Frames and sampling theorem
Let \(\{\varphi(\cdot -n)\}\) be a frame for a closed subspace \(V_0\) of \(L_2(R)\), where \(\varphi\) is a continuous function that is either locally absolutely continuous and \(\varphi^{'} \in L^2(R)\), or \( |\varphi(x)|\leq C(1+|x|)^{-1/2-\varepsilon}\) for some \(\varepsilon > 0\). The authors find a necessary and sufficient condition for the existence of a frame \(\{S(\cdot -n)\}\) for \(V_0\) such that for every \(f \in V_0\), \( f(t) = \sum_k f(k) S(t-k)\), where the convergence is in \(L^2(R)\). Among the corollaries of this result is a sampling theorem of \textit{G. G. Walter} [IEEE Trans. Inf. Theory 38, No. 2/II, 881-884 (1992; Zbl 0744.42018)]. For a smaller set of functions \(\varphi\), the authors also obtain a result on irregular sampling that yields, as a particular case, a theorem of \textit{Y. N. Liu} and \textit{G. G. Walter} [J. Fourier Anal. Appl. 2, No. 2, 181-189 (1995; Zbl 0886.42025)]. A number of examples illustrate the results obtained.
scientific article; zbMATH DE number 1329976