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Positive Matrices in the Hardy Space with Prescribed Boundary Representations via the Kaczmarz Algorithm

For a singular probability measure $μ$ on the circle, we show the existence of positive matrices on the unit disc which admit a boundary representation on the unit circle with respect to $μ$. These positive matrices are constructed in several different ways using the Kaczmarz algorithm. Some of these positive matrices correspond to the projection of the Szegő kernel on the disc to certain subspaces of the Hardy space corresponding to the normalized Cauchy transform of $μ$. Other positive matrices are obtained which correspond to subspaces of the Hardy space after a renormalization, and so are not projections of the Szegő kernel. We show that these positive matrices are a generalization of a spectrum or Fourier frame for $μ$, and the existence of such a positive matrix does not require $μ$ to be spectral.