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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Algorithms for matrix extension and orthogonal wavelet filter banks over algebraic number fields


Authors: Bin Han and Xiaosheng Zhuang
Journal: Math. Comp. 82 (2013), 459-490
MSC (2010): Primary 42C40, 41A05; Secondary 42C15, 65T60
Published electronically: May 24, 2012
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Abstract: As a finite dimensional linear space over the rational number field $ \mathbb{Q}$, an algebraic number field is of particular importance and interest in mathematics and engineering. Algorithms using algebraic number fields can be efficiently implemented involving only integer arithmetics. We observe that all known finitely supported orthogonal wavelet low-pass filters in the literature have coefficients coming from an algebraic number field. Therefore, it is of theoretical and practical interest for us to consider orthogonal wavelet filter banks over algebraic number fields. In this paper, we formulate the matrix extension problem over any general subfield of $ \mathbb{C}$ (including an algebraic number field as a special case), and we provide step-by-step algorithms to implement our main results. As an application, we obtain a satisfactory algorithm for constructing orthogonal wavelet filter banks over algebraic number fields. Several examples are provided to illustrate the algorithms proposed in this paper.


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Additional Information

Bin Han
Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1
Email: bhan@math.ualberta.ca

Xiaosheng Zhuang
Affiliation: Institut für Mathematik, Technische Universität, 10623 Berlin, Germany
Email: xzhuang@math.tu-berlin.de

DOI: http://dx.doi.org/10.1090/S0025-5718-2012-02618-4
PII: S 0025-5718(2012)02618-4
Keywords: Matrix extension, matrix factorization, algebraic number fields, symmetry, multiwavelets, orthogonal wavelet filter banks, algebraic wavelet filters.
Received by editor(s): April 9, 2011
Received by editor(s) in revised form: September 7, 2011
Published electronically: May 24, 2012
Additional Notes: This research was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC Canada) under Grant RGP 228051.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.