A new proof of a Paley—Wiener type theorem for the Jacobi transform

Abstract

which generalizes the Mehler-Fok transform, was studied by Titchmarsh [23, w 17], Olevskii [21], Braaksma and Meulenbeld [2], Flensted--Jensen [9], [11, w and w and Flensted--Jensen and Koornwinder [12]. Some papers by Ch6bli [3], [4], [5] deal with a larger class of integral transforms which includes the Jacobi transform. An even more general class was considered by Braaksma and De Shoo [24]. In the present paper short proofs will be given of a Paley--Wiener type theorem and the inversion formula for the Jacobi transform. The L2-theory, i.e. the Plancherel theorem, is then an easy consequence. These results were earlier obtained by Flensted--Jensen [9], [11, w and by Ch6bli [5]. However, to prove the Paley-Wiener theorem these two authors needed the L2-theory, which can be obtained as a corollary of the Weyl S t o n e T i t c h m a r s h K o d a i r a theorem about the spectral decomposition of a singular Sturm--Liouville operator (cf. for instance Dunford and Schwartz [6, Chap. 13, w The proofs presented here exploit the properties of Jacobi functions as hypergeometric functions and no general theorem needs to be invoked. Furthermore, it turns out that the Paley--Wiener theorem, which was proved by Flensted---Jensen [11, w for real c~, fi, ~ > 1, holds for all complex values of ~ and ft. The key formula in this paper is a generalized Mehler formula