Frames in spaces with finite rate of innovation

Abstract

Signals with finite rate of innovation are those signals having finite degrees of freedom per unit of time that specify them. In this paper, we introduce a prototypical space \(V_q(\Phi, \Lambda)\) modeling signals with finite rate of innovation, such as stream of (different) pulses found in GPS applications, cellular radio and ultra wide-band communication. In particular, the space \(V_q(\Phi, \Lambda)\) is generated by a family of well-localized molecules \(\Phi\) of similar size located on a relatively separated set \(\Lambda\) using \(\ell^q\) coefficients, and hence is locally finitely generated. Moreover that space \(V_q(\Phi, \Lambda)\) includes finitely generated shift-invariant spaces, spaces of non-uniform splines, and the twisted shift-invariant space in Gabor (Wilson) system as its special cases. Use the well-localization property of the generator \(\Phi\), we show that if the generator \(\Phi\) is a frame for the space \(V_2(\Phi, \Lambda)\) and has polynomial (sub-exponential) decay, then its canonical dual (tight) frame has the same polynomial (sub-exponential) decay. We apply the above result about the canonical dual frame to the study of the Banach frame property of the generator \(\Phi\) for the space \(V_q(\Phi, \Lambda)\) with \(q\ne 2\), and of the polynomial (sub-exponential) decay property of the mask associated with a refinable function that has polynomial (sub-exponential) decay.