Schauder-type expansions of continuous functions on the unit interval

AbstractLet the space of continuous functions on [0, 1] which vanish at 0 be denoted by C. It will be shown that for any complete orthonormal set of functions {αi(s)} of bounded variation and such that αi(1) = 0, there is a simply described linear combination of the continuous functions {∝0tαi(s) ds} which converges uniformly to x(t) for almost all x ϵ C (“almost all” in the sense of Wiener measure)