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This volume presents recent developments in the fractional calculus of functions of one and several real variables, and shows the relation of this field to a variety of areas in pure and applied mathematics. Beyond some basic properties of fractional integrals in one and many dimensions, it contains a mathematical theory of certain important weakly singular integral equations of the first kind arising in mechanics, diffraction theory and other areas of mathematical physics. The author focuses on explicit inversion formulae that can be obtained by making use of the classical Marchaudís approach and its generalization, leading to wavelet type representations.
Table of Contents
Preface Notation Chapter 1 Generalities Chapter 2 One-dimensional fractional integrals Chapter 3 Fractional integro-differentiation via wavelet transforms Chapter 4 Riesz potentials on R Chapter 5 Oscillatory potentials on R Chapter 6 Potentials on a half-space Chapter 7 Riesz potentials on a ball Chapter 8 Fractional integrals on a sphere Appendix 1 Appendix 2 Appendix 3 Bibliography
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