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Spectral gaps for sets and measures. (English) Zbl 1245.42008

The uncertainty principle in harmonic analysis has motivated much research in harmonic analysis. The principle roughly states that it is not possible for both a function (or measure) and its Fourier transform to be small, unless it is identically zero. There are many ways to formulate problems arising from this principle by considering different interpretations of ‘small’. In this paper, smallness is understood in terms of the sparseness of the supports of the measure and its transform, the so-called gap problem. The objective of the gap problem is to obtain quantitative estimates relating the size of gaps in the support of \(\widehat{\mu }\) with the size of the support of \(\mu \). A classical theorem of this type is Beurling’s gap theorem which says that if the sequence of gaps in the support of \(\mu \) is ‘long’, then the support of \(\widehat{\mu }\) cannot have any gaps, unless \(\mu =0\).
Given \(X\) a closed subset of \(\mathbb{R}\), denote by \(G_{X}\) the supremum of the size of the gap in the support of \(\widehat{\mu }\), taken over all finite measures \(\mu \) supported on \(X\). The author introduces a characteristic of \(X\), called \(C_{X}\), which incorporates both a density and an energy condition. The main result is the proof that \(G_{X}=2\pi C_{X}\). The proof uses Toeplitz operators and other important ideas from analytic function theory, such as the Beurling-Malliavin multiplier theorem.
The paper is long and technical, but the ideas are well motivated, both mathematically and physically. The author also gives various formulations and discusses related problems such as de Brange’s result on the existence of a measure with a given spectral gap, the problem of uniform convergence of continuous functions by trigonometric polynomials, and the type problem, the problem of finding the least \(a\) such that the exponential functions \(e^{i\lambda x}\), with \(\lambda \in [0,a]\), span \(L^{2}(\mu )\). He concludes with a Toeplitz version of de Branges’ theorem.

MSC:

42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
43A05 Measures on groups and semigroups, etc.
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
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