Skip to Main Content

Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Generalized matrix spectral factorization and quasi-tight framelets with a minimum number of generators
HTML articles powered by AMS MathViewer

by Chenzhe Diao and Bin Han HTML | PDF
Math. Comp. 89 (2020), 2867-2911 Request permission

Abstract:

As a generalization of orthonormal wavelets in $L_2({\mathbb {R}})$, tightframelets (also called tight wavelet frames) are of importance in wavelet analysis and applied sciences due to their many desirable properties in applications such as image processing and numerical algorithms. Tight framelets are often derived from particular refinable functions satisfying certain stringent conditions. Consequently, a large family of refinable functions cannot be used to construct tight framelets. This motivates us to introduce the notion of a quasi-tight framelet, which is a dual framelet but behaves almost like a tight framelet. It turns out that the study of quasi-tight framelets is intrinsically linked to the problem of the generalized matrix spectral factorization for matrices of Laurent polynomials. In this paper, we provide a systematic investigation on the generalized matrix spectral factorization problem and compactly supported quasi-tight framelets. As an application of our results on generalized matrix spectral factorization for matrices of Laurent polynomials, we prove in this paper that from any arbitrary compactly supported refinable function in $L_2({\mathbb {R}})$, we can always construct a compactly supported one-dimensional quasi-tight framelet having the minimum number of generators and the highest possible order of vanishing moments. Our proofs are constructive and supplemented by step-by-step algorithms. Several examples of quasi-tight framelets will be provided to illustrate the theoretical results and algorithms developed in this paper.
References
Similar Articles
Additional Information
  • Chenzhe Diao
  • Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1
  • Email: diao@ualberta.ca
  • Bin Han
  • Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1
  • MR Author ID: 610426
  • Email: bhan@ualberta.ca
  • Received by editor(s): June 9, 2018
  • Published electronically: June 5, 2020
  • Additional Notes: This research was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC)
  • © Copyright 2020 American Mathematical Society
  • Journal: Math. Comp. 89 (2020), 2867-2911
  • MSC (2010): Primary 42C40, 42C15, 47A68, 41A15, 65D07
  • DOI: https://doi.org/10.1090/mcom/3523
  • MathSciNet review: 4136550