Event Details

FIRST TALK: 11:30h

Author : Rabia Aktas-Karaman, Ankara University, Turkey

Title : Fourier transforms of orthogonal polynomials on the cone and on the paraboloid

Abstract : The purpose of this paper is to obtain Fourier transforms of multivariate orthogonal polynomials on the cone such as Laguerre polynomials on the cone and Jacobi polynomials on the cone and to define two new families of multivariate orthogonal functions by using Parseval's identity. Also, the obtained results are expressed in terms of the continuous Hahn polynomials. Moreover, we present similar results for the orthogonal polynomials on the paraboloid.

References:
1. Koelink, H.T. On Jacobi and continuous Hahn polynomials, Proc. Amer. Math. Soc., 124(3), 887-898, 1996.
2. Aktas Karaman, R., Area, I. Fourier transforms of orthogonal polynomials on the cone, Numerical Algorithms, https://doi.org/10.1007/s11075-025-02112-x , 2025.
3. Güldogan Lekesiz, E., Aktas, R., Area, I. Fourier transform of the orthogonal polynomials on the unit ball and continuous Hahn polynomials, Axioms, 11(10), 558, 2022.
4. Cetin, H.O., Aktas Karaman, R. Fourier transforms of orthogonal structures on the paraboloid, submitted for publication

SECOND TALK:

Author : Miguel Piñar, Universidad de Granada

Title : A fully diagonalized spectral method on the unit ball

Abstract : Our main objective in this work is to demonstrate how orthogonal Sobolev polynomials emerge as a useful tool within the framework of spectral methods for boundary-value problems. The solution of a boundary-value problem for a stationary Schr\"odinger equation on the unit ball can be studied from a variational perspective. In this variational formulation, a Sobolev inner product naturally arises. As test functions, we consider the linear space of polynomials satisfying the boundary conditions on the sphere, and a basis of mutually orthogonal polynomials with respect to the Sobolev inner product is provided. The basis of the proposed method is provided in terms of spherical harmonics and univariate Sobolev orthogonal polynomials. The connection formula between these orthogonal Sobolev polynomials and classical orthogonal polynomials on the ball is established. Consequently, the Sobolev Fourier coefficients of a function satisfying the boundary value problem are recursively derived. Finally, numerical experiments were presented.

Location: Seminario 2, IMAG.
When: Thursday, September 11th 2025