Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Hankel transforms and GASP


Author: Stanton Philipp
Journal: Trans. Amer. Math. Soc. 176 (1973), 59-72
MSC: Primary 44A15
MathSciNet review: 0316978
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [1] Salomon Bochner, Lectures on Fourier integrals. With an author’s supplement on monotonic functions, Stieltjes integrals, and harmonic analysis, Translated by Morris Tenenbaum and Harry Pollard. Annals of Mathematics Studies, No. 42, Princeton University Press, Princeton, N.J., 1959. MR 0107124
  • [2] G. H. Hardy, Divergent Series, Oxford, at the Clarendon Press, 1949. MR 0030620
  • [3] A. C. Offord, On the uniqueness of the representation of a function by a trigonometric integral, Proc. London Math. Soc. (2) 42 (1937), 422-480.
  • [4] P. M. Owen, The Riemannian theory of Hankel transforms, Proc. London Math. Soc. (2) 39 (1935), 295-320.
  • [5] Victor L. Shapiro, The uniqueness of functions harmonic in the interior of the unit disk, Proc. London Math. Soc. (3) 13 (1963), 639–652. MR 0155983
  • [6] Victor L. Shapiro, The uniqueness of solutions of the heat equation in an infinite strip, Trans. Amer. Math. Soc. 125 (1966), 326–361. MR 0201847, 10.1090/S0002-9947-1966-0201847-1
  • [7] E. C. Titchmarsh, Introduction to the theory of Fourier integrals, Clarendon Press, Oxford, 1937.
  • [8] G. N. Watson, A treatise on the theory of Bessel functions, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1995. Reprint of the second (1944) edition. MR 1349110
  • [9] E. T. Whittaker and G. N. Watson, A course of modern analysis, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996. An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions; Reprint of the fourth (1927) edition. MR 1424469
  • [10] A. Zygmund, On trigonometric integrals, Ann. of Math. (2) 48 (1947), 393–440. MR 0021612
  • [11] -, Trigonometrical series. Vol. 1, 2nd rev. ed., Cambridge Univ. Press, New York, 1959. MR 21 #6498.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 44A15

Retrieve articles in all journals with MSC: 44A15


Additional Information

DOI: http://dx.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