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A new proof of a Paley-Wiener type theorem for the Jacobi transform. (English) Zbl 0303.42022

MSC:
42A38Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
44A15Special transforms (Legendre, Hilbert, etc.)
26A33Fractional derivatives and integrals (real functions)
33C05Classical hypergeometric functions, 2 F 1
References:
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[2]Braaksma, B. L. J., andMeulenbeld, B., Integral transforms with generalized Legendre functions as kernels.Compositio Math.,18 (1967), 235–287.
[3]Chébli, H., Sur la positivité des opératerurs de ”translation généralisée” associés à un opérateur de Sturm-Liouville sur [0, .C. R. Acad. Sci. Paris, A,275 (1972), 601–604.
[4]Chébli, H., Positivité des opérateurs de translation généralisée associés à un opérateur de Sturm-Liouville sur ]0, .Séminaire de Théorie Spectrale, année 1972–73 Institut de Recherche Mathématique Avancée, Strasbourg.
[5]Chébli, H., Sur un théorème de Paley-Wiener associé à la decomposition spectrale d’un opérateur de Sturm-Liouville sur ]0, .J. Functional Anal.,17 (1974), 447–461. · Zbl 0288.47040 · doi:10.1016/0022-1236(74)90052-4
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[10]Flensted-Jensen, M., Spherical functions on rank one symmetric spaces and generalizations.Proceedings Symposia Pure Mathematics, Vol. 26, American Mathematical Society, Providence (R. I.), 1973.
[11]Flensted-Jensen, M., The spehrical functions on the universal covering ofSU(n,1)/SU(n).Københavns Universitet Mat. Institut Preprint Series, (1973), No. 1.
[12]Flensted-Jensen, M. andKoornwinder, T. H., The convolution structure for Jacobi function expansions.Ark. Mat.,11 (1973), 245–262. · Zbl 0267.42009 · doi:10.1007/BF02388521
[13]Flensted-Jensen, M. andRagozin, D. L., Spherical functions are Fourier transforms ofL 1-functions.Ann. Sci. Ecole Norm. Sup. (4),6 (1973), 457–458.
[14]Gasper, G., Formulas of the Dirichlet-Mehler type.Prepublication (1973).
[15]Helgason, S., An analogue of the Paley-Wiener theorem for the Fourier transform on certain symmetric spaces.Math. Ann.,165, (1966), 297–308. · Zbl 0178.17101 · doi:10.1007/BF01344014
[16]Helgason, S., A duality for symmetric spaces with applications to group representations.Advances in Math.,5 (1970), 1–154. · Zbl 0209.25403 · doi:10.1016/0001-8708(70)90037-X
[17]Hörmander, L.,Linear partial differential operators. Springer-Verlag, Berlin, 1963.
[18]Koornwinder, T. H., The addition formula for Jacobi polynomials, I. Summary of results.Nederl. Akad. Wetensch. Proc. A,75=Indag. Math.,34 (1972), 188–191.
[19]Koornwinder, T. H., Jacobi polynomials, II. An analytic proof of the product formula.SIAM J. Math. Anal.,5 (1974), 125–137. · Zbl 0269.33015 · doi:10.1137/0505014
[20]Mehler, F. G., Ueber eine mit den Kugel und Cylinderfunctionen verwnadte Function und ihre. Anwendung in der Theorie der Elektricitätsvertheilung.Math. Ann.,18 (1881), 161–194. · doi:10.1007/BF01445847
[21]Olevskiį, M. N., On the representation of an arbitrary function in the form of an integral with a kernel containing a hypergeometric function.Dokl. Akad. Nauk. SSSR.,69 (1949), 11–14 (Russian).
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