Localized bases in \(L^2(0,1)\) and their use in the analysis of Brownian motion (Q2481509)

The main result in this paper concerns bases in the Hilbert spaces \(L^2(X,\mu)\) defined from self-similar measures. The results of the author are spelled out in more detail for \(L^2((0,1); \text{Lebesgue})\), where the classical Walsh system is a special case. The construction uses ideas from dynamical systems and operator algebras. Up to two specific reflection symmetries the scheme produces infinite families of orthonormal bases in \(L^2((0,1); \text{Lebesgue})\).