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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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
Keywords
- Hypergeometric function transform
- Matrix-valued orthogonal polynomials
- Second-order differential operator
- Five-term operator
- Spectral analysis