Some smooth compactly supported tight framelets associated to the quincunx matrix

We construct several families of tight wavelet frames in L2(R2) associated to the quincunx matrix. A couple of those families has five generators. Moreover, we construct a family of tight wavelet frames with three generators. Finally, we show families with only two generators. The generators have co...

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Bibliographic Details
Published in:Journal of mathematical analysis and applications 2016-05, Vol.437 (1), p.35-50
Main Authors: San Antolín, A., Zalik, R.A.
Format: Article
Language:English
Subjects:
Fourier transform
Quincunx matrix
Refinable function
Tight framelet
Unitary Extension Principle
Vanishing moments
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Summary:We construct several families of tight wavelet frames in L2(R2) associated to the quincunx matrix. A couple of those families has five generators. Moreover, we construct a family of tight wavelet frames with three generators. Finally, we show families with only two generators. The generators have compact support, any given degree of regularity, and any fixed number of vanishing moments. Our construction is made in Fourier space and involves some refinable functions, the Oblique Extension Principle and a slight generalization of a theorem of Lai and Stöckler. In addition, we will use well known results on construction of tight wavelet frames with two generators on R with the dyadic dilation. The refinable functions we use are constructed from the Daubechies low pass filters and are compactly supported. The main difference between these families is that while the refinable functions associated to the five generators have many symmetries, the refinable functions used in the construction of the others families are merely even.
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2015.12.022