n ). Indeed, using Fubini’s theorem and the Fourier inversion formula, we get   −n σk (ψ)  (2π ) σˆ (q, p) eq,p (x, ξ )ψ(x, ξ )dx dξ dq dp Qk Cn ˇ σˆ (q, p)ψ(q, p)dq dp, ψ ∈ S(R2n ), (30.11)  Qk

and hence, by the Lebesgue dominated convergence theorem and the adjoint formula for the Fourier transform, σk (ψ) → σ (ψ)

148

30. Symplectic Invariance of Weyl Transforms

as k → ∞ for all ψ in S(R2n ). So, by continuity, (Aσk )(ϕ) → (Aσ )(ϕ) in S  (Rn ) as k → ∞ for all ϕ in S(Rn ). Now, by (30.8), (2π)−n (ρ(q, p)ϕ)(x)σˆ (q, p)dq dp  lim Sl (x), l→∞

Qk

(30.12)

x ∈ Rn ,

(30.13)

where Sl (x) is a Riemann sum of the form
Sl (x)  (2π)−n (ρ(qj , pj )ϕ)(x)σˆ (qj , pj )δj  ((Asl )ϕ)(x),

x ∈ Rn , (30.14) where (qj , pj ) is a point in Qk chosen in such a way that each coordinate in qj is nonzero;
sl (x, ξ )  (2π)−n eqj ,pj (x, ξ )σˆ (qj , pj )δj , x, ξ ∈ Rn , (30.15) and −n



sl (x, ξ ) → (2π )

Qk

eq,p (x, ξ )σˆ (q, p)dq dp, x, ξ ∈ Rn ,

as l → ∞. But by (30.11) and (30.15), we get  
sl (ψ)  (2π)−n σˆ (qj , pj ) eqj ,pj (x, ξ )ψ(x, ξ )dx dξ δj Cn
ˇ j , pj )δj  σˆ (qj , pj )ψ(q ˇ → σˆ (q, p)ψ(q, p)dq dp  σk (ψ) (30.16) Qk

as l → ∞ for all ψ in S(R2n ). Thus, by (30.16), sl → σk in S  (R2n ) as l → ∞, and hence, by continuity, (Asl )(ϕ) → (Aσk )(ϕ)

(30.17)

in S  (Rn ) as l → ∞ for all ϕ in S(Rn ). Next, for all ψ in S(Rn ), we get, by (2.3), (30.14), and Fubini’s theorem,  
Sl (ψ)  (2π)−n σˆ (qj , pj ) (ρ(qj , pj )ϕ)(x)ψ(x)dx δj Rn   → (2π )−n σˆ (q, p) (ρ(q, p)ϕ)(x)ψ(x)dx dq dp Q Rn k (ρ(q, p)ϕ)(·)σˆ (q, q)dq dp)(ψ)  ((2π )−n Qk

as l → ∞, and hence Sl → (2π )−n

Qk

(ρ(q, p)ϕ)(·)σˆ (q, q)dq dp

(30.18)

30. Symplectic Invariance of Weyl Transforms

in S  (Rn ) as l → ∞. So, by (30.14), (30.17), and (30.18), (Aσk )(ϕ)  (2π)−n (ρ(q, p)ϕ)(·)σˆ (q, q)dq dp.

149

(30.19)

Qk

Thus, by (30.10), (30.12), and (30.19), (Aσ )(ϕ)  Wσ ϕ,

ϕ ∈ S(Rn ),

and hence Aσ  Wσ ,

σ ∈ S(R2n ).

(30.20)

Finally, let σ ∈ S (R ). Then, by Proposition 1.18, there is a sequence {σj }∞ j 1 of functions in S(R2n ) such that σj → σ in S  (R2n ) as j → ∞. So, for all ϕ and ψ in S(Rn ), we get, by (12.1) and (30.20), 

2n

¯ (Wσ ϕ)(ψ)  (2π)−n/2 σ (W (ϕ, ψ))

¯  lim (2π)−n/2 σj (W (ϕ, ψ)) j →∞

 lim (Wσj ϕ)(ψ) j →∞

 lim ((Aσj )ϕ)(ψ), j →∞

and hence (Aσj )(ϕ) → Wσ ϕ 

n



n

(30.21)

in S (R ) as j → ∞. But by continuity, (Aσj )(ϕ) → (Aσ )(ϕ)

(30.22)

in S (R ) as j → ∞. So, by (30.21) and (30.22), Aσ  Wσ , and the proof is complete. ✷

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References

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Notation Index

B(L2 (Rn )), 22

L(S(Rn ), S  (Rn )), 146

C0∞ , 2

R λ , 39 Raλ , 138

ej,k , 101 en , 94 fˆ, 4 fˇ, 6 Ff , 4 F −1 f , 6 f ∗ g, 3 f ∗λ g, 43 f ⊗ g, 29 G, 39 H n , 10 Hn , 87 hn , 93 lk , 121 LF , 76 Lαn , 95 Lr∗ , 71

S, 2 Sh , 25 Sp(n, R), 137 Tσ , 8 V (f, g), 10 Wσ , 19 W (f ), 13 W (f, g), 15 (W H )n , 75 ∂α, 2 ρ(q, p), 9 σ (x, D), 24 ψj,k , 113 [z, w], 37 ∗ , 22 H S , 35

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Index

Adjoint formula, 5, 44, 142, 147 admissible wavelet, 76 annihilation operator, 93, 103, 106 Bracket operation, 38 Canonical commutation relations, 38 classical mechanics, 23, 133 commutator, 38 convolution, 2, 3 creation operator, 93, 103, 106 Differential operator, 1, 2 dilation operator, 5 Dirac delta, vi, 62 Eigenfunction, vi, 93, 106, 131, 133 eigenvalue, vi, 93, 106, 131, 133

H ∗ -algebra, 33–35 harmonic oscillator, 93, 103, 106 Hausdorff–Young inequality, 71–73, 125 Heisenberg group, vi, 37, 38, 138 Hermite function, vi, 87, 93, 101, 103, 106, 107, 109 Hermite polynomial, 87, 90 Hilbert–Schmidt operator, 25, 26, 32, 34, 57, 58, 74, 85 Inverse Fourier transform, 6, 30, 83, 109 irreducible representation, 40, 42, 75, 76, 138, 140 Joint probability distribution, 13 Kernel, 25, 26, 32, 57

Fourier inversion formula, 5, 8, 62, 68, 125, 147 Fourier series, 127, 128 Fourier transform, vi, 4–6, 13, 30, 44, 71, 72, 79, 81, 83, 119, 121 Fourier–Wigner transform, 9, 10, 13, 16, 101 functional calculus, 24, 25, 33

Laguerre polynomial, 87, 95, 98, 107, 109, 132 Laplace transform, 119 Laplacian, 106 left-invariant vector field, 38 Lie algebra, 38 localization operator, vi, 77–79, 83, 85

158

Index

Mehler’s formula, 107, 109 metaplectic representation, 142 modulation operator, 5 momentum, 1, 13, 23 Moyal identity, 15, 16, 101 multiplication operator, 23 Nondeterministic statistical dynamics, 15

spectrum, 133 square-integrable representation, vi, 76 state, 13 Stone–von Neumann theorem, 42 structure equations, 38 symbol, vi, 2, 8, 19–21, 29, 33, 37, 43, 55, 59, 75, 77, 83, 113–115 symplectic form, 37, 38, 43, 135 symplectic group, vi, 135, 137 symplectic invariance, 145

Observable, 1, 24, 35, 133, 145 Partial differential operator, 2, 8, 20, 87, 106 phase space, 23, 77, 133 Plancherel’s theorem, 6, 15, 46, 71, 101 Planck’s constant, 1, 23 position, 13, 23 projective representation, 10 pseudo-differential operator, v, vi, 7, 8 Quantization, vi, 19, 23, 35, 133, 145 quantum mechanics, v, 15, 24, 37, 38, 133

Tempered distribution, vi, 6, 7, 59, 61, 131 tempered function, 7, 114, 115, 128 tensor product, 29, 57 three lines theorem, 47 translation operator, 5 twisted convolution, vi, 37, 43, 44 twisting operator, 30 Unitarily equivalent representations, 42 unitary representation, 39, 40, 42, 44, 75, 76, 138, 140 Vector field, 103

Radial symbol, 113–115 regularization, 3 representation, vi, 19, 76, 142 resolution of the identity, 76, 77, 85 Riemann–Lebesgue lemma, 4, 125 Riesz–Thorin theorem, 47, 49, 57, 71, 78 Schr¨odinger representation, 10, 39 signal analysis, 75, 77

Weyl calculus, 33 Weyl–Heisenberg group, 75 Weyl transform, v, vi, 7, 9, 19, 21, 29, 32, 35, 37, 43, 44, 55, 59, 63, 71, 83–85, 87, 113, 114, 123, 127, 128, 131, 145 Wigner transform, vi, 9, 13, 15, 21, 59 Young’s inequality, 3, 129