Some questions about Hardy functions

CHAPTER 3 PROBLEMS FROM PROBABILITY THEORY In this chapter we have collected the problems which have a probabilistic origin and which are formulated in terms of the spectral theory of functions. Nowadays, probability theory is used very frequently in harmonic analysis. As an example, we indicate J.-P. Kahane's book: "SEries de Fourier Ale~toires." The stationary Gaussian processes form a bridge between the spectral theory of functions and probability theory and this bridge rests upon the mathematical model of a filter, due to N. Wiener. Indeed, if one considers the predecessors of those problems which are now considered by the spectral theory of functions, then one arrives without difficulty at the harmonic analysis problem of the convolution operators f § K*f on ~ , related firmly with Wiener's name. On the other hand, practically any stationary Gaussian process can be considered as the result of the action of some filter, i.e., of the convolution operator K*, on the "white noise" process. Roughly speaking, all the stochastic information is contained in the "white noise" model and the problem consists in the investigation of its redistribution under the action of the filter K*. Therefore, it is possible that the language of harmonic analysis, Hardy classes, entire functions, briefly, the language in which this entire collection is written, has become the language of many important purely probabilistic investigations. Taking into account the historical importance of the theory of stationary Gaussian processes, we have decided to start this relatively small chapter with problems which arise in the mentioned theory. The problems of Secs. 1.3 and 2.3 complete each other in an organic manner. The formulation of the problems in Sec. 1.3 is not distinguished by a concrete character; they form rather a list of very interesting directions of research. The reader interested in the problems of this section should turn to the excellent book by Dym and McKean (see [I] in the reference list of Sec. 1.3). On the contrary, Sec. 2.3 contains a series of concrete analytic problems related basically to the problem of "Past and Future." In connection with the problems of Sec. 3.3 we mention two papers which may turn out to be useful here. M. Benedicks (Royal Inst. of Technology, Stockholm, 1976, preprint) has shown that if f and f have Lebesgue supports of finite Lebesgue measure, then f ~ 0. On the other hand, W. O. Amrein and A. M. Berthier have proved [J. Funct. Anal., 24, No. 3, pp. 258267 (]977)] that for any pair A, B of measurable subsets of the space ~ condition

the space of functions i ~ : ~ 6 ~ ( ~ ) , $ - ~ A = ~ ,

~:%8= ~}

Here XE is the characteristic function of the set E. is considered with these results.

, satisfying the

has infinite dimension in ~ ( ~ ) Clearly,

.

the conjecture of Sec. 3.3

We conclude this short introduction by indicating some problems placed in other chapters. In Sec. 4.6 one poses the question of the approximation by trigonometric polynomials with restrictions on the spectrum. The probabilistic meaning of such problems is well known (see the already cited book by Dym and McKean). Also Chap. 9 is related to probability theory. It is sufficient to recall that the theory of the spaces H ~ and BMO, constructed by E. M. Stein and Ch. Fefferman in their well-known paper [Acta Math., 129, pp. 137-193 (1972)] by means of classical harmonic analysis, has first appeared as a result of the investigations of specialists in the theory of probability. The connection with Chap. 9 is attested also by D. Sarason's result, formulated in Sec. 2.3 (Theorem I).

2131

1.3.

SOME QUESTIONS ABOUT HARDY FUNCTIONS~

The theory of Gaussian-distributed noise leads to a variety of substantial mathematical questions about Hardy functions. I will put the questions in a purely mathematical way; the reader is referred to [I] for the statistical interpretation and/or additional information. I.

Let A, A ~ 0, be summable on the line and let I

R

~A

4+~z

~__oo -

ixt ~ ixT tials e , t ~ 0, span L ( ~ , ~ ) , but how is e for fixed T > 0 (by these functions)? See [I, Sec. 4.2]. 2.

Let ~, ~ 6 H ~

efficientIu

approximated

, be outer, let T, T > 0, be fixed, and let

9 K =-.kqiF I

cb t,

T

~)

Then the exponen"

being the inverse transform ~-L~I C ~ ( ~ ) ~

.

What can be said about K

?

~K~,

cannot

R

be <~ for all small T; also,

Kv

can be enormously singular; see [I, Sec. 4.4].

3. Let ~ , ~ m ~ , be outer. The question is to explain what makes the phase function h*/h the ratio of two inner functions or the reciprocal of an inner function; see [I, Sec. 4.6]. eixT'h*/h is itself an inner function if and only if h is integral of exponential type ~T. 4. Let ~ , ~ r See [I, Sec. 4.12]. 5.

, be outer.

When does h*/h belong to the span of eixtH ~, t ~ 0, in L ?

The following conditions are equivalent for outer h, ~ H %

: a) e 2ixT -h*/h is the

ratio of a function of class H$ and a function of' class H~; b) ~ I ~ / ~ l ~ + tegral function f of expQnential type ~T; c) [ 6~mT

~+~ ~_{ - K ~

~ i

oo for some in-

for some

~,~+~

;

see [I, Sec. 4.13]. What can be said about such functions h? Note that b) is a problem of "multiplying down" the function I/h in the style of [2]. What outer functions satisfy a), b), c) for every T, T > 0? for no T, T > 0? Note that h cannot satisfy c) for T, T = 0. 6. The phase function h*/h is ubiquitous. outer function h, h ~ H~?

What can be said about it for the general

LITERATURE CITED I 9

2.

H. Dym and H. P. McKean, Gaussian Processes, Function Theory, and the Inverse Spectral Problem, Academic Press, New York (1976). A. Beurling and P. Malliavin, "On Fourier transforms of measures with compact support," Acta Math., 107, 291-302 (1962).

~H. P. McKEAN. N.Y. I001k.

2132

New York University, Courant Institute of Mathematical Sciences, New York,