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

TL;DR

This paper constructs positive matrices in the Hardy space with boundary representations related to a singular measure using the Kaczmarz algorithm, generalizing spectral concepts without requiring spectral measures.

Contribution

It introduces new methods to construct positive matrices with boundary representations in the Hardy space, extending the concept of spectral measures and Fourier frames.

Findings

Positive matrices admit boundary representations with respect to singular measures

Construction methods include projections of the Szeg"H{o} kernel and renormalized subspaces

Existence of such matrices does not require the measure to be spectral

Abstract

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.