International Journal of Mathematics and Mathematical Sciences
Volume 10 (1987), Issue 2, Pages 267-286
doi:10.1155/S0161171287000334

The Mejer transformation of generalized functions

E.L. Koh1 , E.Y. Deeba2 and M.A. Ali3

1Department of Mathematics and Statistics, University of Regina, Regina S4S 0A2, Canada
 2Department of Applied Mathematical Sciences, University of Houston-Downtown, Houston 77002, Texas, USA
 31598, Way 510, Muharraq 205, Bahrain

Abstract

This paper extends the Meijer transformation, Mμ, given by (Mμf)(p)=2pΓ(1+μ)∫0∞f(t)(pt)μ/2Kμ(2pt)dt, where f belongs to an appropriate function space, μ ϵ (−1,∞) and Kμ is the modified Bessel function of third kind of order μ, to certain generalized functions. A testing space is constructed so as to contain the Kernel, (pt)μ/2Kμ(2pt), of the transformation. Some properties of the kernel, function space and its dual are derived. The generalized Meijer transform, M¯μf, is now defined on the dual space. This transform is shown to be analytic and an inversion theorem, in the distributional sense, is established.