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

 

An algorithm for
matrix extension and wavelet construction


Authors: W. Lawton, S. L. Lee and Zuowei Shen
Journal: Math. Comp. 65 (1996), 723-737
MSC (1991): Primary 41A15, 41A30, 15A54, 65D07, 65F30
MathSciNet review: 1333319
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Abstract | References | Similar Articles | Additional Information

Abstract: This paper gives a practical method of extending an $n\times r$ matrix $P(z)$, $r \leq n $, with Laurent polynomial entries in one complex variable $z$, to a square matrix also with Laurent polynomial entries. If $P(z)$ has orthonormal columns when $z$ is restricted to the torus $\mathbf{T}$, it can be extended to a paraunitary matrix. If $P(z)$ has rank $r$ for each $z\in \mathbf{T}$, it can be extended to a matrix with nonvanishing determinant on $\mathbf{T}$. The method is easily implemented in the computer. It is applied to the construction of compactly supported wavelets and prewavelets from multiresolutions generated by several univariate scaling functions with an arbitrary dilation parameter.


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Additional Information

W. Lawton
Affiliation: Institute of Systems Science, National University of Singapore, Heng Mui Keng Terrace, Kent Ridge, Singapore 0511
Email: wlawton@iss.nus.sg

S. L. Lee
Affiliation: Department of Mathematics, National University of Singapore, 10 Kent Ridge Crescent, Singapore 0511
Email: matleesl@math.nus.sg

Zuowei Shen
Email: matzuows@math.nus.sg

DOI: http://dx.doi.org/10.1090/S0025-5718-96-00714-4
PII: S 0025-5718(96)00714-4
Keywords: Wavelets, prewavelets, matrix extension, splines
Received by editor(s): February 15, 1994
Received by editor(s) in revised form: October 4, 1994, and January 30, 1995
Article copyright: © Copyright 1996 American Mathematical Society