On Multiplicative Orthogonal Functions

The purpose of this paper is to reconstruct the Rademacher and Walsh systems and to generalize them by using a certain time series. In §§ 1-3, we discuss a method to construct the Rademacher and Walsh functions. Let t-space be a set of positive integers 1, 2, ... and x(t) be a discrete time series taking 1 or -1 with probabilities 1/2, for each t=1, 2, ……. Then by means of the analogous method to Wiener we can map these time series onto the unit interval. That is to say, every time series x(t) is identified with a point α of the interval 〔0, 1〕 . We call this unit interval α-space and we denote this time series by x(t, α). Put ψ_n(α)=x(n, α). Then we know by the construction of correspondence between time series and α-space ψ_n(α) are nothing but the Rademacher functions. Next we define homogeneous functions and nonhomogeneous functions on t-space by (2.1), (2.3), (2.3)' and (2.20). By means of (2.4) we can map L^2(t) linearly and isometrically into L^2(α). Using (2.8), (2.11) an