2. VARIETY OF MATHEMATICAL FORMS OF UP 7

Theorem 1. [Beurling’s Gap Theorem] Let μ ≡ 0 be a finite complex measure

on R. If the complement of the support of μ is long then the support of ˆ μ does not

have any gaps, i.e. there is no interval on R where ˆ μ is identically 0.

We will further discuss this result in chapter 2. Beurling’s Gap Theorem is one

of the original results on the Gap Problem, the problem of finding the maximal size

of the gap in the support of ˆ μ in terms of the metric characteristics of the support

of μ. The Gap Problem is one of the main topics of these notes.

2.3. Decay/Decay. This version of UP says that f and

ˆ

f cannot both decay

fast near infinity. A classical result of this type was proved by Hardy in 1933 [59].

Assume that

|f(x)| C(1 +

|x|n)e−aπx2

, and |

ˆ(x)|

f C(1 +

|x|n)e−bπx2

for some a, b, C 0, n ∈ N. If ab 1 then f ≡ 0. If ab = 1 then f =

pe−aπx2

for

some polynomial p of degree at most n.

A year later Morgan [110] replaced the right-hand sides of the inequalities

with

eaxp

and

ebxq

respectively and proved that if

1

p

+

1

q

1 then f ≡ 0. He

also described the constants a, b that give the same conclusion when 1

p

+ 1

q

= 1.

More general functions in the right-hand sides, with integral inequalities in place

of pointwise ones were later considered by Dzhrbashyan [44].

In the same section of UP one may mention another classical result by Beurling

that says that if f ∈

L1(R)

and

R×R

|f(x)||

ˆ(y)|e|x||y|dxdy

f ∞

then f ≡ 0. An extension of this result was obtained by Hedenmalm in 2012 [62].

2.4. Decay/Support or Support/Decay. If f decays fast near infinity then

ˆ

f cannot have small support. One of the classical theorems in this direction belongs

to Levinson [97]. This result will be further discussed in chapter 2.

Theorem 2. [Levinson’s Gap Theorem] Let μ be a finite measure on R. Denote

M(x) = |μ|((x, ∞)).

Suppose that log M is not Poisson-summable on R+. If ˆ μ vanishes on an interval

then μ ≡ 0.

This statement was later improved by Beurling [14] who replaced the interval

with any set of positive Lebesgue measure. A theorem by Volberg [146, 147, 23]

showed that under some regularity conditions on M, the conclusion can be further

strengthened to say that log |ˆ| μ must be Poisson summable. Further results in this

direction are due to Borichev [20, 23].

Using the fact that the Fourier transform of

ˆ

f is f(−x), one can now approach

the same problem from the opposite direction and look for statements which say

that if f has small support then

ˆ

f cannot decay too fast. Another consequence

of the F. and M. Riesz theorem tells us that if supp f is semibounded then log |

ˆ|

f

is Poisson-summable. The theorem by Volberg mentioned in the last subsection is

a vast generalization of this statement. Once again, smallness of the support can

be understood in many different ways. For instance, a special place in this part of

UP is occupied by problems about measures whose ‘essential’ support has measure