# The Generalized H- and G-transforms

## Abstract

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 < γ^{*} ≤ 0, *p* ≠ *q*; b) *c*^{*} = 0, γ^{*} > 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).

## Keywords

Integral Equation Important Method Similar Fact Inversion Formula Integral Transform## Preview

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