The Titchmarsh integral transformation by the index of a Bessel function

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:118

Article

Yakubovich, SB

关键词: the Titchmarsh transform;    the Kontorovich-Lebedev transform;    Bessel functions;   

DOI  :  10.1016/S0377-0427(00)00298-3

来源: Elsevier

【 摘 要 】

This paper deals with an index integral transformation using Bessel functions as kernels. It was introduced and studied by Titchmarsh in 1946 as an example of a continuous spectrum Bessel-function expansions in Sturm-Liouville boundary value problems. Later in the second edition of his book (Titchmarsh, Eigenfunction Expansions Associated with Second-order Differential Equations, Part I, 2nd Edition, Clarendon Press, Oxford, 1946) in 1962 he corrected his expansion by adding an additional term, which contains a combination of an integral and series. In this paper the Titchmarsh formula is simplified and contains just integrals with Bessel and Lommel functions as kernels, which generate a pair of Titchmarsh integral transformations. By using the composition properties of the Titchmarsh transform and its relationship with the Kontorovich-Lebedev transform, L-p-properties of the Titchmarsh transform are investigated and inversion theorems are proved. The question of the correctness of Titchmarsh's formulas is completely closed by this discussion. (C) 2000 Elsevier Science B.V. All rights reserved. MSC. 44A15; 44A20.

【 授权许可】

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