The aim of this paper is to construct sup-exponentially localized kernels and frames in the context of classical orthogonal expansions, namely, expansions in Jacobi polynomials, spherical harmonics, orthogonal polynomials on the ball and simplex, and Hermite and Laguerre functions.

We establish asymptotics for Christoffel functions, and universality limits, associated with multivariate orthogonal polynomials, on the boundary of the unit ball in Rd. Orthogonal Polynomials, Universality Limits, Christoffel functions. 42C05, 42C99, 42B05, 60B20

A set of orthogonal polynomials with 8 independent “q’s” is defined which generalizes the Laguerre polynomials. The moments of the measure for these polynomials are the generating functions for permutations according to eight different statistics. Specializing these statistics gives many other well-known sets of combinatorial objects and relevant statistics. The specializations are studied, wit...

We investigate monotonicity properties of extremal zeros of orthogonal polynomials depending on a parameter. Using a functional analysis method we prove the monotonicity of extreme zeros of associated Jacobi, associated Gegenbauer and q-Meixner-Pollaczek polynomials. We show how these results can be applied to prove interlacing of zeros of orthogonal polynomials with shifted parameters and to d...

In this paper we study special systems of orthogonal polynomials on the unit circle. More precisely, with a view to the recurrence relations fulfilled by these orthogonal systems, we analyze a link of non-negative arithmetic to harmonic sequences as a main subject. Here, arithmetic sequences appear as coefficients of orthogonal polynomials and harmonic sequences as corresponding Szegő parameters.

Let μ be a probability measure on the real line with finite moments of all orders. Apply the Gram-Schmidt orthogonalization process to the system {1, x, x, . . . , xn, . . . } to get orthogonal polynomials Pn(x), n ≥ 0, which have leading coefficient 1 and satisfy (x − αn)Pn(x) = Pn+1(x) + ωnPn−1(x). In general it is almost impossible to use this process to compute the explicit form of these po...