Spherical and Cherednik-Opdam transforms of Jacobi-type polynomials
J.G.M. van der Klein (TU Delft - Electrical Engineering, Mathematics and Computer Science)
W.G.M. Groenevelt – Mentor (TU Delft - Analysis)
JMAM Van Neerven – Graduation committee member (TU Delft - Analysis)
F.M. De Oliveira Filho – Graduation committee member (TU Delft - Discrete Mathematics and Optimization)
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Abstract
The spherical transform maps the orthogonal basis of symmetric Jacobi-type polynomials to an orthogonal basis of (symmetric) Wilson polynomials. The spherical transform is closely related to the Cherednik-Opdam transform, as it is essentially its symmetric version. The symmetric Jacobi-type polynomials can be composed from the non-symmetric Jacobi-type polynomials. These relations, between the symmetric and non-symmetric theory, give an incentive to consider the Cherednik-Opdam transform of non-symmetric Jacobi-type polynomials. This work gives an overview of the symmetric theory about the spherical transform of Jacobi-type polynomials and lays down the groundwork for the Cherednik-Opdam transform of the non-symmetric Jacobi-type polynomials.