Necessary and sufficient conditions for the representation of a function by a Weierstrass transform

where R and S tend to + oo independently. In (1), the function 0(y) is absolutely integrable, and in (2), a(y) is of bounded variation, both in every finite interval. We abbreviate (1) and (2) by f(x) = W[qS(x) ] and f(x) = WT [a(x) ], respectively. Our aim is to characterize the of functions f(x) which have the representation (2) with increasing a(y). It is well known that Weierstrass used an integral of type (1) to give the first proof that a continuous function can be uniformly approximated by a polynomial. Hence the terminology. This name for the transform was used by Pollard [1946; 317](1). Other names used in the literature are Gauss transform [Pollard 1943; 59], Gauss-Weierstrass transform [Hille 1948; 3711 and IHille transform [Gonzailez Domlnguez 1941; 34]. Hille [1926] made an extensive study of the functions f(x) and +k(x) of (1), referring to them as a class of reciprocal functions. A symbolic inversion of the transform (1) was first given by Eddington [1914] as follows: