#### Title

Frames in spaces with finite rate of innovation

#### Abbreviated Journal Title

Adv. Comput. Math.

#### Keywords

frame; Banach frame; localized frame; signals with finite rate of; innovation; space of homogenous type; matrix algebra; refinable; function; wavelets; SHIFT-INVARIANT SPACES; TIGHT WAVELET FRAMES; MULTIRESOLUTION ANALYSIS; INFINITE MATRICES; HOMOGENEOUS TYPE; P-FRAMES; RECONSTRUCTION; SIGNALS; DECOMPOSITIONS; LOCALIZATION; Mathematics, Applied

#### 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 wideband communication. In particular, the space V-q( Phi,Lambda) is enerated by a family of well- localized molecules Phi of similar size located on a relatively separated set Lambda using l(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 not equal 2, and of the polynomial ( sub- exponential) decay property of the mask associated with a refinable function that has polynomial ( sub- exponential) decay.

#### Journal Title

Advances in Computational Mathematics

#### Volume

28

#### Issue/Number

4

#### Publication Date

1-1-2008

#### Document Type

Article

#### Language

English

#### First Page

301

#### Last Page

329

#### WOS Identifier

#### ISSN

1019-7168

#### Recommended Citation

"Frames in spaces with finite rate of innovation" (2008). *Faculty Bibliography 2000s*. 1031.

https://stars.library.ucf.edu/facultybib2000/1031

## Comments

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