Bases of Wavelets and Multiresolution in Analysis on Wiener Space

Abstract

The multiresolution analysis is applied into the space of square integrable Wiener functionals for extending well-known constructions of orthonormal wavelets in L2(R) to this space denoted by L2 (μ), μ being the Wiener measure, as for instance Mallat’s construction or furthermore Goodman–Lee and Tang construction. We also extend the Calderon–Zygmund decomposition theorem into the L1(μ) framework. Even if L1-spaces do not have unconditional bases, wavelets still outperform Fourier analysis in some sense. We illustrate this by introducing periodized Wiener wavelets.