Compactly supported wavelet bases for Sobolev spaces

Research output: Journal Publications and Reviews (RGC: 21, 22, 62)21_Publication in refereed journalpeer-review

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Author(s)

  • Rong-Qing Jia
  • Jianzhong Wang
  • Ding-Xuan Zhou

Related Research Unit(s)

Detail(s)

Original languageEnglish
Pages (from-to)224-241
Journal / PublicationApplied and Computational Harmonic Analysis
Volume15
Issue number3
Publication statusPublished - Nov 2003

Abstract

In this paper we investigate compactly supported wavelet bases for Sobolev spaces. Starting with a pair of compactly supported refinable functions φ and φ̃ in L2(ℝ) satisfying a very mild condition, we provide a general principle for constructing a wavelet ψ such that the waveletsΨjk: = 2 j/2ψ(2j·- k) (j, k ∈ ℤ) form a Riesz basis for L2 (ℝ). If, in addition, φ lies in the Sobolev space Hm(ℝ), then the derivatives 2j/2ψ(m)(2j· -k) (j, k ∈ ℤ) also form a Riesz basis for L2(ℝ)- Consequently, {ψjk: j, k ∈ ℤ} is a stable wavelet basis for the Sobolev space Hm (ℝ). The pair of φ and φ̃ are not required to be biorthogonal or semi-orthogonal. In particular, φ and φ̃ can be a pair of B-splines. The added flexibility on φ and φ̃ allows us to construct wavelets with relatively small supports. © 2003 Published by Elsevier Inc.

Research Area(s)

  • Multiresolution analysis, Riesz bases, Sobolev spaces, Spline wavelets, Stable wavelet bases, Wavelets

Citation Format(s)

Compactly supported wavelet bases for Sobolev spaces. / Jia, Rong-Qing; Wang, Jianzhong; Zhou, Ding-Xuan.
In: Applied and Computational Harmonic Analysis, Vol. 15, No. 3, 11.2003, p. 224-241.

Research output: Journal Publications and Reviews (RGC: 21, 22, 62)21_Publication in refereed journalpeer-review