The Hypergeometric Approach to Integral Transforms and Convolutions

1 Preliminaries.- 1.1 Some special functions.- 1.2 Integral transforms.- 2 Mellin Convolution Type Transforms With Arbitrary Kernels.- 2.1 General Fourier kernels.- 2.2 Examples of the Fourier kernels.- 2.3 Watson type kernels.- 2.4 Bilateral Watson transforms.- 2.5 Multidimensional Watson transforms.- 3 H- and G-transforms.- 3.1 Mellin convolution type transform with Fox's H-function as a kernel.- 3.2 Mellin convolution type transforms with Meijer's G-function as a kernel.- 3.3 The Erdelyi-Kober fractional integration operators.- 4 The Generalized H- and G-transforms.- 4.1 The generalized H-transform.- 4.2 The generalized G-transform.- 4.3 Composition structure of generalized H- and G-transforms.- 5 The Generating Operators of Generalized H-transforms.- 5.1 Generating operators in the space ?Mc,??1.- 5.2 Examples of the generating operators.- 6 The Kontorovich-Lebedev Transform.- 6.1 The Kontorovich-Lebedev transform: notion, existence and inversion theorems in Mc,??1 (L) spaces.- 6.2 The Kontorovich-Lebedev transform in weighted L-spaces.- 6.3 The Kontorovich-Lebedev transform in weighted L2 spaces.- 6.4 The Kontorovich-Lebedev transform of distributions.- 6.5 The Kontorovich-Lebedev transform in Lp-spaces.- 7 General W-transform and its Particular Cases.- 7.1 General G-transform with respect to an index of the Kontorovich-Lebedev type.- 7.2 General W-transform and its composition structure.- 7.3 Some particular cases of W-transform and their properties.- 7.4 F3-transform.- 7.5 L2-theory of the Kontorovich-Lebedev type index transforms.- 8 Composition Theorems of Plancherel Type for Index Transforms.- 8.1 Compositions with symmetric weight.- 8.2 Compositions with non-symmetric weight.- 8.3 Constructions of index transforms in terms of Mellin integrals.- 9Some Examples of Index Transforms and Their New Properties.- 9.1 The Kontorovich-Lebedev like composition transforms.- 9.2 Some index transforms with symmetric kernels.- 9.3 The $$ \Re $$ and $$ \Im - $$ index transforms.- 10 Applications to Evaluation of Index Integrals.- 10.1 Some useful representations and identities.- 10.2 Some general index integrals.- 11 Convolutions of Generalized H-transforms.- 11.1 H-convolutions.- 11.2 Examples of H-convolutions.- 12 Generalization of the Notion of Convolution.- 12.1 Generalized H-convolutions.- 12.2 Generalized G-convolutions.- 13 Leibniz Rules and Their Integral Analogues.- 13.1 General Leibniz rules.- 13.2 Modified Leibniz rule.- 13.3 Leibniz rule for the Erdelyi-Kober fractional differential operator.- 13.4 Modification of the Leibniz rule for the Erdelyi-Kober fractional differential operator.- 13.5 Integral analogues of Leibniz rules.- 14 Convolutions of Generating Operators.- 14.1 Convolutions in the Dimovski sense. General results.- 14.2 Examples of convolutions in the Dimovski sense.- 15 Convolution of the Kontorovich-Lebedev Transform.- 15.1 Definition and some properties of a convolution for the Kontorovich-Lebedev transform.- 15.2 The basic property of convolution. Analogues with the Parseval equality.- 15.3 On the inversion of the Kontorovich-Lebedev transform in the ring L?.- 15.4 The space L? as the commutative normed ring of functions with exponential growth.- 16 Convolutions of the General Index Transforms.- 16.1 Convolutions of the Kontorovich-Lebedev type transforms.- 16.2 The convolutions for the Mehler-Fock and the Lebedev-Skalskaya transforms.- 16.3 The convolution of the Wimp-Yakubovich type index transform.- 17 Applications of the Kontorovich-Lebedev type Convolutions to Integral Equations.- 17.1Kontorovich-Lebedev convolution equations of the second kind.- 17.2 General composition convolution equations.- 17.3 Some results on the homogeneous equation.- 18 Convolutional Ring C?.- 18.1 Multiple Erdelyi-Kober fractional integrodifferential operators.- 18.2 Convolutional ring C?.- 19 The Fields of the Convolution Quotients.- 19.1 Extension of the ring (C?,?*,+).- 19.2 Extension of the ring (L?,*,+).- 20 The Cauchy Problem for Erdelyi-Kober Operators.- 20.1 General scheme.- 20.2 Differential equations of fractional order.- 20.3 Differential equations of hyper-Bessel type.- 21 Operational Method of Solution of some Convolution Equations.- 21.1 Integral equations of Volterra type.- 21.2 Integral equations of second kind with Kontorovich-Lebedev convolution.- References.- Author Index.- Notations.