Hankel transforms and GASP

Author:
Stanton Philipp

Journal:
Trans. Amer. Math. Soc. **176** (1973), 59-72

MSC:
Primary 44A15

DOI:
https://doi.org/10.1090/S0002-9947-1973-0316978-9

MathSciNet review:
0316978

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Abstract: The inversion of the classical Hankel transform is considered from three viewpoints. The first approach is direct, and a theorem is given which allows inversion in the (*C*, 1) sense under fairly weak hypotheses. The second approach is via Abel summability, and it is shown that inversion is possible if it is known that the Hankel transform is Abel summable and if certain critical growth conditions are satisfied. The third approach rests on the observation that Abel means of Hankel transforms satisfy a variant of the GASP equation in two arguments. In this setting the inversion problem becomes a boundary value problem for GASP in a quadrant of the plane with boundary values on one of the axes; a uniqueness theorem for this problem is proved which is best possible in several respects.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1973-0316978-9

Keywords:
Hankel transform,
GASP,
locally integrable,
kernel,
inversion,
smooth,
generalized second derivative,
Bessel function,
Legendre function of second kind,
hypergeometric function,
binomial coefficient

Article copyright:
© Copyright 1973
American Mathematical Society