## Abstract

Matrix extension with symmetry is to find a unitary square matrix **P** of 2*π*-periodic trigonometric polynomials with symmetry such that the first row of **P** is a given row vector **p** of 2*π*-periodic trigonometric polynomials with symmetry satisfying
\(\mathbf {p}\overline{\mathbf{p}}^{T}=1\)
. Matrix extension plays a fundamental role in many areas such as electronic engineering, system sciences, wavelet analysis, and applied mathematics. In this paper, we shall solve matrix extension with symmetry by developing a step-by-step simple algorithm to derive a desired square matrix **P** from a given row vector **p** of 2*π*-periodic trigonometric polynomials with complex coefficients and symmetry. As an application of our algorithm for matrix extension with symmetry, for any dilation factor *M*, we shall present two families of compactly supported symmetric orthonormal complex *M*-wavelets with arbitrarily high vanishing moments. Wavelets in the first family have the shortest possible supports with respect to their orders of vanishing moments; their existence relies on the establishment of nonnegativity on the real line of certain associated polynomials. Wavelets in the second family have increasing orders of linear-phase moments and vanishing moments, which are desirable properties in numerical algorithms.

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Communicated by Stephan Dahlke.

*Dedicated to Professor Wolfgang Dahmen on the occasion of his 60th birthday.*

Research supported in part by NSERC Canada under Grant RGP 228051.

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Han, B. Matrix Extension with Symmetry and Applications to Symmetric Orthonormal Complex *M*-wavelets.
*J Fourier Anal Appl* **15**, 684–705 (2009). https://doi.org/10.1007/s00041-009-9084-y

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DOI: https://doi.org/10.1007/s00041-009-9084-y

### Keywords

- Matrix extension with symmetry
- Trigonometric polynomials
- Symmetry
- Orthonormal complex
*M*-wavelets - General dilation factors
- Linear-phase moments
- Vanishing moments