Complete interpolating sequences for Fourier transforms supported by convex symmetric pol\-y\-gons
type: Item
instance of: scholarly article
title: Complete interpolating sequences for Fourier transforms supported by convex symmetric pol\-y\-gons
zbMATH Open document ID: 1038.42011
DOI: 10.1007/BF02384495
author: Alexander Rashkovskii
author: Yurii Lyubarskii
published in: Arkiv för Matematik
publication date: 2004-03-28T00:00:00Z
review text: Sampling and interpolation for two-dimensional Fourier transforms are studied in this article. Given a convex domain \(M\subset \mathbb R^2\), consider the corresponding Paley-Wiener space \[ PW_M = \left\{f:f(z) =\int_Me^{i\langle z,\xi\rangle}\phi(\xi)\, dm_\xi,\quad\phi\in L^2(M)\right\} \] endowed with the \(L^2(\mathbb R^2)\)-norm. Here \(\mathbb R^2\) is considered as the real plane in \(\mathbb C^2\), \(dm\) stands for the plane Lebesgue measure, and \(\langle\cdot,\cdot\rangle\) is the \(\mathbb C^2\)-scalar product: \(\langle z,\zeta\rangle=z_1\overline\zeta_1+z_2\overline\zeta_2\) for \(z=(z_1, z_2)\), \(\zeta=\zeta_1,\zeta_2\in\mathbb C^2\). A sequence \(\Omega=\{\omega\}\in\mathbb R^2\) is interpolating for \(PW_M\) if for each \(a=\{a_\omega\}_{\omega\in\Omega}\in l^2(\Omega)\) there exists \(f\in PW_M\) solving the interpolation problem \[ f(\omega)=a_\omega,\quad \omega\in\Omega.\tag{1} \] If the solution to this problem is always unique we say that \(\Omega\) is a complete interpolating sequence for \(PW_M\). The authors construct entire functions with plane zeros in \(\mathbb C^2\), generating complete interpolating sequences for the space \(PW_M\). Being of complex dimension 1, the zero set \(\mathbb Z\) of an entire function in \(\mathbb C^2\) itself cannot form such a sequence (in contrast to the one-dimensional case), however it produces a discrete set \(\Omega\in \mathbb Z\) which fits the needs, namely the collection of all pairwise intersections of the zero hyperplanes. In the construction \(\Omega\in\mathbb R^2\) and, if being uniformly separated (i.e., the distance between each pair of distinct points is uniformly bounded off zero), it forms the desired complete interpolating sequence. If \(\Omega\) is not uniformly separated, the interpolation problem (1) can have no solution. One has to modify it by introducing a corresponding block interpolation procedure with simultaneous interpolation at bunches of points which are located close to eac…
Mathematics Subject Classification ID: 42B10
Mathematics Subject Classification ID: 42B05
Mathematics Subject Classification ID: 32A60
Mathematics Subject Classification ID: 32A15
Mathematics Subject Classification ID: 32E30
zbMATH DE Number: 2062422
zbMATH Keywords: two-dimensional Fourier transform
zbMATH Keywords: Paley-Wiener space
zbMATH Keywords: complete interpolating sequence
reviewed by: Eleonora A. Storozhenko
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2062422