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

For a singular probability measure $\mu$ 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 $\mu$. These positive matrices are constructed in several different ways using the Kaczmarz algorithm. Some of these positive matrices correspond to the projection of the Szeg\H{o} kernel on the disc to certain subspaces of the Hardy space corresponding to the normalized Cauchy transform of $\mu$. Other positive matrices are obtained which correspond to subspaces of the Hardy space after a renormalization, and so are not projections of the Szeg\H{o} kernel. We show that these positive matrices are a generalization of a spectrum or Fourier frame for $\mu$, and the existence of such a positive matrix does not require $\mu$ to be spectral.