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Zbl 0838.42016
Ron, Amos; Shen, Zuowei
Frames and stable bases for shift-invariant subspaces of $L\sb 2(\bbfR\sp d)$. (English)
[J] Can. J. Math. 47, No.5, 1051-1094 (1995). ISSN 0008-414X; ISSN 1496-4279/e

Summary: Let $X$ be a countable fundamental set in a Hilbert space $H$, and let $T$ be the operator $$T: \ell_2 (X)\to H: c\mapsto \sum_{x\in X} c(x) x.$$ Whenever $T$ is well-defined and bounded, $X$ is said to be a Bessel sequence. If, in addition, $\text {ran } T$ 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). \par This paper considers the above three properties for subspaces $H$ of $L_2 (\bbfR^d)$, and for sets $X$ of the form $$X= \{\varphi (\cdot -\alpha): \varphi\in \Phi,\ \alpha\in \bbfZ^d\},$$ with $\Phi$ 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.

MSC 2000:: *42C15 Series and expansions in general function systems
47A15 Invariant subspaces of linear operators

Keywords: principal shift-invariant; finitely generated shift-invariant; shift invariant bases; wavelets; splines; PSI spaces; FSI spaces; frames; shift-invariant subspaces; Bessel sequence; stable basis; Riesz basis

Cited in: Zbl 1026.42031 Zbl 0986.46018 Zbl 0892.42017 Zbl 0891.42018

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