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

摘要

Let ϕ be a function in the Wiener amalgam space W_¥(L₁)emph{W}_{infty}(L_1) with a non-vanishing property in a neighborhood of the origin for its Fourier transform [^(f)]widehat{phi}, t={t_n}_{n Î mathbb Z}{bf tau}={tau_n}_{nin {{mathbb Z}}} be a sampling set on ℝ and V_ftV_phi^{bf tau} be a closed subspace of L₂(mathbbR)L_2(hbox{ensuremath{mathbb{R}}}) containing all linear combinations of τ-translates of ϕ. In this paper we prove that every function f Î V_ftfin V_phi^{bf tau} is uniquely determined by and stably reconstructed from the sample set L_ft(f)={ò_mathbbR f(t)[`(f(t-t_n))] dt}_{n Î mathbb Z}L_phi^{bf tau}(f)=Big{int_{hbox{ensuremath{mathbb{R}}}} f(t) overline{phi(t-tau_n)} dtBig}_{nin {{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 ϕ we provide an estimate to the corresponding error.