An algorithm for matrix extension and wavelet construction

Lawton, W.; Lee, S.; Shen, Zuowei

doi:10.1090/S0025-5718-96-00714-4

An algorithm for matrix extension and wavelet construction
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by W. Lawton, S. L. Lee and Zuowei Shen PDF

Math. Comp. 65 (1996), 723-737 Request permission

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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