The Mejer transformation of generalized functions (Q1092378)

This paper extends the Meijer transformation, \(M_{\mu}\), given by \[ (M_{\mu}f)(p)=\frac{2p}{\Gamma (1+\mu)}\int^{\infty}_{0}f(t)(pt)^{\mu /2}K_{\mu}(2\sqrt{pt})dt, \] where f belongs to an appropriate function space, \(\mu\in (-1,\infty)\) and \(K_{\mu}\) is the modified Bessel function of third kind of order \(\mu\), to certain generalized functions. A testing space is constructed so as to contain the kernel, \((pt)^{\mu /2}K_{\mu}(2\sqrt{pt})\), of the transformation. Some properties of the function space and its dual are derived. The generalized Meijer transform, \(\bar M{}_{\mu}f\), is now defined on the dual space. This transform is shown to be analytic and an inversion theorem, in the distributional sense, is established. In a later paper to appear, the authors apply the transform to some boundary value problems with distributional conditions.