NDLI: Some Smooth Compactly Supported Tight Wavelet Frames with Vanishing Moments

Content Provider	Springer Nature Link
Author	San Antolín, A. Zalik, R. A.
Copyright Year	2015
Abstract	Let $$A \in \mathbb {R}^{d \times d}$$ , $$d \ge 1$$ be a dilation matrix with integer entries and $$\| \det A\|=2$$ . We construct several families of compactly supported Parseval framelets associated to A having any desired number of vanishing moments. The first family has a single generator and its construction is based on refinable functions associated to Daubechies low pass filters and a theorem of Bownik. For the construction of the second family we adapt methods employed by Chui and He and Petukhov for dyadic dilations to any dilation matrix A. The third family of Parseval framelets has the additional property that we can find members of that family having any desired degree of regularity. The number of generators is $$2^d+d$$ and its construction involves some compactly supported refinable functions, the Oblique Extension Principle and a slight generalization of a theorem of Lai and Stöckler. For the particular case $$d=2$$ and based on the previous construction, we present two families of compactly supported Parseval framelets with any desired number of vanishing moments and degree of regularity. None of these framelet families have been obtained by means of tensor products of lower-dimensional functions. One of the families has only two generators, whereas the other family has only three generators. Some of the generators associated with these constructions are even and therefore symmetric. All have even absolute values.
Ending Page	909
Page Count	23
Starting Page	887
File Format	PDF
ISSN	10695869
e-ISSN	15315851
Journal	Journal of Fourier Analysis and Applications
Issue Number	4
Volume Number	22
Language	English
Publisher	Springer US
Publisher Date	2015-11-02
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Abstract Harmonic Analysis Mathematical Methods in Physics Fourier Analysis Signal, Image and Speech Processing Refinable function Approximations and Expansions Fourier transform Dilation matrix Unitary Extension Principle Partial Differential Equations Tight framelet Wavelets and other special systems
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Analysis

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