Integral transforms with Bessel and hypergeometric type kernels
by
Anatoly A. Kilbas
Belarusian State University, Minsk, Belarus

We consider the integral transform

(Kf)(x)= ó õ \infty 0  k(xu)f(u)du (x > 0),

(1)

its modification in the Mellin setting

(K1f)(x)= ó õ \infty 0  k(\fracxu)f(u)\fracduu (x > 0),

(2)

and other modifications and generalizations of (1) and (2), the kernels k(z) of which contain functions of bessel and hypergeometric type. The transforms considered include classical Hankel transform, extended and generalized Hankel transforms; Hankel-Schwartz and Bessel-Clifford transforms; Struve and Y_\eta transforms, Meyer K_\eta-transform; Bessel and modified Bessel transforms; Hardy, Titchmarsh and Hardy-Titchmarsh transforms as well as Laplace type transforms; Verma and Mejer transforms; Whitaker and general Whittaker transforms; D_\eta-transform; hypergeometric ₁F₁, ₁F₂, ₂F₁ and general _pF_q-transforms, Whrite and Mittag-Leffler transforms, etc.

We present a general approach to the study of the above transforms in the spaces L_{\nu, r} (\nu in R=(-\infty, \infty),  1 <= r <= \infty) of Lebesgue measurable functions f on R₊=(0, \infty) such that

ó õ \infty 0  |t\nuf(t)|\fracdtt < \infty (1 <= r < \infty),  ess supt > 0|t\nuf(t)| < \infty (r=\infty).

This approach is based on the representations of the considered transforms as particular cases of more general transforms containing H-functions as kernels. The properties such as the boundedness, including the one-to-one property of the map, the representation and the range of these transforms are established, and their inversion formulas are proved.

The results for other spaces of functions are also discussed.

Date received: May 23, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cahk-87.