Complete interpolating sequences for Fourier transforms supported by convex symmetric pol\-y\-gons
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 each other. In this procedure, the assumption \(\{a_\omega\}\in l^2\) should be replaced by a more complicated one. In particular, the authors obtain a special integral representation for rational functions which seems to be of some independent interest.
scientific article; zbMATH DE number 2062422
scientific article