Measures on the unit circle with slowly decaying reflection coefficients and Fourier series

The relation between the theory of orthogonal polynomials on the unit circle and the spectral theory of a class of matrix difference equations known as the Szegő equations is under the investigation. The key role is played by the matrix form of the Szegő recurrences, which are completely determined by a sequence of complex numbers from the open unit disk (reflection coefficients). The structure of measures (absolutely continuous and singular parts) with slowly decaying reflection coefficients is studied via the theory of uniformly convergent Fourier series.