2. VARIETY OF MATHEMATICAL FORMS OF UP 9
Connections with adjacent fields, such as problems on convergence of trigono-
metric polynomials, made description of all R-measures an important classical prob-
lem. A solution suggested by
Shre˘
ider in 1950 [141] involved the so-called Weyl
sets. The statement that he published without a proof, following the style of the
Soviet magazine “Doklady,” said that R-measures are exactly those measures that
do not see Weyl sets. The proof was given by Lyons in 1983 [101].
Lyons’ theorem answers a long standing question on the possibility of a de-
scription of R-measures in terms of a collection of sets, as stated above. Such
a set-theoretic answer was needed in particular due to the connection with con-
vergence problems for trigonometric polynomials. It is still unclear if an analytic
condition, in terms of local symmetry and mass distribution of the measure, can be
found.
For more on R-measures see [81, 83, 102].
2.5. Support/Sampling sets. In this version of UP we claim that if f has
small support then
ˆ
f cannot be small in
L2
norm on a fixed subset of
Rn,
if that
subset is large enough.
More precisely, a set S
Rn
is determining, if for any bounded Σ
Rn
there
exists C = C(Σ) 0 such that
(1.4) supp f Σ ||
ˆ||2
f
2
C
S
|
ˆ|2dmn.
f
A natural problem is to describe all determining sets. The following theorem
gives such a description.
A set S
Rn
is relatively dense if there exists r, δ 0 such that
B(x, r) S δ
for all x.
Theorem 5 (Paneah 1961, Logvinenko-Sereda, 1974). S is determining if and
only if it is relatively dense.
Paneah [121] proved this result in one dimension and one of the implications
in all dimensions, Logvinenko and Sereda finished the theorem [98].
If the support of f : R R is contained in [−1, 1] then its Fourier transform
is an entire function of Paley-Wiener class and ||
ˆ||L2
f can be calculated as the
l2-
norm of
ˆ(n)
f by Parseval’s theorem. The Paneah-Logvinenko-Sereda theorem may
be viewed as an extension of that property. A natural question about sequences
that can replace Z in Parseval’s theorem, with an equivalence of norms or a one
sided inequality, becomes a question on frames, sampling sequences and Riesz bases
in Paley-wiener spaces. Naturally, similar questions can be considered in other
spaces of analytic functions. For examples of deep results in this direction and
further references see a paper by Hruschev, Nikolski and Pavlov on Riesz bases of
reproducing kernels and exponential functions [68] or a paper by Ortega-Cerda and
Seip on Fourier frames [118]. The problem on olya sequences discussed in chapter
3 can be viewed as an L∞-version of the same problem.
2.6. Support/Zero sets. Uniqueness and completeness problems. If
the support of a non-zero function f is bounded then
ˆ
f is an entire function whose
zero set cannot be too large. The problem of obtaining quantative estimates on
the density and balance of such zero sets is treated in many classical texts, see for
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