A generalized Laplace-Stieltjes transformation
Garner McCrossen · Proceedings of the American Mathematical Society · 1957
GARNER McCROSSEN1. Introduction.Gustav Doetsch1 has recently introduced a generalized Laplace transform of order k, k^.0.It is the object of this paper to introduce a generalized Laplace-Stieltjes transform of order k.We assume in what follows that k is positive and integral.We also assume that 7" denotes a complex-valued function of a real variable, which is defined, except perhaps on a set of isolated points, and is of bounded variation on every closed, finite interval [0, 7"], 7^0.By {7"} we mean the class of all functions F defined above.All integrals in this paper are to be interpreted as Riemann or Riemann-Stieltjes integrals.The symbol (*) will denote a convolution.2. Definition of the transform.Let 5 be complex.Let Mk(s, t) be the Cesaro mean of order k of J e-$udF(u), i.e., Mk(s, t) = -| f e->«dF(u)\* f-A .If for some s and some k Lim Mk(s, t) (-►+00 exists, then this limit defines a value fk(s) of a function/*: fk(s) = Lim Mk(s, t).We call/i the generalized Laplace-Stieltjes transform of order k of F.3. Existence and integral representations of the transform.We