Title

Frames In Spaces With Finite Rate Of Innovation

Keywords

Banach frame; Frame; Localized frame; Matrix algebra; Refinable function; Signals with finite rate of innovation; Space of homogenous type; Wavelets

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 Vq(Φ Λ) 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 Vq(Φ Λ) is generated by a family of well-localized molecules Φ of similar size located on a relatively separated set Λ using ℓq coefficients, and hence is locally finitely generated. Moreover that space Vq(Φ Λ) 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 Φ, we show that if the generator Φ is a frame for the space V2(Φ Λ) 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 Φ for the space Vq(Φ Λ) with q ≠ 2, and of the polynomial (sub-exponential) decay property of the mask associated with a refinable function that has polynomial (sub-exponential) decay. © 2006 Springer Science+Business Media B.V.

Publication Date

5-1-2008

Publication Title

Advances in Computational Mathematics

Volume

28

Issue

4

Number of Pages

301-329

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.1007/s10444-006-9021-4

Socpus ID

41549127940 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/41549127940

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