A new proof of a Paley—Wiener type theorem for the Jacobi transform
A new proof of a Paley—Wiener type theorem for the Jacobi transform
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雅可比变换佩利-维纳型定理的新证明
DOI：
10.1007/bf02386203
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发表时间：
1975
期刊：
影响因子：
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通讯作者：
T. Koornwinder
中科院分区：
文献类型：
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作者：
T. Koornwinder
Titchmarsh [23，w 17]，Olevskii [21]，Braaksma和Meulenbeld [2]，Flensted-詹森[9]，[11，w和w以及Flensted-詹森和Koornwinder [12]研究了Mehler-Fok变换。Ch 6 bli [3]，[4]，[5]等的一些文章讨论了一类更大的积分变换，其中包括Jacobi变换。Braaksma和De Shoo [24]考虑了一个更一般的类。本文将给出Paley-Wiener型定理和Jacobi变换的反演公式的简短证明。L2理论，即Plancherel定理，是一个简单的推论。这些结果是由Flensted-詹森[9]，[11]和Ch 6 bli [5]较早得到的。然而，为了证明Paley-Wiener定理，这两位作者需要L2-理论，它可以作为Weyl Ston e Tit c h m ar s h K o d a i r a定理的推论获得，该定理关于奇异Sturm-Liouville算子的谱分解（参见[1]）。例如，Dunford和Schwartz [6，Chap. 13，w这里给出的证明利用了Jacobi函数作为超几何函数的性质，并且不需要调用一般定理。进一步证明了Flensted-詹森[11，w对真实的c~，fi，~ > 1证明的Paley-Wiener定理对~和ft的所有复值都成立。本文的核心公式是一个推广的Mehler公式
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