Matrix Extension with Symmetry and Applications to Symmetric Orthonormal Complex M-wavelets
Birkhäuser Boston · 2009
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 sat- isfying pp 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 symme- try. 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 desir- able properties in numerical algorithms.