Functions in Sampling Spaces

2005/05/09 by Cristina Blanco, Blanco, Cristina, Carlos Cabrelli +3
Mathematics · #39A10 #41A15 #42C40 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #math.CA #msc:39A10 #msc:41A15 #msc:42C40

paper · pdf · doi:10.48550/arxiv.math/0505147

20 pages, to appear in "The Journal of Sampling Theory in Signal and Image Processing", typos corrected

arxiv created 2005/07/21 · arxiv updated 2009/12/01

Abstract

Sampling theory in spaces other than the space of band-limited functions has recently received considerable attention. This is in part because the band-limitedness assumption is not very realistic in many applications. In addition, band-limited functions have very slow decay which translates in poor reconstruction. In this article we study the sampling problem in general shift invariant spaces. We characterize the functions in these spaces and provide necessary and sufficient conditions for a function in L²(\R) to belong to a sampling space. Furthermore we obtain decompositions of a sampling space in sampling subspaces. These decompositions are related with determining sets. Some examples are provided.

Related

• Shift-invariant sampling in two-sided small Fock spaces
• Relevant sampling in finitely generated shift-invariant spaces
• Sampling formulas involving differences in shift-invariant subspaces: a unified approach
• Sampling theorems with derivatives in shift-invariant spaces generated by periodic exponential B-splines
• Reconstruction of functions in principal shift-invariant subspaces of mixed Lebesgue spaces