Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-15T04:29:31.838Z Has data issue: false hasContentIssue false
  • English
  • Français

A spectral approach to an integral equation

Published online by Cambridge University Press:  18 May 2009

W. Lamb
W. Lamb
Affiliation:
Department of Mathematics, University of Strathclyde, Glasgow
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In a recent paper [7], Rooney used a technique involving the Mellin transform to obtain solutions in certain spaces ℒμ, ρ of an integral equation which had been studied previously by Šub-Sizonenko [9]. The integral equation in question can be written as

where I denotes the identity operator and G0.1/2 is given by

with the inversion formula obtained by Rooney taking the form

Rooney verified that (1.1) and (1.2) formed an inversion pair in ℒμ, ρ for 1 ≤ p < ∞ and μ > 0.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 26 , Issue 1 , January 1985 , pp. 83 - 89
Copyright
Copyright © Glasgow Mathematical Journal Trust 1985

References

REFERENCES

1.Balakrishnan, A. V., Fractional powers of closed operators and semigroups generated by them, Pacific J. Math. 10 (1960), 419–437.CrossRefGoogle Scholar
2.Erdélyi, A. et al. , Higher transcendental functions I (McGraw-Hill, 1953).Google Scholar
3.Hövel, H. W. and Westphal, U., Fractional powers of closed operators, Studia Math. 42 (1972), 177–194.CrossRefGoogle Scholar
4.Lamb, W., Fractional powers of operators on Fréchet spaces with applications (Strathclyde Univ. Ph.D. Thesis, 1980).Google Scholar
5.Lamb, W., Fractional powers of operators defined on a Fréchet space, Proc. Edinburgh Math. Soc. (2), (to appear).Google Scholar
6.Love, E. R., Some inequalities for fractional integrals, Linear spaces and approximation, Proc. Conf., Math. Res. Int. Oberwolfach, 1977, Internat. Ser. Numer. Math., 40 (Birkhauser, 1978).Google Scholar
7.Rooney, P. G., On an integral equation of Šub-Sizonenko, Glasgow Math. J. 24 (1983), 207–210.CrossRefGoogle Scholar
8.Rooney, P. G., Multipliers for the Mellin transformation, Canad. Math. Bull. 25 (1982), 257–262.CrossRefGoogle Scholar
9.Šub-Sizonenko, J. A., Inversion of an integral operator by the method of expansion with respect to orthogonal Watson operators, Siberian Math. J. 20 (1979), 318–321.CrossRefGoogle Scholar