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arXiv · 2508.13861

The Kaczmarz Algorithm in Hilbert $C^{*}$-modules

Daniel Alpay·Chad Berner·Eric S. Weber
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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 $C^*$-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 $C(X)$-modules can be generated by the Kaczmarz algorithm and realized as orbits of bounded operators.

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Daniel Alpay, Chad Berner, Eric S. Weber. 2025-08-19. The Kaczmarz Algorithm in Hilbert $C^{*}$-modules. https://arxiv.org/abs/2508.13861

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