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On integral transforms whose kernels are solutions of singular Sturm–Liouville problems

Published online by Cambridge University Press:  14 November 2011

Ahmed I. Zayed
Ahmed I. Zayed
Affiliation:
Mathematics Department, California Polytechnic State University, San Luis Obispo, CA 93407, U.S.A.
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Synopsis

In this paper we investigate integral transforms of type , where φ(x, s) is the solution of the singular Sturm–Liouville problem: y″ + (s2 – q(x))y = 0, 0≦x <∞ with y(0) cos α + y′(0)sin α = 0, y(x) is bounded at ∞, and dp is the spectral measure. If F(s) = sk for some k = 0, 1, 2, …, then f(x) may not exist since, in general, φ(x, s) is not even in . One aim of this paper is to investigate the Abel summability of these integrals. In the special case where q(x) = 0 and α = π/2, then φ(x, s) = cos sx and dp = ds, while if α = 0, then φ(x, s) = −sin sx/s and dp = s2ds. It is known that

where the values of these integrals are interpreted as the Abel limits of these integrals or as the Fourier transform of some tempered distributions. Another aim of this paper is to derive the analogue of these results for the general kernel φ(x, s), and then apply that to the theory of asymptotic expansions.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 108 , Issue 3-4 , 1988 , pp. 201 - 228
Copyright
Copyright © Royal Society of Edinburgh 1988

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