Localized bases in L(0, 1) and their use in the analysis of Brownian motion

Motivated by problems on Brownian motion, we introduce a recursive scheme for a basis construction in the Hilbert space L2(0, 1) which is analogous to that of Haar and Walsh. More generally, we find a new decomposition theory for the Hilbert space of square-integrable functions on the unit-interval, both with respect to Lebesgue measure, and also with respect to a wider class of self-similar measures μ. That is, we consider recursive and orthogonal decompositions for the Hilbert space L2(μ) where μ is some self-similar measure on [0, 1]. Up to two specific reflection symmetries, our scheme produces infinite families of orthonormal bases in L2(0, 1). Our approach is as versatile as the more traditional spline constructions. But while singly generated spline bases typically do not produce orthonormal bases, each of our present algorithms does.