An approximation of Daubechies wavelet matrices by perfect reconstruction filter banks with rational coefficients

It is described how the coefficients of Daubechies wavelet matrices can be approximated by rational numbers in such a way that the perfect reconstruction property of the filter bank be preserved exactly.     

On Approximate Spectral Factorization of Matrix Functions

It is proved that if positive definite matrix functions (i.e. matrix spectral densities) S _n , n=1,2,… , are convergent in the L ₁-norm, ||S_n-S||_L1? 0\|S_{n}-S\|_{L_{1}}\to 0, and ò₀^2plogdetS_n(e^iq) dq?ò₀^2plogdetS(e^iq) dq\int_{0}^{2\pi}\log \mathop{\mathrm{det}}S_{n}(e^{i\theta})\,d\theta\to\int_{0}^{2\pi}\log \mathop{\mathrm{det}}S(e^{i\theta})\,d\theta, then the corresponding (canonical) spectral factors are convergent in L ₂, ||S⁺_n-S⁺||_L2? 0\|S^{+}_{n}-S^{+}\|_{L_{2}}\to 0. The formulated logarithmic condition is easily seen to be necessary for the latter convergence to take place.     

On Compact Wavelet Matrices of Rank m and of Order and Degree N

A new parametrization (one-to-one onto map) of compact wavelet matrices of rank $m$ and of order and degree $N$ is proposed in terms of coordinates in the Euclidian space $\mathbb {C}^{(m-1)N}$ . The developed method depends on Wiener–Hopf factorization of corresponding unitary matrix functions and allows to construct compact wavelet matrices efficiently. Some applications of the proposed method are discussed.     

A Simple Proof of the Matrix-Valued Fejér-Riesz Theorem

A very short proof of the Fejér-Riesz lemma is presented in the matrix case.      

On Factorization and Partial Indices of Unitary Matrix-Functions of One Class

An effective factorization and partial indices are found for a class of unitary matrix functions.     

Matrix spectral factorization and wavelets

In this paper, recently published results on matrix spectral factorization is reviewed, and their connection to wavelet matrices is revealed.