The Kaczmarz Algorithm in Hilbert $C^{*}$-modules
Functional Analysis
2025-08-20 v1
Abstract
The Kaczmarz algorithm in Hilbert spaces is a classical iterative method for stably recovering vectors from inner product data. In this paper, we extend the algorithm to the setting of Hilbert -modules and establish analogues of its effectiveness in both finite-dimensional and stationary cases. Consequently, we demonstrate that continuous families of elements in a Hilbert space can be uniformly recovered using the Kaczmarz algorithm. Additionally, we develop a normalized Cauchy transform for continuous families of measures and use it to provide sufficient conditions under which standard frames in Hilbert -modules can be generated by the Kaczmarz algorithm and realized as orbits of bounded operators.
Cite
@article{arxiv.2508.13861,
title = {The Kaczmarz Algorithm in Hilbert $C^{*}$-modules},
author = {Daniel Alpay and Chad Berner and Eric S. Weber},
journal= {arXiv preprint arXiv:2508.13861},
year = {2025}
}