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Title: Fractional calculus, fractional powers of operators and Mellin multiplier transforms
Author: McBride, Adam C.
Awarding Body: University of Edinburgh
Current Institution: University of Edinburgh
Date of Award: 1994
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We shall present a theory of fractional calculus for generalised functions on (0,∞) and use this theory as a basis for extensions to some related areas. In the first section, appropriate spaces of test-functions and generalised functions on (0,∞) are introduced and the properties of operators of fractional calculus obtained relative to these spaces. Applications are given to hypergeometric integral equations, Hankel transforms and dual integral equations of Titchmarsh type. In the second section, the Mellin transform is used to define fractional powers of a very general class of operators. These definitions include standard operators as special cases. Of particular interest are powers of differential operators of Bessel or hyper-Bessel type which are related to integral operators with special functions, notably G-functions, as kernels. In the third section, we examine operators whose Mellin multipliers involve products and/or quotients of Γ-functions. There is a detailed study of the range and invertibility of such operators in weighted LP-spaces and in appropriate spaces of smooth functions. The Laplace and Stieltjes transforms give two particular examples. In the final section, we show how our theory of fractional calculus on (0,∞) can be used to develop a corresponding theory on IRn in the presence of radial symmetry. In this framework the mapping properties of multidimensional radial integrals and Riesz potentials are obtained very precisely.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (D.Sc.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available