Preliminary Note on the Inversion of the Laplace Integral

VOL. 18, 1932

MA THEMA TICS: D. V. WIDDER

181

PRELIMINARY NOTE ON THE INVERSION OF THE LAPLACE INTEGRAL By D. V. WIDDER DEPARTMENT

OF

MATHEMATICS, HARVARD UNIVERSITY

Communicated January 15, 1932

In a previous note in these PROCEEDINGS' we stated the following result: If (p(t) is continuous in the interval 0 < t < o, if lim (p(t) = a, and if 1=

X

co

f(x) = J'ext1p(t)dt, then kim [x^ +I ( 1)kf (,x) -

ip(p

)]=°

(1)

uniformly in the interval 0 _ x < a. This result was obtained in order to discuss the zeros of gp(t) in terms of those of the derivatives of f(x). The result has, however, an important interest in itself, since it enables us to invert the Laplace integral (1) under the conditions described in the theorem. We have, in fact,

(o(t)

=

urn

(_ 1)k f(k) (k/t)k

+1

uniformly for 0 < t < c. This result was obtained earlier by E. Post2 under the restriction of continuity on sp(t) but without the condition that p(t) approaches a limit as t becomes infinite. On the other hand the uniformity of the approach was not obtained by Post. In the present note we sketch a theory whereby the integral equation (1) may be solved with no restriction on so(t) as to continuity. We simply impose the natural condition that s(t) shall be integrable in the sense of Lebesgue and (so that the integral (1) may converge for x sufficiently large) the condition Ip(t)

<Mea

0O
where M and a are constants. We then turn to the problem of inverting integrals of the form

f(x)

exda(t) e

(2)

where no restrictions are imposed on a(t) except those necessary to make the improper Stieltjes integral have a meaning: a(t) is of bounded varia-

182

PROC. N. A. S.

MA THEMA TICS: D. V. WIDDER

tion in every finite interval and is such that (2) converges for some x sufficiently large. It is important to note that the absolute convergence of (2) is not required. The result obtained is the following:

2(+ 2(+ 2 (-

fw(l) k

k1 Xc[ok =(Lo) +ikit ~~k= [

)k+ lUkf(k +1) (U)

l

du]

(3)

These results form a most powerful tool for the solution of the type of problem considered by the author in an earlier paper.3 For example, consider Bernstein's theorem proved there. It states that if f(x) has its derivatives alternately positive and negative in the interval 0 < x < co, then f(x) can be represented in the form (2) with a(t) a non-decreasing function. This result could be predicted at once from (3). For under the assumed conditions on f(x) the integrand in (3) is positive so that the quantity in brackets is quite obviously an increasing function of t. Hence the limit a(t) will be non-decreasing if it exists. We are able to show further that the limit does exist and thus give an elementary proof of Bernstein's theorem involving no use of the moment problem. Further applications of the result are made to the determination of necessary and sufficient conditions for the representation of f(x) in the form (2) where a(t) is of prescribed form. The following theorems serve to sketch the theory. THEOREM 1. The limit k +1 (*2+ limkim eek/[i ./1 + + + ~

* ] + +,()]

exists and has the value 0

/2 1

0 1.

This theorem is proved by obtaining an integral expression for the quantity in brackets and then applying known methods to the determination of an asymptotic expression for it. THEOREM 2. If co f(x) =Je-Xt(p(t)dt.,

where the integral converges for x sufficiently large and p(t) is of bounded variation in every finite interval, then li

1 (_1)kf(k)(k/l)kk+ k! tk +

ko

p(t+) +2 (t)

MA THEMA TICS: D. V. WIDDER

VOL. 18, 1932

183

By use of this theorem we are able to prove

THEOREM 3. If e-xtda(t),

f(x) =

where a(t) is of bounded variation in every finite interval, the integral converging for x sufficiently large, then

lif[f(X+k1)

ukf

u

k

du]

= a(t+)

It is possible to obtain another expression for a(t) values of f(k) (X + c), where c is any constant. It is

+ a(t-)

depending

on

the

U ecklu f(1 (U + c)du] a(t+) + a(-) klim jf() + (_1)k+lf 2/ k!

k=

This clearly reduces to the result of Theorem 3 when c is zero. THEOREM 4. Let

f(x)

e-xt(t)dt,

=

where cp(t) is integrable in the sense of Lebesgue and uniformly bounded the interval (0, co). Then (_1)kf(k (k/t)k

r

ki!

in

= ( p(t)t

tk+1

almost everywhere. This theorem can easily be extended to the case in which sp(t) is not uniformly bounded but satisfies a condition of the type described in the introduction. We give next two uniqueness theorems. THEOREM 5. Let f(x) satisfy the condition

k+i' If(k)(X) I< Mk'

;

k

=

0, 1, 2,

Then

lim (_ 1)k kco

=

c e

k! tk+

o

1

dt = f(x), x >

0.

THEOREM 6. Let f(x) satisfy the condition

|y

kl!(+l

(u)du

J

< M,

x >

O; k

=

O, 1, 2, ..

MA THEMA TICS: D. V. WIDDER

184

PROe. N. A. S.

and let (Pk

(u)du + f(ok) .

I)=(+

Then, if SPk(t) is of bounded variation in every finite interval, i

f

k= co

e`xd(pk(t)

=

f(x), x> 0.

Both of these theorems can be made more general so as to obtain an expression for f(x) in an arbitrary interval. By use of these theorems we can prove the following results. THEOREM 7. A necessary and sufficient condition that f(x) should be completely monotonic for x > c is that

f(x) =

,co e"-x da(t),

where a (t) is non-decreasing and the integral converges for x > c. THEOREM 8. A necessary and suicient condition thai f(x) can be represented in the form C*

f (x)

=

e-Xtda(t),

the integral converging absolutely for x > 0, is that Jc

uk-| f(k+l)(U)

I du < M

x > 0; k = O, 1, 2.

For earlier proofs of Theorem 7 and Theorem 8 reference may be made to the author's paper already cited.3 The proofs now obtained have distinct advantages over those given earlier in that they involve no reference to the moment problem. 1 D. V. Widder, "On the Changes of Sign of the Derivatives of a Function Defined by a Laplace Integral," Proc. Nat. Acad. Sci., 18, 112-114(1932). 2 E. L. Post, "Generalized Differentiation," Trans. Am. Math. Soc., 32, 723 (1930). 3 D. V. Widder, "Necessary and Sufficient Conditions for the Representation of a Function as a Laplace Integral," Trans. Am. Math. Soc., 33, 851 (1931).