A simple device, based on the factorization of invertible matrix polynomials, enabling to identify the possibility of fast implementation of linear transforms is presented. Its applicability is demonstrated in the case of Hadamard matrices and their generalization, Hadamard matrix polynomials.
Discrete wavelets are viewed as linear algebraic transforms given by banded orthogonal matrices which can be built up from small matrix blocks satisfying certain conditions. A generalization of the finite support Daubechies wavelets is discussed and some special cases promising more rapid signal reduction are derived.
We present algorithms for the determination of polynomials orthogonal with respect to a positive weight function multiplied by a polynomial with simple roots inside the interval of integration. We apply these algorithms to search for and calculate all possible sequences of imbedded quadratures of maximal polynomials order of precision for the generalized Laguerre and Hermite weight functions.
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