Abstract:The notion of $K$-balancing was introduced a few years ago as a condition for the construction of orthonormal scaling function vectors and multi-wavelets to ensure the property of preservation/annihilation of scalar-valued discrete polynomial data of order $K$ (or degree $K-1$), when decomposed by the corresponding matrix-valued low-pass/high-pass filters. While this condition is indeed precise, to the best of our knowledge only the proof for $K=1$ is known. In addition, the formulation of the $K$-balancing condition for $K\ge 2$ is so prohibitively difficult to satisfy that only a very few examples for $K=2$ and vector dimension 2 have been constructed in the open literature. The objective of this paper is to derive various characterizations of the $K$-balancing condition that include the polynomial preservation property as well as equivalent formulations that facilitate the development of methods for the construction purpose. These results are established in the general multivariate and bi-orthogonal settings for any $K\ge 1$.
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- Charles K. Chui
- Affiliation: Department of Mathematics & Computer Science, University of Missouri–St. Louis, St. Louis, Missouri 63121 and Department of Statistics, Stanford University, Stanford, California 94305
- Email: firstname.lastname@example.org
- Qingtang Jiang
- Affiliation: Department of Mathematics & Computer Science, University of Missouri–St. Louis, St. Louis, Missouri 63121
- Email: email@example.com
- Received by editor(s): May 18, 2003
- Received by editor(s) in revised form: January 9, 2004
- Published electronically: June 7, 2004
- Additional Notes: The first author was supported in part by NSF Grants #CCR-9988289 and #CCR-0098331, and ARO Grant #DAAD 19-00-1-0512
The second author was supported in part by a University of Missouri–St. Louis research award
- © Copyright 2004 American Mathematical Society
- Journal: Math. Comp. 74 (2005), 1323-1344
- MSC (2000): Primary 42C40, 65T60; Secondary 94A08
- DOI: https://doi.org/10.1090/S0025-5718-04-01681-3
- MathSciNet review: 2137005