  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.
