# Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.78.

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## The structure of balanced multivariate biorthogonal multiwavelets and dual multiframeletsHTML articles powered by AMS MathViewer

by Bin Han
Math. Comp. 79 (2010), 917-951 Request permission

## Abstract:

Multiwavelets and multiframelets are of interest in several applications such as numerical algorithms and signal processing, due to their desirable properties such as high smoothness and vanishing moments with relatively small supports of their generating functions and masks. In order to process and represent vector-valued discrete data efficiently and sparsely by a multiwavelet transform, a multiwavelet has to be prefiltered or balanced. Balanced orthonormal univariate multiwavelets and multivariate biorthogonal multiwavelets have been studied and constructed in the literature. Dual multiframelets include (bi)orthogonal multiwavelets as special cases, but their fundamental prefiltering and balancing property has not yet been investigated in the literature. In this paper we shall study the balancing property of multivariate multiframelets from the point of view of the discrete multiframelet transform. This approach, to our best knowledge, has not been considered so far in the literature even for multiwavelets, but it reveals the essential structure of prefiltering and the balancing property of multiwavelets and multiframelets. We prove that every biorthogonal multiwavelet can be prefiltered with the balancing order matching the order of its vanishing moments; that is, from every given compactly supported multivariate biorthogonal multiwavelet, one can always build another (essentially equivalent) compactly supported biorthogonal multiwavelets with its balancing order matching the order of the vanishing moments of the original one. More generally, we show that if a dual multiframelet can be prefiltered, then it can be equivalently transformed into a balanced dual multiframelet with the same balancing order. However, we notice that most available dual multiframelets in the literature cannot be simply prefiltered with its balancing order matching its order of vanishing moments and they must be designed to possess high balancing orders. The key ingredient of our approach is based on investigating some properties of the subdivision and transition operators acting on discrete vector polynomial sequences, which play a central role in a discrete multiframelet transform and are of interest in their own right. We also establish a new canonical form of a matrix mask, which greatly facilitates the investigation and construction of multiwavelets and multiframelets. In this paper, we obtain a complete criterion and the essential structure for balanced or prefiltered dual multiframelets in the most general setting. Our investigation of the balancing property of a multiframelet deepens our understanding of the multiframelet transform in signal processing and scientific computation.
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• Bin Han
• Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1
• MR Author ID: 610426
• Email: bhan@math.ualberta.ca
• Received by editor(s): July 21, 2008
• Received by editor(s) in revised form: April 22, 2009
• Published electronically: December 10, 2009
• Additional Notes: This research was supported in part by NSERC Canada under Grant RGP 228051