The Kaczmarz Algorithm in Hilbert <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e27" altimg="si3.svg"> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>∗</mml:mo> </mml:mrow> </mml:msup> </mml:

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.