On inversion of H‐transform in 𝔏_v,r‐space

Abstract

ABSTRACT. The paper is devoted to study the inversion of the integral transform (Hf)(x) fo H,' [xt (a"a')l" f(t)dt involving the H-function as the kernel in the space ,, of functions jr such that fo itf(t)l dt -<x (l<r<o, ).

A generalized Hankel transform on Lν,r–spaces

“…In this section we repeat some results of the theory of the H-transform on L ν,r -spaces proved in [4], [7] and [19]. Following these papers we use the notation…”

“…Mapping properties such as the boundedness, the representation and the range of the H-transform were investigated in [4], [7] - [9], while its inversion was established in [19]. These results are applied to investigate the generalized Hankel transform (1.1).…”

MODIFIED H-TRANSFORMS IN £v,r-SPACES

“…The conditions for the validity of these formulas depend on the inversion formulas (6.1) and (6.2) for the //-transform (1.7) given in [24]. These conditions are different in the cases when a* = 0, and A = 0 or A ^ 0.…”

“…It is known [24] that, if a* = 0 under some additional conditions, the inversion of the ii-transform in £"iT.-space in the respective form (3.7) or (3.10) can be found. Namely…”

“…Many authors investigated the properties of the ii-transforms in Z>i(0, oo), ¿2(0, 00) and some special function spaces (see a short survey and bibliography in [11]- [13]). Mapping properties such as the boundedness, the representation and the range of the if-transform (1.7) were proved independently by the authors together with Shlapakov in [11]- [13] and with Shlapakov and Glaeske in [6]- [7] and by Betancor and Jerez Diaz [1], The invertibility of (1.7) in £ Uir was given by the authors together with Shlapakov in [24]. We note that the investigation of the boundedness, representation and range of the if-transform (1.7) is based on the technique of Mellin transform developed by Rooney [20] for the G-transform, while the invertibility of such a transform is due to the asymptotic behavior of the function H™ q n (z) in (1.4) considered by the authors in [9].…”