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Frames in spaces with finite rate of innovation

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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.

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  1. Department of Mathematics, University of Central Florida, Orlando, FL, 32816, USA

    Qiyu Sun

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Sun, Q. Frames in spaces with finite rate of innovation. Adv Comput Math 28, 301–329 (2008). https://doi.org/10.1007/s10444-006-9021-4

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