Localized bases in [formula omitted] and their use in the analysis of Brownian motion

Abstract

Motivated by problems on Brownian motion, we introduce a recursive scheme for a basis construction in the Hilbert space L 2 ( 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 L 2 ( 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 L 2 ( 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.