Discrete Transforms and Orthogonal Polynomials of (Anti)symmetric Multivariate Sine Functions

Discrete Cosine and Sine Transforms

The discrete fractional cosine and sine transforms

This paper is concerned with the definitions of the discrete fractional cosine transform (DFRCT) and the discrete fractional sine transform (DFRST). The definitions of DFRCT and DFRST are based on the eigen decomposition of DCT and DST kernels. This is the same idea as that of the discrete fractional Fourier transform (DFRFT); the eigenvalue and eigenvector relationships between the DFRCT, DFRS...

MOPS: Multivariate orthogonal polynomials (symbolically)

In this paper we present a Maple library (MOPs) for computing Jack, Hermite, Laguerre, and Jacobi multivariate polynomials, as well as eigenvalue statistics for the Hermite, Laguerre, and Jacobi ensembles of Random Matrix theory. We also compute multivariate hypergeometric functions, and offer both symbolic and numerical evaluations for all these quantities. We prove that all algorithms are wel...

Discrete entropies of orthogonal polynomials

Let pn, n ∈ N, be the nth orthonormal polynomial on R, whose zeros are λ j , j = 1, . . . , n. Then for each j = 1, . . . , n, ~ Ψj def = ( Ψ1j , . . . ,Ψ 2 nj ) with Ψij = p 2 i−1(λ (n) j ) ( n−1 ∑ k=0 pk(λ (n) j ) ) −1 , i = 1, . . . , n, defines a discrete probability distribution. The Shannon entropy of the sequence {pn} is consequently defined as Sn,j def = − n ∑ i=1 Ψij log ( Ψij ) . In t...

Multivariate Orthogonal Polynomials and Operator Theory

The multivariate orthogonal polynomials are related to a family of commuting selfadjoint operators. The spectral theorem for these operators is used to prove that a polynomial sequence satisfying a vector-matrix form of the three-term relation is orthonormal with a determinate measure.