Index Transforms

This book deals with the theory and some applications of integral transforms that involve integration with respect to an index or parameter of a special function of hypergeometric type as the kernel (index transforms). The basic index transforms are considered, such as the Kontorovich-Lebedev transform, the Mehler-Fock transform, the Olevskii Transform and the Lebedev-Skalskaya transforms. The Lp theory of index transforms is discussed, and new index transforms and convolution constructions are demonstrated. For the first time, the essentially multidimensional Kontorovich-Lebedev transform is announced. General index transform formulae are obtained. The connection between the multidimensional index kernels and G and H functions of several variables is presented. The book is self-contained, and includes a list of symbols with definitions, author and subject indices, and an up-to-date bibliography.This work will be of interest to researchers and graudate students in the mathematical and physical sciences whose work involves integral transforms and special functions. 1. Preliminaries. 1.1. The spaces Lp and Lp(p). 1.2. Special functions of the hypergeometric type. 1.3. Fourier's, Mellin's and Laplace's transforms. 1.4. General Mellin convolution type integral transforms. 1.5. Notion of the index transforms -- 2. The Kontorovich-Lebedev transform. 2.1. Definition, inversion in L[symbol]. 2.2. Note on the K-L-summability of integrals. 2.3. The space L[symbol]. Parseval's relation. Plancherel's theorem. 2.4. Composition representations of the K-L transform. A convolution Hilbert space. 2.5. Representations through the Mellin transform. Watson's type lemma. 2.6. The index-convolution Kontorovich-Lebedev transform -- 3. The Mehler-Fock transform. 3.1. Definition. Inversion in Zp-space. 3.2. The composition theorem of inversion. 3.3. The generalized Mehler-Fock transform. 3.4. Parseval's equality. 3.5. An index-convolution transform related to the Mehler-Fock integral -- 4. Convolution of the Kontorovich-Lebedev transform. 4.1. Definition of the convolution. Useful estimates. 4.2. The factorization property. Parseval's type equality. 4.3. The space L[symbol] as a normed ring. 4.4. Convolution Hilbert spaces. 4.5. On the Kontorovich-Lebedev convolution integral equations. 4.6. Other convolution constructions -- 5. General index transforms. 5.1. The Kontorovich-Lebedev transform in a complex domain. Analogs of the Hardy type spaces and the Paley-Wiener theorem. 5.2. Composition theorems for general index transforms. 5.3. Watson's type kernels. 5.4. Compositions with the Mellin-Barnes integrals -- 6. Index transforms of the Lebedev-Skalskaya type. 6.1. Useful representations and estimates. 6.2. The Lebedev-Skalskaya transforms. 6.3. L2-theory of the Lebedev-Skalskaya transforms. 6.4. Convolution representations -- 7. Index transforms with hypergeometric functions in the kernel. 7.1. Index transforms of the Olevskii type. 7.2. General R-transforms. 7.3. Note on the essentially multidimensional Kontorovich-Lebedev transform This book deals with the theory and some applications of integral transforms that involve integration with respect to an index or parameter of a special function of hypergeometric type as the kernel (index transforms). The basic index transforms are considered, such as the Kontorovich-Lebedev transform, the Mehler-Fock transform, the Olevskii Transform and the Lebedev-Skalskaya transforms. The Lp theory of index transforms is discussed, and new index transforms and convolution constructions are demonstrated. For the first time, the essentially multidimensional Kontorovich-Lebedev transform is announced. General index transform formulae are obtained. The connection between the multidimensional index kernels and G and H functions of several variables is presented. The book is self-contained, and includes a list of symbols with definitions, author and subject indices, and an up-to-date bibliography. This work will be of interest to researchers and graudate students in the mathematical and physical sciences whose work involves integral transforms and special functions This work deals with the theory and some applications of integral transforms that involve integration with respect to an index or parameter of a special function of the hypergeometric type as the kernel (index transforms). We assume that the reader is familiar with Lebesgue measurability of functions and the Lebesgue integral.