Universal sampling and interpolation of band-limited signals
Let \(S\subset {\mathbb R}\) be a bounded (measurable) set and let \(PW_S\) denote the Paley--Wiener space of all functions \(f\in L^2({\mathbb R})\) which are the Fourier transforms of functions from \(L^2(S)\). Given a sequence of distinct real numbers \(\Lambda=\{\lambda_j, j\in {\mathbb Z}\}\), one associates with each \(f \in PW_S\) its ``samples'' \(f\rightarrow \{f(\lambda)\}\), \(\lambda\in \Lambda\). One can raise the following three problems: A) When can the signal \(f\) be recovered (i.e. uniquely defined) from the samples? B) When do the sample provide a ``stable reconstruction'' of the signal, that is, the frame condition \(A \|f\|_{L^2({\mathbb R})}\leq (\sum_{\lambda\in\Lambda} | f(\lambda) |^2)^{1/2}\leq B \|f\|_{L^2({\mathbb R})}\) holds with some positive constants \(A\) and \(B\) for every \(f\) in \(PW_S\)? C) For every data \(\{c_j, j\in {\mathbb Z}\}\in \ell^2({\mathbb Z})\) when is there a signal \(f\in PW_S\) satisfying \(f(\lambda_j)=c_j\), \(j\in {\mathbb Z}\), i.e. \(\Lambda\) is the set of interpolation for \(PW_S\)? In this paper, the authors are interested in the following version of problems A) and B): Is it possible to define a ``univeral'' sampling sequence \(\Lambda\) of a given density which provides reconstruction or stable reconstruction of any signal whose spectrum \(S\) has a sufficiently small Lebegue measure? A similar variant of C) is: Does there exist \(\Lambda\) of positive density which is a set of interpolation for every space \(PW_S\) with sufficiently large mes \(S\)? It is proved that the universal uniqueness can be achieved in the general situation. On the other hand, for stable reconstruction it is crucial to know whether the spectrum is compact or dense. A similar result is true for the interpolation. Use is made of Beurling's and Kahane's densities.
scientific article; zbMATH DE number 5565915