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A Hypergeometric Function Transform and Matrix-Valued Orthogonal Polynomials

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Abstract

The spectral decomposition for an explicit second-order differential operator T is determined. The spectrum consists of a continuous part with multiplicity two, a continuous part with multiplicity one, and a finite discrete part with multiplicity one. The spectral analysis gives rise to a generalized Fourier transform with an explicit hypergeometric function as a kernel. Using Jacobi polynomials, the operator T can also be realized as a five-diagonal operator, leading to orthogonality relations for 2×2-matrix-valued polynomials. These matrix-valued polynomials can be considered as matrix-valued generalizations of Wilson polynomials.

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Authors and Affiliations

  1. DIAM, Technische Universiteit Delft, PO Box 5031, 2600 GA, Delft, The Netherlands

    Wolter Groenevelt

  2. IMAPP, FNWI, Radboud Universiteit Nijmegen, Heyendaalseweg 135, 6525 AJ, Nijmegen, The Netherlands

    Erik Koelink

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  1. Wolter Groenevelt
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Correspondence to Wolter Groenevelt.

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Communicated by Tom H. Koornwinder.

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Groenevelt, W., Koelink, E. A Hypergeometric Function Transform and Matrix-Valued Orthogonal Polynomials. Constr Approx 38, 277–309 (2013). https://doi.org/10.1007/s00365-013-9207-1

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  • DOI: https://doi.org/10.1007/s00365-013-9207-1

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