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The Hypergeometric Approach to Integral Transforms and Convolutions

The aim of this book is to develop a new approach which we called the hyper­ geometric one to the theory of various integral transforms, convolutions, and their applications to solutions of integro-differential equations, operational calculus, and evaluation of integrals. We hope that this simple ap...

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Bibliographic Details
Main Authors: Yakubovich, S.B., Luchko, Yury (Author)
Format: eBook
Language:English
Published: Dordrecht Springer Netherlands 1994, 1994
Edition:1st ed. 1994
Series:Mathematics and Its Applications
Subjects:
Integral Equations
Special Functions
Mathematical Analysis
Integral Transforms And Operational Calculus
Online Access:
Collection: Springer Book Archives -2004 - Collection details see MPG.ReNa
Table of Contents:
  • 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
  • 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
  • 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

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