On two new characterizations of Stieltjes transforms for distributions (Q1084300)

For a function f, a complex number p, and a positive integer n, the transformations, dilation \(u_ n\) and exponential shift \(T^{-p}\) are respectively defined as: \[ u_ nf(t)=nf(nt)\quad and\quad T^{- p}f(t)=e^{-pt}f(t). \] Following the technique used by Gasztelyi for defining the Laplace transform, the present author has characterized the Stieltjes transform of distributions in the following two ways: I. If f is a Stieltjes transformable distribution (in a certain space), then the Stieltjes transform of f is defined as: \[ S[f](p)=(1/\phi (0))\lim_{j\to \infty}<u_ jT^{-p} f,\phi > \] where \(f(t)=S[\xi](p)\), \(0\leq t\leq R\) for every positive R, \(\phi\) is a test function in D, with \(\phi\) (0)\(\neq 0\) and \(\xi\) is another distribution. II. Another characterization of the Stieltjes transform of a distribution f is given as: \[ S[f](p)=<T^{-p} f,1> \] where \(f(t)=S[\xi](p)\), \(0\leq t\leq R\) for any positive R and \(\xi\) is another distribution. The standard theorem on analyticity, uniqueness and invertibility of Stieltjes transform using the above definitions are proved.