Transformations of Weiner Integrals Under Translations

In his paper on Generalized Harmonic Analysis (in which references to his earlier work are given) N. Wiener I defines an average or integral over the space C of all functions x(t) continuous in 0 ? t ? 1 and vanishing at t = 0. The integral proved to be very useful in the theory developed there. More recently, the present authors II have shown that the solution of a fairly general non-linear equation can be expressed in terms of certain Wiener integrals. In view of these rather distinct problems in which the Wiener integrals have been useful, it seems worthwhile to develop further certain aspects of the Wiener integral. In the present paper we show how the integral transforms under a translation, and we consider special cases of translations which seem to lead to rather interesting results. Our first (restricted) transformation theorem is as follows: THEOREM 1. Let Fy be a functional defined and Wiener summable over C, let Fy be bounded in y(.) for y(-) in any uniformly bounded set, and let Fy be continuous in the sense that if Iy(n)(t) is any sequence of functions of C which converge uniformly in 0 < t ? 1 to y(t) then