Model subspaces techniques to study Fourier expansions in L^2 spaces associated to singular measures

Abstract

Let μ be a probability measure on T that is singular with respect to the Haar measure. In this paper we study Fourier expansions in L2(T,μ) using techniques from the theory of model subspaces of the Hardy space. Since the sequence of monomials {zn}n∈N is effective in L2(T,μ), it has a Parseval frame associated via the Kaczmarz algorithm. Our first main goal is to identify the aforementioned frame with boundary values of the frame Pφ(zn) for the model subspace H(φ)=H2⊖φH2, where Pφ is the orthogonal projection from the Hardy space H2 onto H(φ). The study of Fourier expansions in L2(T,μ) also leads to consider positive kernels in the Hardy space. Our second main goal is to study the set of measures μ which reproduce a kernel contained in a model subspace. We completely characterize this set when the kernel is the reproducing kernel of a model subspace, and we study the consequences of this characterization.