Semen B. Yakubovich

The hypergeometric approach to integral transforms and convolutions

Preface. 1. Preliminaries. 2. Mellin Convolution Type Transforms with Arbitrary Kernels. 3. H- and G-Transforms. 4. The Generalized H- and G-Transforms. 5. The Generating Operators of Generalized H-Transforms. 6. The Kontorovich--Lebedev Transform. 7. General W-Transform and its Particular Cases. 8. Composition Theorems of Plancherel Type for Index Transforms. 9. Some Examples of Index Transforms and their New Properties. 10. Applications to Evaluation of Index Integrals. 11. Convolutions of Generalized H-Transforms. 12. Generalization of the Notion of Convolution. 13. Leibniz Rules and their Integral Analogues. 14. Convolutions of Generating Operators. 15. Convolution of the Kontorovich--Lebedev Transform. 16. Convolutions of the General Index Transforms. 17. Applications of the Kontorovich--Lebedev Type Convolutions to Integral Equations. 18. Convolutional Ring Calpha. 19. The Fields of the Convolution Quotients. 20. The Cauchy Problem for Erdelyi--Kober Operators. 21. Operational Method of Solution of Some Convolution Equations. References. Author Index. Subject Index. Notations.

Convolutions of Generalized H-transforms

The term “convolution” is commonly used in analysis. Best known examples of convolutions are the convolution of the Fourier, Laplace, and Mellin transforms. Broadly speaking, a convolution is always conceived as a bilinear, commutative and associative operation in a linear space, i.e., it is a multiplication in a linear space, such that the space itself becomes a commutative ring. There is a great variety of papers, articles, and books, in which the problems connected with investigation and using the convolutions are studied. We refer to L. Berg (1976), I.H. Dimovski (1982), I.H. Dimovski and V.S. Kiryakova (1984), I.H. Dimovski and S.L. Kalla (1988), H.-J. Glaeske and A. Heγ (1986), (1987), (1988), M. Goldberg (1961), S.I. Grozdev (1980), V.A. Kakichev (1967), (1990), V.S. Kiryakova (1989), Yu. F. Luchko and S.B. Yakubovich (1991), (1991a), N.A. Meller (1960), J. Mikusinski and G. Ryll-Nardzewski (1952), (1953), Nguyen Thanh Hai and S.B. Yakubovich (1992), J. Rodrigues (1990), M. Saigo and S.B. Yakubovich (1991), S.B. Yakubovich (1987a), (1987b), (1990), (1991a), (1991b), (1992), S.B. Yakubovich and Yu. F. Luchko (1991a), (1991b), S.B. Yakubovich and Nguyen Thanh Hai (1991), S.B. Yakubovich et al. (1992).

Operational Method of Solution of some Convolution Equations

It is well known that the Mikusinski’s operational calculus as well as other operational calculi for hyper-Bessel type differential operators were used for the solution of ordinary differential equations both with constant and variable coefficients and for the solution of some partial differential equations (see, for example, V.A. Ditkin and A.P. Prudnikov (1963), Yu. F. Luchko and S.B. Yakubovich (1993), (1994), N.A. Meller (1960), J. Mikusinski (1959), J. Rodrigues (1989), K. Yosida (1984)). It is worth mentioning that operational method may be used for the solution of integral equations as well. This idea was realized, in particular, in R.G. Buschman (1972), A. Erdelyi (1962), D. Nikolic-Despotovic (1975) for the case of integral equations of the first kind. In this chapter, we will investigate an application of operational calculi developed in the Chapter 19 to the solution of some classes integral equations of the second kind.

The Kontorovich-Lebedev Transform

In the previous Chapters 1–5, we considered the so-called Mellin convolution type integral transforms and some of their important particular cases. However, the family of one-dimensional integral transforms is very different. Namely, there are representations of arbitrary functions, where integration has been realized with respect to an index of special function of hypergeometric type from the kernels. The basic examples of such transforms are the Kontorovich-Lebedev and Mehler-Fock transforms (see M.I. Kontorovich and N.N. Lebedev (1938), F.G. Mehler (1881) and V. A. Fock (1943)). Our hypergeometric approach makes it possible not only to investigate these known integral transforms from the new point of view, but, according to Wimp (see J. Wimp (1964)), also to construct some generalizations and inversions. For instance, an inversion of the Wimp transform with respect to the index of Meijer’s G-function has been simplified by the first author in 1983 using the theory of generalized H-transform (4.1) (see S.B. Yakubovich (1985)). Thus, these so-called index transform classes are very interesting and very important in applications.

Convolutions of Generating Operators

We considered convolutions of the generalized H-transforms in Chapter 11. The main property of these convolutions is the following {fy205-1} where (H a f)(x) is the generalized H-transform with the power weight (11.1). It follows from this relation that the H-convolution (f * a )(x) is connected with some integral transform and this connection is reflected in the names of other convolutions (Laplace convolution, Mellin convolution, et c.). A different approach to the definition of convolution, which connects some other operator with a convolution has been proposed by I.H. Dimovski ((1966)–(1981)). His definition is more suitable in developing a Mikusinski type operational calculus. In this chapter we will deal with convolutions in the Dimovski’s sense.

The Generalized H- and G-transforms

We already considered Mellin convolution type transforms with the H- and G-functions as kernels in the Chapter 3. As we have seen, the inversion formulae depend on the values of the k, μ (a) k = 0, μ > 1; b) k > 0 in Theorem 3.3 and c*, γ* (a) c* = 0, 1/2 0; c) c* > 0 in Theorem 3.4) which are determined by the parameters of the H- and G-functions. This fact does not cause any troubles if we consider the particular cases of the H- and G-transforms. However, if we will investigate the H- and G- transforms as united objects, we will need new approach, which unites the studied cases with the new ones (k = 0, μ ≤ 1; k < 0; c* = 0, γ* ≤ - 1/2; c* < 0). This approach is based on the Parseval formula (3.45) and was studied by Vu Kim Tuan et al. (1986), Vu Kim Tuan (1987), S.G. Samko et al. (1987), Nguyen Thanh Hai and S.B. Yakubovich (1992).

Convolutions of the General Index Transforms

The aim of this chapter is to generalize the notion of convolution (15.1) for the Kontorovich-Lebedev transform on the index transforms with the similar structure, which have been considered in Chapter 7. In this chapter we will study transforms like the compositions of the Kontorovich-Lebedev transform (6.52) and the general Mellin convolution type transforms (2.1), following S.B. Yakubovich (1993) and S.B. Yakubovich and A.I. Moshinskii (1993). Interesting particular cases of these convolutions related to known index transforms will be established.

The Generating Operators of Generalized H-transforms

It is well known that the Laplace transform (3.50) possesses the following property

General W-transform and its Particular Cases

L\left\{ {\frac{d} {{dt}}f\left( t \right);x} \right\} = xL\left\{ {f\left( t \right);x} \right\} - f\left( 0 \right),