Journal Article10.1007/S11425-007-0056-X
Biorthogonal multiple wavelets generated by vector refinement equation
15
TL;DR: This paper provides a general method for the construction of compactly supported biorthogonal multiple wavelets by refinable function vectors which are the solutions of vector refinement equations of the form of a finitely supported sequence of r × r matrices called the refinement mask.
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Abstract: Biorthogonal multiple wavelets are generated from refinable function vectors by using the multiresolution analysis. In this paper we provide a general method for the construction of compactly supported biorthogonal multiple wavelets by refinable function vectors which are the solutions of vector refinement equations of the form
$$\varphi (x) = \sum\limits_{\alpha \in \mathbb{Z}^s } {a(\alpha )\varphi (Mx - \alpha ), x \in \mathbb{R}^s } ,$$
where the vector of functions ϕ = (ϕ
1, …, ϕ
r)T is in
$$(L_2 (\mathbb{R}^s ))^r ,a = :(a(\alpha ))_{\alpha \in \mathbb{Z}^s } $$
is a finitely supported sequence of r × r matrices called the refinement mask, and M is an s × s integer matrix such that lim
n→∞
M
−n
= 0. Our characterizations are in the general setting and the main results of this paper are the real extensions of some known results.
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Citations
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Vector refinement equations with infinitely supported masks
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TL;DR: The purpose of this paper is to provide a necessary and sufficient condition to guarantee the sequence (Q"a^nf)"n"="1","2","... to converge in L"2-norm", and characterize biorthogonal multiple refinable functions.
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Riesz multiwavelet bases generated by vector refinement equation
Song Li,ZhiSong Liu +1 more
TL;DR: In this article, a compactly supported Riesz multi-wavelet sequence for L2(ℝs) matrices has been characterized, where the bracket product [f, g] of two vectors of functions f, g in (L2(β, g) is defined.
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Subdivision schemes with polynomially decaying masks
TL;DR: This paper gives a complete characterization of convergence of the sequence of vector refinement equations with polynomially decaying masks and a general dilation matrix in L2-norm and obtains a characterization of smoothness of solutions of refinement equation mentioned above for the case r = 1.
3
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