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Frames and Stable Bases for Shift-Invariant Subspaces of L2(ℝd)

Published online by Cambridge University Press:  20 November 2018

Amos Ron  and
Zuowei Shen
Amos Ron
Affiliation:
Computer Science Department, University of Wisconsin-Madison, 1210 West Dayton Street, Madison, Wisconsin 53706, U.S.A. e-mail: amos@cs.wisc.edu
Zuowei Shen
Affiliation:
Department of Mathematics, National University of Singapore, 10 Kent Ridge Crescent, Singapore 0511 e-mail: matzuows@leonis.nus.sg
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Abstract

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Let X be a countable fundamental set in a Hilbert space H, and let T be the operator Whenever T is well-defined and bounded, X is said to be a Bessel sequence. If, in addition, ran T is closed, then X is a frame. Finally, a frame whose corresponding T is injective is a stable basis (also known as a Riesz basis). This paper considers the above three properties for subspaces H of L2(ℝd), and for sets X of the form with Φ either a singleton, a finite set, or, more generally, a countable set. The analysis is performed on the Fourier domain, where the two operators TT* and T* T are decomposed into a collection of simpler "fiber" operators. The main theme of the entire analysis is the characterization of each of the above three properties in terms of the analogous property of these simpler operators.

Keywords

42C15Riesz basesstable basesshift-invariant basesPSI spacesFSI spacesframesBessel sequenceswaveletssplines
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 47 , Issue 5 , 01 October 1995 , pp. 1051 - 1094
Copyright
Copyright © Canadian Mathematical Society 1995

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