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Symmetric interpolatory dual wavelet frames


Author: A. V. Krivoshein
Translated by: the author
Original publication: Algebra i Analiz, tom 28 (2016), nomer 3.
Journal: St. Petersburg Math. J. 28 (2017), 323-343
MSC (2010): Primary 42C40; Secondary 65T60
DOI: https://doi.org/10.1090/spmj/1453
Published electronically: March 29, 2017
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Abstract: For any symmetry group $ {\mathcal H}$ and any appropriate matrix dilation (compatible with $ {\mathcal H}$), an explicit method is given for the construction of $ {\mathcal H}$-symmetric interpolatory refinable masks that obey the sum rule of an arbitrary order $ n$. Moreover, a description of all such masks is obtained. This type of mask is the starting point for the construction of symmetric wavelets and interpolatory subdivision schemes preserving symmetry properties of the initial data. For any given $ {\mathcal H}$-symmetric interpolatory refinable mask, an explicit technique is suggested for the construction of dual wavelet frames such that the corresponding wavelet masks are mutually symmetric and have vanishing moments up to the order $ n$. For an Abelian symmetry group $ {\mathcal H}$, this technique is modified so that all the resulting wavelet masks have the $ {\mathcal H}$-symmetry property.


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Additional Information

A. V. Krivoshein
Affiliation: St. Petersburg State University, Universitetskaya nab. 7/9, 199034 St. Petersburg, Russia
Email: a.krivoshein@spbu.ru, KrivosheinAV@gmail.com

DOI: https://doi.org/10.1090/spmj/1453
Keywords: Dual wavelet frames, matrix extension principle, interpolatory mask, symmetry group, subdivision schemes
Received by editor(s): September 10, 2015
Published electronically: March 29, 2017
Additional Notes: Supported by Saint-Petersburg State University (research grant no. 9.38.198.2015) and by RFBR (project no. 15-01-05796 a).
Article copyright: © Copyright 2017 American Mathematical Society