Model subspaces techniques to study Fourier expansions in L2 spaces associated to singular measures

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 P φ (zⁿ) for the model subspace H (φ) = H² ⊖ φ H² , where P φ 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.Facultad de Ciencias Exacta