A Hypergeometric Function Transform and Matrix-Valued Orthogonal Polynomials
- 158 Downloads
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.
KeywordsHypergeometric function transform Matrix-valued orthogonal polynomials Second-order differential operator Five-term operator Spectral analysis
Mathematics Subject Classification33C45 47B25 47E05
- 14.Koelink, E.: Spectral theory and special functions. In: Laredo Lectures on Orthogonal Polynomials and Special Functions. Adv. Theory Spec. Funct. Orthogonal Polynomials, pp. 45–84. Nova Sci., Hauppauge (2004) Google Scholar
- 16.Koornwinder, T.H.: Special orthogonal polynomial systems mapped onto each other by the Fourier–Jacobi transform. In: Orthogonal Polynomials and Applications, Bar-le-Duc, 1984. Lecture Notes in Math., vol. 1171, pp. 174–183. Springer, Berlin (1985) Google Scholar
- 17.Neretin, Yu.A.: Some continuous analogues of the expansion in Jacobi polynomials, and vector-valued orthogonal bases. Funkc. Anal. Prilozh. 39(2), 31–46, 94 (2005) (in Russian). Translation in Funct. Anal. Appl. 39(2), 106–119 (2005) Google Scholar