  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.
