Compactly supported wavelet bases for Sobolev spaces |
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Authors: | Rong-Qing Jia Jianzhong Wang Ding-Xuan Zhou |
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Institution: | a Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Canada T6G 2G1;b Department of Mathematics and Statistics, Sam Houston State University, Huntsville, TX 77341, USA;c Department of Mathematics, City University of Hong Kong, Hong Kong, China |
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Abstract: | In this paper we investigate compactly supported wavelet bases for Sobolev spaces. Starting with a pair of compactly supported refinable functions φ and
in
satisfying a very mild condition, we provide a general principle for constructing a wavelet ψ such that the wavelets ψjk:=2j/2ψ(2j·−k) (
) form a Riesz basis for
. If, in addition, φ lies in the Sobolev space
, then the derivatives 2j/2ψ(m)(2j·−k) (
) also form a Riesz basis for
. Consequently,
is a stable wavelet basis for the Sobolev space
. The pair of φ and
are not required to be biorthogonal or semi-orthogonal. In particular, φ and
can be a pair of B-splines. The added flexibility on φ and
allows us to construct wavelets with relatively small supports. |
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Keywords: | Wavelets Spline wavelets Multiresolution analysis Stable wavelet bases Riesz bases Sobolev spaces |
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