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Annihilator, Completeness and Convergence of Wavelet System

Published online by Cambridge University Press:  11 January 2016

Kwok-Pun Ho
Kwok-Pun Ho*
Affiliation:
Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China, makho@ust.hk
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Abstract

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We show that if is a frame and {ψＱ}Ｑ∈Q ∈ ∩ Mα(ℝn) is its dual frame (for the definition of Mα(ℝn), see Definition 2.1), where Q is the collection of dyadic cubes, then for any f ∈ S′(ℝn), there exists a sequence of polynomials, PL,L′,L″, such that

(0.1)

in the topology of S′(ℝn), where δ(i) = max(2i, 1). We prove this result by explicitly constructing the polynomials PL,L′,L″. Furthermore, using the above result, we assert that the linear span of the one-dimensional wavelet system is dense in a function space if and only if the dual space of this function space has an trivial intersection with the set of polynomials. This is proved by using the annihilator of the one-dimensional wavelet system.

Keywords

42B3542C1542C4046E3547B38
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 188 , December 2007 , pp. 59 - 105
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2007

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