Palle E. T. Jorgensen, Anilesh Mohari
Published 2005-12-30, updated 2007-03-18Version 3
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.
Comments: 26 pages, LaTeX2e "elsart" document class, 2 figures comprising 33 EPS graphics. v2: corrections for consistency, more explanation about the tools which might not be too widely known, and clearer presentation of the results in the application to Brownian motion in the final section. v3: corrections for consistency and to meet the requirements from refree's report for the J. Approx. Theory
Martingales, endomorphisms, and covariant systems of operators in Hilbert space
Brownian motion in a ball in the presence of spherical obstacles
Matrix representations for some self-similar measures on $\mathbb{R}^{d}$
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