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Hankel Transforms

In: Introduction to Hyperfunctions and Their Integral Transforms

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  • Urs Graf

Abstract

First we show that the conventional Hankel transform pair arises in a natural way when, in the two-dimensional Fourier transformation, polar coordinates are introduced. Unfortunately, no firm convention about the definition of the Hankel transform pair is established. We shall use the most widespread one. In order to lay the groundwork for the theory of Hankel transformation of hyperfunctions, we present a concise exposition of the various cylinder functions, the integrals of Lommel and MacRobert’s proof of the inversion formula. The Hankel transform of a hyperfunction defined on the positive part of the real line is then defined by using the Hankel functions for the kernel. Along the line of MacRobert’s proof and using the integrals of Lommel, we then prove that the defined Hankel transform of a hyperfunction is a self-reciprocal transformation when restricted to the strictly positive part of the real axis. The operational rules known for the Hankel transformation of ordinary functions are then carried over to the Hankel transformation of hyperfunctions. The chapter closes with a few applications about problems of mathematical physics.

Suggested Citation

  • Urs Graf, 2010. "Hankel Transforms," Springer Books, in: Introduction to Hyperfunctions and Their Integral Transforms, chapter 0, pages 337-372, Springer.
    • Handle: RePEc:spr:sprchp:978-3-0346-0408-6_7
    DOI: 10.1007/978-3-0346-0408-6_7
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