10 1. MATHEMATICAL SHAPES OF UNCERTAINTY

instance [97]. Basic results in this direction use the growth estimates of Paley-

Wiener functions as functions of exponential type. Originating from this approach

is a whole branch of complex analysis dealing with distributions of zero sets of

entire functions in different classes of growth near infinity, see the well-known book

by Levin [94].

The study of zero sets is closely related to the study of uniqueness sets via a

trivial observation that if F is not uniquely determined in its class by its values on

Λ = {λn} ⊂ C, then Λ is in the zero set of F − G for some G from the same class.

Another close relative is the set of completeness problems.

We say that a system of vectors in a Banach space is complete if finite linear

combinations of these vectors are dense in the space. Given a space and a system of

vectors, an important problem that comes from many applications is to determine

if the system is complete in the space. The space usually appears as a space of

functions and the system is formed by some special or elementary functions often

called harmonics.

A general problem of completeness of harmonics in a Banach space gave Har-

monic Analysis its name. The area of UP contains many completeness problems,

some of which will be discussed in detail in this text.

If a system is complete, then the existence of a function orthogonal to all of

the harmonics violates the uncertainty principle in some form. To give an example

of this relation let us consider one of the most celebrated of such problems, the

Beurling–Malliavin problem.

Let Λ = {λn} ⊂ C be a sequence of complex numbers and let EΛ =

{e−2πiλn

}

be the system of complex exponentials with frequencies from Λ. One of the funda-

mental questions of analysis asks when is EΛ complete in

L2(−a,

a). By the duality

of

L2,

one may equivalently ask when does there exist a function in f ∈

L2(−a,

a)

orthogonal to all exponentials from EΛ. If such a function f exists, then it’s Fourier

transform is an entire function of the Paley-Wiener class PWa vanishing on Λ.

Conversely, the Paley-Wiener theorem says that any PWa-function vanishing on

Λ is a Fourier transform of such an f. Hence, the Beurling–Malliavin problem of

completeness of EΛ in L2(−a, a) is equivalent to a problem of UP on zero sets of

entire functions with bounded spectrum.

The Beurling–Malliavin problem was solved by Beurling and Malliavin in a

series of paper in the early 1960’s. In these notes we discuss this problem together

with its recent extensions and further completeness problems, see chapters 5, 7 and

8.

3. Function theoretic background

In this section we provide the minimal function theoretic background necessary

for the rest of the text. The definitions and statements are given in a very condensed

form. A researcher who wants to work in the presented area of UP will certainly

need a deeper knowledge of these basics. In that regard let us mention two books

on

Hp-theory,

[52, 84], a book on model spaces KΘ, [116], a survey paper on Clark

theory, [126], and a book and a chapter on de Branges’ spaces [26], [43, Chapter

6]. A text that will include all these topics at once is currently in preparation [105].

A standard subject of complex function theory is the study of linear spaces of

analytic functions in a fixed complex domain. In view of the Riemann conformal

mapping theorem, all simply connected domains split into two (unequal) classes