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The Kaczmarz Algorithm in Hilbert C ∗ -modules

Year: 2025

Type: article

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.

Source: Expositiones Mathematicae

DOI: 10.1016/j.exmath.2025.125738

Authors: Daniel Alpay, Chad Berner, Eric S. Weber

Institutions: Chapman University, Missouri University of Science and Technology, Iowa State University

Language: English

Topic: Advanced Banach Space Theory

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,500

OpenAlex ID: W4416612334

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Cites: 15

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