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Matrix Extension with Symmetry and Applications to Symmetric Orthonormal Complex M-wavelets

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Abstract

Matrix extension with symmetry is to find a unitary square matrix P of 2π-periodic trigonometric polynomials with symmetry such that the first row of P is a given row vector p of 2π-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 2π-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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Correspondence to Bin Han.

Additional information

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

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