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Localized bases in L^2(0,1) and their use in the analysis of Brownian motion

Palle E. T. Jorgensen·Anilesh Mohari
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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 (mu). That is, we consider recursive and orthogonal decompositions for the Hilbert space L^2(mu) where mu 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.

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Palle E. T. Jorgensen, Anilesh Mohari. 2007-03-18. Localized bases in L^2(0,1) and their use in the analysis of Brownian motion. https://arxiv.org/abs/math/0512659

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