Generalized ShiftInvariant Systems

Authors: Amos Ron Zuowei Shen

Publish Date: 2004/10/08

Volume: 22, Issue: 1, Pages: 1-45

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Abstract

A countable collection X of functions in L 2mboxfootnotesizebf R is said to be a Bessel system if the associated analysis operator txsXL 2mboxsmallbf Rdto ell 2X fmapsto inprofx xin X is welldefined and bounded A Bessel system is a fundamental frame if txsX is injective and its range is closed This paper considers the above two properties for a generalized shiftinvariant system X By definition such a system has the form X=bigcup jin J Y j where each Y j is a shiftinvariant system ie is comprised of lattice translates of some functions and J is a countable or finite index set The definition is general enough to include wavelet systems shiftinvariant systems Gabor systems and many variations of wavelet systems such as quasiaffine ones and nonstationary ones The main theme of this paper is the “fiberization” of txsX which allows one to study the frame and Bessel properties of X via the spectral properties of a collection of finiteorder Hermitian nonnegative matrices

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