Hostname: page-component-cb9f654ff-nr592 Total loading time: 0 Render date: 2025-08-27T01:26:42.922Z Has data issue: false hasContentIssue false
  • English
  • Français

The Hankel transform of some classes of generalized functions and connections with fractional integration*†

Published online by Cambridge University Press:  14 November 2011

Adam C. McBride
Adam C. McBride
Affiliation:
Department of Mathematics, University of Strathclyde
Get access Rights & Permissions [Opens in a new window]

Synopsis

In previous papers [11,12,13], certain spaces of generalized functions were studied from the point of view of fractional calculus. In this paper, we show how a Hankel transform Hv of order v can be defined on for all complex numbers v except for those lying on a countable number of lines of the form Re v = constant in the complex v-plane. The mapping properties of Hv on are obtained. Various connections between Hv (or modifications of Hv) and operators of fractional integration are examined.

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 81 , Issue 1-2 , 1978 , pp. 95 - 117
Copyright
Copyright © Royal Society of Edinburgh 1978

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

1Braaksma, B. L. J. and Schuitman, A.Some classes of Watson transforms and related integral equations for generalized functions. SIAM J. Math. Anal. 7 (1976), 771–798.Google Scholar
2Snoo, H. S. V. deA note on Watson transforms. Proc. Cambridge Philos. Soc. 73 (1973), 83–85.CrossRefGoogle Scholar
3Dube, L. S. and Pandey, J. N.On the Hankel transformation of distributions. Tohoku Math. J. 27 (1975), 337–354.CrossRefGoogle Scholar
4Erdélyi, A.On fractional integration and its application to the theory of Hankel transforms. Quart. J. Math. Oxford Ser. 11 (1940), 293–303.Google Scholar
5Erdélyi, A. et al. Higher transcendental functions 2 (New York: McGraw-Hill, 1953).Google Scholar
6Kober, H.On fractional integrals and derivatives. Quart. J. Math. Oxford Ser. 11 (1940), 193–211.Google Scholar
7Kober, H. and Erdélyi, A.Some remarks on Hankel transforms. Quart J. Math. Oxford Ser. 11 (1940), 212–221.Google Scholar
8Lee, W. Y. K.On spaces of type Hu and their Hankel transformations. SIAM J. Math. Anal. 5 (1974), 336–347.CrossRefGoogle Scholar
9Lee, W. Y. K.On Schwartz's Hankel transformation of certain spaces of distributions. SIAM J. Math. Anal. 6 (1975), 427–432.CrossRefGoogle Scholar
10McBride, A. C. A Theory of Fractional Integration for Generalised Functions with Applications (Edinburgh Univ. Ph.D. Thesis, 1971).Google Scholar
11McBride, A. C.A theory of fractional integration for generalised functions. SIAM J. Math. Anal. 6 (1975), 583–599.Google Scholar
12McBride, A. C.A theory of fractional integration for generalised functions, II. Proc. Roy. Soc. Edinburgh Sect. A, 77 (1977), 335–349.Google Scholar
13McBride, A. C.A note on the spaces F'p,μ. Proc. Roy. Soc. Edinburgh Sect. A, 77 (1976), 39–47.CrossRefGoogle Scholar
14Okikiolu, G. O.On integral operators with kernels involving Bessel functions. Proc. Cambridge Philos. Soc. 62 (1966), 477–484.CrossRefGoogle Scholar
15Okikiolu, G. O.On integral operators with kernels involving Bessel function's (Correction and Addendum). Proc. Cambridge Philos. Soc. 67 (1970), 583–586.Google Scholar
16Rall, L. B. (Ed.). Seminar on non-linear functional analysis and applications (New York: Academic Press, 1971).Google Scholar
17Rooney, P. G.On the ranges of certain fractional integrals. Canad. J. Math. 24 (1972), 1198–1216.Google Scholar
18Rooney, P. G.A technique for studying the boundedness and extendability of certain types of operators. Canad. J. Math. 25 (1973), 1090–1102.CrossRefGoogle Scholar
19Sneddon, I. N.Mixed boundary value problems in potential theory (Amsterdam: North-Holland, 1966).Google Scholar
20Titchmarsh, E. C.The theory of Fourier integrals (Oxford: Univ. Press, 1948).Google Scholar
21Treves, F.Topological vector spaces, distributions and kernels (New York: Academic Press, 1967).Google Scholar
22Watson, G. N.Theory of Bessel functions (Cambridge: Univ. Press, 1966).Google Scholar
23Zemanian, A. H.Generalized integral transformations (New York: Interscience, 1968).Google Scholar