Reconstruction of band-limited functions from local averages
Given \(\Omega>0\), the set of functions in \(L^2(R)\) whose Fourier transform is supported on \([-\Omega,\Omega]\) is denoted by \(B_\Omega\). Let \(P\) denote the orthogonal projection of \(L^2(R)\) onto \(B_\Omega\). It is proved that every band-limited function can be reconstructed from local averages near certain points. More precisely, the following Theorem is proved: Theorem 1: Assume that \(\{x_k\}_{k\in Z}\) is a real sequence for which \[ \lim_{k\to \pm \infty}x_k= \pm \infty \;\text{and} \;0< x_{k+1}-x_k \leq \beta< \frac{\pi}{\Omega}. \] Then, for any \(\delta\in ]0, \pi/\Omega - \beta[\) and functions \(u_k\) for which \[ u_k\geq 0, \;u_k\in (L^1\cap L^2)(R), \text{ supp } u_k \subseteq [x_k-\delta_2,x_k+\delta/2], \;\int u_k=1, \] the functions \(\left\{\sqrt{\frac{x_{k+1}-x_k}{2}}Pu_k\right\}_{k\in Z}\) constitute a frame for \(B_\Omega\). As a consequence of Theorem 1, there exist a collection of functions \(\{S_k\}_{k\in Z}\) (the dual frame) in \(B_\Omega\) such that each \(f\in B_\Omega\) has a representation \[ f(x)=\sum_{k\in Z}\sqrt{\frac{x_{k+1}-x_k}{2}} <f,u_k>S_k(x) \] with convergence in \(L^2(R)\) and uniformly on \(R\). It is proved that the result is no longer valid if \(\delta>\pi/\Omega - \beta\). Theorem 1 generalizes a previous result by Gröchenig.
scientific article; zbMATH DE number 1761142