# Stability of the Kaczmarz Reconstruction for Stationary Sequences

## TL;DR

This paper investigates the stability of the Kaczmarz algorithm for reconstructing vectors from inner products with stationary sequences, proposing a relaxed version that mitigates noise effects and ensures stable reconstruction in various settings.

## Contribution

It introduces a relaxation of the Kaczmarz algorithm, akin to Abel summation, that stabilizes reconstruction in noisy environments for stationary sequences.

## Key findings

- Relaxed Kaczmarz algorithm stabilizes reconstruction under noise.
- Full noise removal achieved for certain function spaces.
- Stabilizes Fourier series reconstruction in singular measures.

## Abstract

The Kaczmarz algorithm is an iterative method to reconstruct an unknown vector $f$ from inner products $\langle f , \varphi_{n} \rangle $. We consider the problem of how additive noise affects the reconstruction under the assumption that $\{ \varphi_{n} \}$ form a stationary sequence. Unlike other reconstruction methods, such as frame reconstructions, the Kaczmarz reconstruction is unstable in the presence of noise. We show, however, that the reconstruction can be stabilized by relaxing the Kaczmarz algorithm; this relaxation corresponds to Abel summation when viewed as a reconstruction on the unit disc. We show, moreover, that for certain noise profiles, such as those that lie in $H^{\infty}(\mathbb{D})$ or certain subspaces of $H^{2}(\mathbb{D})$, the relaxed version of the Kaczmarz algorithm can fully remove the corruption by noise in the inner products. Using the spectral representation of stationary sequences, we show that our relaxed version of the Kaczmarz algorithm also stabilizes the reconstruction of Fourier series expansions in $L^2(\mu)$ when $\mu$ is singular.