Two-dimensional symmetric and antisymmetric generalizations of exponential and cosine functions

Properties of the four families of recently introduced special functions of two real variables, denoted here by $E^\pm$, and $\cos^\pm$, are studied. The superscripts $^+$ and $^-$ refer to the symmetric and antisymmetric functions respectively. The functions are considered in all details required for their exploitation in Fourier expansions of digital data, sampled on square grids of any density and for general position of the grid in the real plane relative to the lattice defined by the underlying group theory. Quality of continuous interpolation, resulting from the discrete expansions, is studied, exemplified and compared for some model functions.Comment: 22 pages, 10 figure

On q-orthogonal polynomials, dual to little and big q-Jacobi polynomials

This paper studies properties of q-Jacobi polynomials and their duals by means of operators of the discrete series representations for the quantum algebra U_q(su_{1,1}). Spectrum and eigenfunctions of these operators are found explicitly. These eigenfunctions, when normalized, form an orthogonal basis in the representation space. The initial U_q(su_{1,1})-basis and the bases of these eigenfunctions are interconnected by matrices, whose entries are expressed in terms of little and big q-Jacobi polynomials. The orthogonality by rows in these unitary connection matrices leads to the orthogonality relations for little and big q-Jacobi polynomials. The orthogonality by columns in the connection matrices leads to an explicit form of orthogonality relations on the countable set of points for {}_3\phi_2 and {}_3\phi_1 polynomials, which are dual to big and little q-Jacobi polynomials, respectively. The orthogonality measure for the dual little q-Jacobi polynomials proves to be extremal, whereas the measure for the dual big q-Jacobi polynomials is not extremal.Comment: 26 pages, LaTeX, the exposition is slightly improved and some additional references have been adde