The convolution structure for Jacobi function expansions. (English) Zbl 0267.42009


42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
44A15 Special integral transforms (Legendre, Hilbert, etc.)
44A35 Convolution as an integral transform
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
43A90 Harmonic analysis and spherical functions
Full Text: DOI


[1] Askey, R., Jacobi polynomials, I. New proofs of Koornwinder’s Laplace type integral representation and Bateman’s bilinear sum. To appear inSIAM J. Math. Anal. · Zbl 0242.33019
[2] Chébli, H., Sur la positivité des opérateurs de “translation généralisée{” associés à un operateur de Sturm-Liouville sur [0, .C. R. Acad. Sci. Paris 275 (1972), 601–604.} · Zbl 0241.47032
[3] Erdélyi, A., et al.,Higher transcendental functions vol. I. McGraw-Hill, New York, 1953. · Zbl 0051.30303
[4] Flensted-Jensen, M., Paley-Wiener type theorems for a differential operator connected with symmetric spaces.Arkiv för Matematik 10 (1972), 143–162. · Zbl 0233.42012
[5] Gasper, G., Positivity and the convolution structure for Jacobi series.Ann. of Math. 93 (1971), 112–118. · Zbl 0208.08101
[6] –, Banach algebras for Jacobi series and positivity of a kernel.Ann. of Math. 95 (1972), 261–280. · Zbl 0236.33013
[7] Helgason, S.,Differential geometry and symmetric spaces. Academic Press, New York, 1962. · Zbl 0111.18101
[8] Koornwinder, T. H., The addition formula for Jacobi polynomials, I. Summary of results.Nederl. Akad. Wetensch. Proc. Ser A 75=Indag. Math. 34 (1972), 188–191. · Zbl 0247.33017
[9] Koornwinder, T. H. Jacobi polyomials, II. An analytic proof of the product formula. To appear inSIAM J. Math. Anal. · Zbl 0242.33020
[10] Kunze, R. A. &Stein, E. M., Uniformly bounded representations and harmonic analysis of the 2{\(\times\)}2 real unimodular group.Amer. J. Math. 82 (1960), 1–62. · Zbl 0156.37104
[11] Tréves, F.,Topological vector spaces, distributions and kernels. Academic Press, New York, 1967. · Zbl 0171.10402
[12] Weinberger, A., A maximum property of the Cauchy problem.Ann. of Math. 64 (1956), 505–512. · Zbl 0072.10003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.