Hostname: page-component-745bb68f8f-cphqk Total loading time: 0 Render date: 2025-01-30T05:36:45.615Z Has data issue: false hasContentIssue false
  • English
  • Français

Related singular problems and the generalized Hilbert transform

Published online by Cambridge University Press:  14 February 2012

John W. Dettman
John W. Dettman
Affiliation:
Department of Mathematical Sciences, Oakland University, Rochester, Michigan, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Synopsis

Abstract versions of the Cauchy problem for the Euler-Poisson-Darboux equation and the Dirichlet problem for the equation of generalized axially symmetric potential theory are related by an integral transformation. In certain special cases, this leads to abstract versions of (1) the Poisson formula for the solution of G.A.S.P.T. in a half-space, (2) pseudo-analytic functions in a half-space, and (3) a generalized Hilbert transform related to the work of Heywood, Kober, and Okikiolu. Some properties of this generalized Hilbert transform are studied including an inversion theorem.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 79 , Issue 1-2 , 1977 , pp. 173 - 182
Copyright
Copyright © Royal Society of Edinburgh 1977

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.)

References

1Bragg, L. R.. Hypergeometric operator series and related partial differential equations. Trans. Amer. Math. Soc. 143 (1969), 319–336.CrossRefGoogle Scholar
2Bragg, L. R. and Dettman, J. W.. Related problems in partial differential equations. Bull. Amer. Math. Soc. 74 (1968), 375–378.CrossRefGoogle Scholar
3Bragg, L. R. and Dettman, J. W.. Related partial differential equations and their applications. SIAMJ. Appl. Math. 16 (1968), 459–467.CrossRefGoogle Scholar
4Bragg, L. R. and Dettman, J. W.. An operator calculus for related partial differential equations. J. Math Anal. 22 (1969) 261–271.Google Scholar
5Bragf, L.R. and Dettman, J.W.. A class of related Dirchlet and initial value promblems.Proc. Amer. Math.Soc. 21 (1969), 50–56Google Scholar
6Butzer, P. L. and Nessel, J. J.. Fourier Analysis and Approximation (New York: Academic Press, 1971).CrossRefGoogle Scholar
7Dettman, J. W.. Initial-boundary value problems related through the Stieltjes transform.J. Math. Anal. Appl. 25 (1969), 341–349.CrossRefGoogle Scholar
8Dettman, J. W.. Related semi-groups and the abstract Hilbert transform, to appear.Google Scholar
9Donaldson, J. A.. A singular abstract Cauchy problem. Proc. Nat Acad. Sci. U.S.A. 66 (1970), 269–274.Google Scholar
10Donaldson, J. A.. An operational calculus for a class of abstract operator equations. J. Math. Anal. Appl. 37 (1972), 167–184.Google Scholar
11Donaldson, J. A.. Some remarks on uniqueness for a class of singular abstract Cauchy problems, to appear.Google Scholar
12Fattorini, H. O.. Ordinary differential equations in linear topological spaces I. J.Differential Equations 5 (1968), 72–105.Google Scholar
13Fattorini, H. O.. Ordinary differential equations in linear topological spaces II. J. Differential Equations 6 (1969), 50–70.Google Scholar
14Goldstein, J. A.. Semi-groups and second order differential equations. J. Functional Analysis 4 (1969), 50–70.Google Scholar
15Goldstein, J. A.. On a connection between first and second order differential equations in Banach spaces. J. Math. Anal. Appl. 30 (1970), 246–251.Google Scholar
16Goldstein, J. A.. An asymptotic property of solutions of wave equations. Proc. Amer. Math.Soc. 23 (1969), 359–363.Google Scholar
17Goldstein, J. A.. An asymptotic property of solutions of wave equations II. J.Math.Anal. Appl. 32 (1970), 392–399.Google Scholar
18Goldstein, J. A.. On the growth of solutions of inhomogeneous abstract wave equations. J. Math. Anal. Appl. 37 (1972), 650–654.CrossRefGoogle Scholar
19Heywood, P.. On a modification of the Hilbert transform. J. London Math. Soc. 42 (1967), 641–645.Google Scholar
20Hille, E. and Phillips, R. S.. Functional analysis and semi-groups. Amer. Math. Soc.Colloq. Publ. 31 (1957).Google Scholar
21Huber, A.. On the uniqueness of generalized axially symmetric potentials. Ann. of Math. 60 (1954), 351–358.Google Scholar
22Kober, H.. A modification of Hilbert transforms, the Weyl integral, and functional equations. J. London Math. Soc. 42 (1967), 42–50.CrossRefGoogle Scholar
23Okikiolu, G. O.. A generalization of the Hilbert transform. J.London Math. Soc. 40 (1965), 27–30.Google Scholar
24Riesz, M.. L'integrale de Riemann-Liouville et le probleme de Cauchy. Acta Math. 81 (1949), 1–223.CrossRefGoogle Scholar
25Shinbrot, M.. Asymptotic behaviour of solutions of abstract wave equations. Proc. Amer. Math. Soc. 19 (1968), 1403–1406.CrossRefGoogle Scholar
26Stein, E. M.. The characterization of functions arising as potentials I. Bull. Amer. Math. Soc. 67 (1961), 102–104.Google Scholar
27Stein, E. M.. The characterization of functions arising as potentials II. Bull. Amer. Math. Soc. 68 (1962), 577–582.CrossRefGoogle Scholar
28Stein, E. M. and Zygmund, A.. On the fractional differentiation of functions. Proc. London Math. Soc. 14A (1965), 249–264.Google Scholar
29Weinstein, A.. Generalized axially symmetric potential theory. Bull. Amer. Math. Soc. 59 (1953), 20–38.Google Scholar