arXiv subjects

Tom Koornwinder

Publications and source records attributed to Tom Koornwinder.

3 recordsLinked to original sources

arXiv2024

Memories of Ian G. Macdonald

This is a slightly edited translation of a paper in Dutch which appeared in Nieuw Archief voor Wiskunde (5) 25 (2024), No.2, 87-90 on the occasion of I.G. Macdonald's death in 2023, and aimed at a very broad mathematical audience. First we review some of Macdonald's most important older results. Then we focus on the period 1985-1995 when Macdonald often visited the Netherlands and there was much interaction between his work, notably the Macdonald polynomials, and the work by the authors. We end with some glimpses about Macdonald as a person.

math.HO↗

arXiv2018

Jacobi Polynomials, Bernstein-type Inequalities and Dispersion Estimates for the Discrete Laguerre Operator

The present paper is about Bernstein-type estimates for Jacobi polynomials and their applications to various branches in mathematics. This is an old topic but we want to add a new wrinkle by establishing some intriguing connections with dispersive estimates for a certain class of Schrödinger equations whose Hamiltonian is given by the generalized Laguerre operator. More precisely, we show that dispersive estimates for the Schrödinger equation associated with the generalized Laguerre operator are connected with Bernstein-type inequalities for Jacobi polynomials. We use known uniform estimates for Jacobi polynomials to establish some new dispersive estimates. In turn, the optimal dispersive decay estimates lead to new Bernstein-type inequalities.

math.CA↗

arXiv2012

Limit transition between hypergeometric functions of type BC and type A

Let $F_{BC}(λ,k;t)$ be the Heckman-Opdam hypergeometric function of type BC with multiplicities $k=(k_1,k_2,k_3)$ and weighted half sum $ρ(k)$ of positive roots. We prove that $F_{BC}(λ+ρ(k),k;t)$ converges for $k_1+k_2\to\infty$ and $k_1/k_2\to \infty$ to a function of type A for $t\in\b R^n$ and $λ\in\b C^n$. This limit is obtained from a corresponding result for Jacobi polynomials of type BC, which is proven for a slightly more general limit behavior of the multiplicities, using an explicit representation of Jacobi polynomials in terms of Jack polynomials. Our limits include limit transitions for the spherical functions of non-compact Grassmann manifolds over one of the fields $\mathbb F= \mathbb R, \mathbb C, \mathbb H$ when the rank is fixed and the dimension tends to infinity. The limit functions turn out to be exactly the spherical functions of the corresponding infinite dimensional Grassmann manifold in the sense of Olshanski.

math.CA↗