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The fact that change of scale is a pathological transformation in Wiener space has long been known. For many problems, this pathology causes no special difficulties. However it is sometimes necessary to consider functions of the form F(λx) where λ varies over the positive reals and x varies over Wiener space. In this setting a variety of conceptual subtleties arise. In this paper we give a framework and several results which prove useful in dealing with these difficulties. In the last section of this paper we discuss several papers in the recent literature in the light of this framework. 1* Notation and terminology; introduction* Let T = [0, 1]and let C 0 (T) denote Wiener space, that is, the space of real-valued continuous functions on T which vanish at t = 0 (The notation C Q (T) will never be abbreviated to C o . The latter notation will be introduced latter for a certain proper subsets of C 0 (T).). Let & denote the Borel measurable subsets of C 0 (T) and let m 1 denote Wiener measure. One can complete (C 0 (Γ), &, mj in the usual way to obtain (C 0 (Γ), S^u m x ) where Sf x is the class of all Wiener measurable sets. Let σ n be the partition 0 = t 0 < t λ < < ί 2 » = 1 where t k = k/2 n for k = 0, 1, -, 2\ Given x in C 0 (T), let S σn (x) = Σf=i [»(«*)α(**-i)] a . For λ ^ 0, let C λ = {x in C 0 (T): lim^ S σ Jx) = λ 2 } and let D = {x in C 0 (T):lim^S σ J>) fails to exist}. Note that \C μ = C λμ .Clearly D and the sets C λ9 λ ^ 0, are all Borel sets and C 0 (T) is the disjoint union of this family of sets.The key to our discussion is the following result due to Levy [35] and independently, but later, to Cameron and Martin [8]. THEOREM 1. m^CJ = 1. 157 158 G. W. JOHNSON AND D. L. SKOUG

Abstract. In this paper we develop an Lp Fourier-Feynman theory for a class of functionals on Wiener space of the form F(x) = f(J0 axdx, ... , /0 a"dx). We then define a convolution product for functionals on Wiener space and show that the Fourier-Feynman transform of the convolution product is a product of Fourier-Feynman transforms.

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In this paper we use generalized Fourier-Hermite functionals to obtain a complete orthonormal set inis a very general function space. We then proceed to give a necessary and sufficient condition that a functional F in L 2 (C a,b [0, T ]) has an integral transform F γ,β F also belonging to L 2 (C a,b [0, T ]).