On a class of non-uniform average sampling expansions and partial reconstruction in subspaces of L2(R)

• Authors: Nikolaos D. Atreas
• Venue: Advances in Computational Mathematics
• Year: 2012

Let 驴 be a function in the Wiener amalgam space $\emph{W}_{\infty}(L_1)$ with a non-vanishing property in a neighborhood of the origin for its Fourier transform $\widehat{\phi}$ , ${\bf \tau}=\{\tau_n\}_{n\in {{\mathbb Z}}}$ be a sampling set on 驴 and $V_\phi^{\bf \tau}$ be a closed subspace of $L_2(\hbox{\ensuremath{\mathbb{R}}})$ containing all linear combinations of 驴-translates of 驴. In this paper we prove that every function $f\in V_\phi^{\bf \tau}$ is uniquely determined by and stably reconstructed from the sample set $L_\phi^{\bf \tau}(f)=\Big\{\int_{\hbox{\ensuremath{\mathbb{R}}}} f(t) \overline{\phi(t-\tau_n)} dt\Big\}_{n\in {{\mathbb Z}}}$ . As our reconstruction formula involves evaluating the inverse of an infinite matrix we consider a partial reconstruction formula suitable for numerical implementation. Under an additional assumption on the decay rate of 驴 we provide an estimate to the corresponding error.