The Kaczmarz Algorithm in Hilbert $C^{*}$-modules
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908494758674432 |
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| author | Alpay, Daniel Berner, Chad Weber, Eric S. |
| author_facet | Alpay, Daniel Berner, Chad Weber, Eric S. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_13861 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Kaczmarz Algorithm in Hilbert $C^{*}$-modules Alpay, Daniel Berner, Chad Weber, Eric S. Functional Analysis 41A65, 46L08 (Primary) 42C15 (Secondary) 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. |
| title | The Kaczmarz Algorithm in Hilbert $C^{*}$-modules |
| topic | Functional Analysis 41A65, 46L08 (Primary) 42C15 (Secondary) |
| url | https://arxiv.org/abs/2508.13861 |