A new factorization technique of the matrix mask of univariate refinable functions

Abstract

A univariate compactly supported refinable function φ can always be written as the convolution product Bk ∗ f , with Bk the B-spline of order k, f a compactly supported distribution, and k the approximation orders provided by the underlying shift-invariant space S(φ). Factorizations of univariate refinable vectors Φ were also studied and utilized in the literature. One of the by-products of this article is a rigorous analysis of that factorization notion, including, possibly, the first precise definition of that process. The main goal of this article is the introduction of a special factorization algorithm of refinable vectors that generalizes the scalar case as closely (and unexpectedly) as possible: the original vector Φ is shown to be ‘almost’ in the form Bk ∗ F , with F still compactly supported and refinable, and k the approximation order of S(Φ): ‘almost’ in the sense that Φ and Bk ∗ F differ at most in one entry. The algorithm guarantees F to retain the possible favorable properties of Φ, such as the stability of the shifts of Φ and/or the polynomiality of the mask symbol. At the same time, the theory and the algorithm are derived under relatively mild conditions and, in particular, apply to Φ whose shifts are not stable, as well as to refinable vectors which are not compactly supported. The usefulness of this specific factorization for the study of the smoothness of FSI wavelets (known also as ‘multiwavelets’ and ‘multiple wavelets’) is explained. The analysis invokes in an essential way the theory of finitely generated shift-invariant (FSI) spaces, and, in particular, the tool of superfunction theory. AMS (MOS) Subject Classifications: Primary 42C15, 42A85, Secondary 41A25, 46E35

DOI: 10.1007/PL00005424

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@article{PlonkaHoch2001ANF, title={A new factorization technique of the matrix mask of univariate refinable functions}, author={Gerlind Plonka-Hoch and Amos Ron}, journal={Numerische Mathematik}, year={2001}, volume={87}, pages={555-595} }