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A theory of fractional integration for generalised functions II*

  • Adam C. McBride (a1)

Synopsis

In a previous paper [2], a theory of fractional integration was developed for certain spaces Fp,μ of generalised functions. In this paper we extend this theory by relaxing some of the restrictions on the various parameters involved. In particular we show how a generalised Erdelyi-Kober operator can be defined on p,μ for 1 ≦ p ≦ ∞ and for all complex numbers μ except for those lying on a countable number of lines of the form Re μ = constant in the complex μ-plane. Mapping properties of these generalised operators are obtained and several applications mentioned.

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1Kober, H.. On fractional integrals and derivatives. Quart. J. Math. Oxford Ser. 11 (1940), 193211.
2McBride, A. C.. A theory of fractional integration for generalised functions. SIAMJ. Math. Anal. 6 (1975), 583599.
3McBride, A. C.. Solution of hypergeometric integral equations involving generalised functions. Proc. Edinburgh Math. Soc. 19 (1975), 265285.
4McBride, A. C.. A note on the spaces Fʹp,μ. Proc. Roy. Soc. Edinburgh Sect. A 11 (1977), 3947.
5Sneddon, I. N.. Mixed boundary value problems in potential theory (Amsterdam: North-Holland 1966).
6Zemanian, A. H.. Generalized integral transformations (New York: Interscience, 1968).
7Erdelyi, A.. On fractional integration and its application to the theory of Hankel transforms. Quart. J. Math. Oxford Ser. 11 (1940), 293303.

A theory of fractional integration for generalised functions II*

  • Adam C. McBride (a1)

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