1. BASIC NOTATIONS 3

sources of results and references. Further references will be given in section 2 and

throughout these notes.

Our goal is to look only at a small sample of classical problems of Fourier anal-

ysis that stem from Wiener’s original ideas of 1920’s, together with their modern

extensions and generalizations. Among them are Beurling’s Gap Problem on the

Fourier transform, classical completeness problems, such as the Beurling–Malliavin

problem and the Type Problem on completeness of trigonometric polynomials in

L2, a problem on oscillations of Fourier integrals with a spectral gap, etc. The

discussion will include a brief outline of the Toeplitz approach to the problems of

UP developed recently in our joint work with Nikolai Makarov [103, 104]. One of

the extensions of UP that we will consider involves the so-called Weyl-Titchmarsh

transform, a generalization of the Fourier transform that appears in spectral prob-

lems for differential operators. Via the Weyl-Titchmarsh transform, the traditional

Payley-Wiener spaces are replaced with de Branges spaces of entire functions or

with model subspaces of the Hardy space in the upper half-plane. The traditional

completeness problems of systems of complex exponentials become problems on

completeness of special functions or spectral problems for Schr¨ odinger equations

and Krein’s canonical systems. The Toeplitz approach allows one to use methods

of complex function theory and harmonic analysis to solve some of such problems.

One of my goals when writing these notes was to make it possible to read each

chapter from 2 to 8 independently. This led to numerous repetitions in definitions

and arguments, for which I apologize in advance. Additionally, after giving it some

thought, I have decided against including a list of open problems. The main reason

for this omission was that most of the problems I had in mind were still raw and

uneven in quality. At the same time, I hope very much that an interested reader

can find plenty of open problems on her/his own while looking through the text,

especially in the last two chapters.

1. Basic notations

If f is a function from L1(R) we denote by

ˆ

f its Fourier transform

(1.1)

ˆ(z)

f =

R

f(t)e−2πiztdt.

The function

ˆ(z)

f is well defined for z ∈ R and, under various additional

conditions on f, may be defined in a larger subset of the complex plane C.

Let M be a set of all finite Borel complex measures on the real line. Similarly,

for μ ∈ M we define

ˆ(z) μ =

R

e−2πiztdμ(t).

Via Parseval’s theorem, the Fourier transform may be extended to be a unitary

operator from

L2(R)

into itself and can be defined for even broader classes of

distributions. It can also be extended to functions (measures, distributions) in

Rn

using the same formula

ˆ(s)

f =

Rn

f(t)e−2πis,tdmn(t),

where mn is the Lebesgue measure in

Rn,

s, t ∈

Rn

and s, t is the standard

scalar product.