Model subspaces techniques to study Fourier expansions in L2 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 L 2 ( T , μ ) using techniques from the theory of model subspaces of the Hardy space. Since the sequence of monomials { z n } n ∈ N is effective in L 2 ( 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 P φ ( z n ) for the model subspace H ( φ ) = H 2 ⊖ φ H 2 , where P φ is the orthogonal projection from the Hardy space H 2 onto H ( φ ) . The study of Fourier expansions in L 2 ( 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.