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 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 2 C S | ˆ|2dm n . 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 can be calculated as the l2- norm of ˆ(n) 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 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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