Compactly supported wavelet bases for Sobolev spaces
نویسندگان
چکیده
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 wavelets ψjk := 2j/2ψ(2j · − k) (j, k ∈ Z) form a Riesz basis for L2(R). If, in addition, φ lies in the Sobolev space H(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 H(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.
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