Fourier Transforms and Approximations

272 Pages·2000·10.477 MB·\272

by  A M Sedletskii (Author)

Fourier Transforms and Approximations
ANALYTICAL METHODS AND SPECIAL FUNCTIONS
An International Series of Monographs in Mathematics
FOUNDING EDITOR: A.P. Prudnikov (Russia)
EDITORS: C.F. Dunkl (USA), H.-J. Glaeske (Germany) and M. Saigo (Japan)
Volume 1
Series of Faber Polynomials
P.K. Suetin
Volume 2
Inverse Spectral Problems For Differential Operators and Their Applications
V.A. Yurko
Volume 3
Orthogonal Polynomials in Two Variables
P.K. Suetin
Volume 4
Fourier Transforms and Approximations
A.M. Sedletskii
Additional Volumes in Preparation
Bessel Functions and Their Applications
B.G. Korenev
Hypersingular Integrals and Their Applications
S.G. Samko
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Fourier Transforms and Approximations
A.M. Sedletskii
Moscow State University, Russia
Translated from the Russian by E.V. Pankratiev
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CONTENTS
Preface vii
Chapter 1. Notation and some preliminaries from analysis 1
1.1. General notation 1
1.2. Slowly varying functions 2
1.3. Systems in Banach spaces 4
1.4. Interpolation of linear operators 5
1.5. Fourier transforms 6
1.6. Hardy spaces IP 1
1.7. Entire functions 8
Comments to Chapter 1 10
Chapter 2. Distribution of zeros of finite Fourier (Laplace) transforms 11
2.1. On zeros of finite Laplace transforms 11
2.2. On zeros of Fourier sine- and cosine-transforms 16
2.3. Bounds for zeros of the finite Laplace transforms 25
2.4. A condition for all the zeros of an entire function of exponential
type to lie in a curvilinear half-plane 31
Comments and supplements to Chapter 2 39
Chapter 3. Estimates of Fourier and Laplace transforms and their
applications 41
3.1. Asymptotic behaviour of finite Laplace transforms 41
3.2. Complex variants of the Abelian theorem for Laplace transforms 52
3.3. Decreasing finite Fourier transforms and their application to the
approximation 55
3.4. Finite Fourier transforms without zeros in a neighbourhood of the
real axis 60
Comments and supplements to Chapter 3 63
Chapter 4. Laplace transforms in the weight spaces LP and their
applications 65
4.1. Laplace transforms as operators in the spaces LPa 65
4.2. On completeness and non-minimality of a system of exponents in
If(- 7t, n) 69
Comments and supplements to Chapter 4 81
Chapter 5. Stability of classes of finite Fourier transforms and its
application 83
5.1. The invariance of the class 71} 83
5.2. The invariance of the class 7 Lq 95
5.3. Entire functions of Bernstein’s class that are not Fourier-Stieltjes
transforms 101
5.4. Excesses of systems of exponents 111
Comments and supplements to Chapter 5 119
vi CONTENTS
Chapter 6. Non-harmonic Fourier series (behaviour on the initial
interval) 121
6.1. Formulae for partial sums 121
6.2. Non-harmonic Fourier series and the condition (Ap) 127
6.3. Equiconvergence and uniform convergence of non-harmonic
Fourier series 137
Comments and supplements to Chapter 6 147
Chapter 7. Non-harmonic Fourier series (behaviour on the real line) 149
7.1. Extension of convergence of quasi-polynomials 149
7.2. Continuation of functions from the initial segment 161
7.3. Convergence and summability of non-harmonic Fourier series in
the If-norm (1 < p < °°) on every segment 164
7.4. Properties of the system (exp(i(n + ft sgn ri)t)) 171
Comments and supplements to Chapter 7 176
Chapter 8. The Miintz-Szasz problem 179
8.1. The case of real exponents and analysis of the problem in the
spaces C and If, p > 2 179
8.2. On zeros of analytic functions in a disk 187
8.3. Analysis of the problem in the weight spaces LP 194
Comments and supplements to Chapter 8 210
Chapter 9. Fourier transforms of rapidly decreasing functions 211
9.1. Theorems of Pitt’s type 211
9.2. Fourier transforms of rapidly decreasing functions on a half-line
and on the line 218
Comments and supplements to Chapter 9 229
Chapter 10. Approximation by translates and exponents on the line 231
10.1. Dense families of translates of a function on a line 231
10.2. Approximative properties of weighted exponents on the whole line 245
Comments and supplements to Chapter 10 254
References 255
Index 261
PREFACE
A more detailed title of this monograph would be ‘Fourier transforms that are analytic
functions and their applications to the function approximation by means of exponents and
powers on subsets of the real axis: on an interval, on a semi-axis, and on the whole axis.’
We consider three classes of Fourier transforms:
1) finite Fourier transforms;
2) Fourier transforms on a semi-axis;
3) Fourier transforms of rapidly decreasing functions (on the whole line).
First, we consider the finite Fourier-Stieltjes transform, i.e., the Fourier transform of the
measure which is concentrated on a finite segment,
b
F{z) = I elztdcr(t), vara(0 < oo, —oo < a < b < oo. (1)
The class of such functions is sufficiently large. It contains, for instance, the functions
sinz, cosz, the Bessel function Jv(z) multiplied by (2/z)v for v > —1/2, the confluent
hypergeometric function F\(a\ c; iz) for 9tc > > 0. Indeed,
eiz — e~iz eiz + e~iz
sinz = -----—------, cosz = -----------,
21 2
© "Mz) - + 1/2) / e"'a ~
-1
1
F/(a; c; z) = r(a^ - a) / ^ ^ > °-
0
The class of functions (1) contains the well-known class W% consisting, by definition, of
entire functions of exponential type < a which belong to L2 on the real axis. By the
Paley-Wiener theorem, the class W2 coincides with the class of functions representable in
the form
a
F(Z) = J eizt f(t)dt, f e L2.
—a
Functions of the form (1) are widely used in spectral theory, in the theory of differential-
difference equations, and in non-harmonic analysis. They have applications in radio-physics,
optics, and communication theory.
vii
viii PREFACE
Functions (1) are entire functions of exponential type. Therefore, their analytical
behaviour is of essential interest. By this we mean the behaviour of functions (1) in the
complex plane, in particular, the distribution of their zeros. The theory of entire functions
gives us some relationships, however, we mean the study of the dependence of these
properties on the function o(t). This problem has been investigated by E.C. Titchmarsh,
G.H. Hardy, G. Polya, M. Cartwright and others.
On one hand, by virtue of the importance of the class of functions (1), the analytical
behaviour is of essential interest in itself. On the other hand, it is closely connected with
the problems of non-harmonic analysis.
Non-harmonic Fourier analysis (or simply non-harmonic analysis) is the theory of
approximation of functions by means of a system of exponents
(exp(/A.nr)), A = (A„) C C (2)
on a finite interval. It includes the study of the following questions: completeness,
minimality, the basis property of systems (2) in different functional spaces, the behaviour of
the biorthogonal series in system (2) on a finite interval, i.e., non-harmonic Fourier series.
These questions have been considered in papers by R. Paley and N. Wiener, N. Levinson, L.
Schwartz, J.-P. Kahane, P. Koosis, R. Redheffer, A. Beurling and P. Malliavin, B.Ya. Levin,
A.F. Leont’ev, S.V. Hruscev, N.K. Nikolskii, and B.S. Pavlov, R.M. Young, and others.
Our interest in the approximate properties of systems (2) on a finite interval is motivated
by a series of factors. We emphasize one of them, preliminarily noting that system (2) is a
generalization of the trigonometric system exp(int), n e Z.
System (2) (or a system of linear combinations of functions (2)) often plays the role of
the system of the eigenfunctions of an operator. Thus, the system of eigenfunctions of the
derivation operator Dy = —iy' with the ‘spreading’ boundary condition
b
j y(t)da(t) = 0, var <r(0 < oo,
a
coincides with system (2), where A is the sequence of zeros of function (1).
One of the aims of this book is to present the analytical aspect of the theory of the finite
Fourier transform and its applications in non-harmonic analysis.
We would then like to consider the Fourier transforms of the functions that are
concentrated on a semi-axis. However, it is more convenient to consider the Laplace
transforms after some rotations in the complex plane. So, let
F(z) J e~z‘f(t)dt, 9tz > 0, (3)
K+
where f(t) is a locally integrable function such that the function F(z) is analytic in the
half-plane 91z > 0. The class of functions (3) contains the Hardy class H2 in the right
half-plane; by the Paley-Wiener theorem the class H2 coincides with the class of functions
(3), where f e L2.
If / € Lpy 1 < p < 2, then by the Hausdorff-Young theorem F{z) e Hq,\/p-\-l/q = 1.
Hence, the zeros zn of the function F(z) satisfy the well-known Blaschke condition
PREFACE ix
However, considering the case / € Lp, p > 2 changes the situation essentially. We present
the results on the distribution of zeros of function (3) which concern this case as well as a
more general one, when f e La, where La is a Lp-space on the semi-axis R+ with respect
to the measure tadt. Moreover, we study the mapping f(t) -> F(rel°), r > 0, for a fixed
0 e (—7t/2, 7t/2) as an operator acting in the spaces L&.
We need the description of zeros of functions (3) with f e Lp in connection, in particular,
with the MUntz-Szasz problem.
The well-known theorem of Muntz states if 0 < \jl\ < • • • < iin < ... , then the
completeness of the system of powers
(***■) (4)
in the space Co[0, 1 ] = (/ e C[0, 1] : /(0) = 0) is equivalent to the condition
V - = oo.
O. Szasz has considered complex exponents with the condition 9\p,n > —1/2. The Szasz
theorem states that the completeness of system (4) in L2(0, 1) is equivalent to the condition
+ 1/2 _
1 + I Mn + 1/2|2
By the Muntz-Szasz problem we mean the problem of description of complete systems of
powers (4) in the spaces Lp(0, 1), 1 < p < ooandCo[0, 1]. Let us note that all the functions
of system (4) belong to the space Lp(0, l)(Co[0, 1]) if and only if 9l/x„ > —1 /p(> 0).
If we consider C[0,1] instead of Co[0,1], then we should add to system (4) the function
which is identically equal to one.
The change of variable jc = exp(—t) allows us to reformulate the Muntz-Szasz problem
as the problem of describing complete systems of the exponents
(exp(-A„0), 9U„ > 0
in the spaces Lp{R+), p > 1 and Co(M+) = (/ € C[0, oo) : /(oo) = 0), where
Xn = fxn -f- 1/p in the case of Lp and Xn = pn in the case Co - Thanks to this reformulation,
the distribution of zeros of functions (3) with / € Lq becomes the analytic equivalent of
the Miintz-Szasz problem (for spaces Lp).
We present the current status of the Miintz-Szasz problem.
Finally, we consider the Fourier transforms of rapidly decreasing functions, i.e., of
functions of the form
exp(—a\t\a)f(t), a > 0, a > 1, (5)
where the factor f(t) is relatively small for large |/, so that the Fourier transform of function
(5)
F(z) = j elzt exp(—a\t\a)f(t)dt, a > 0, a > 1, (6)
R
is an entire function. As an example, we write the known identity
e*P{ - l ) = :k f e‘aCXt’(J 2) ‘“- Z€C-
R