Stability of the Kaczmarz Reconstruction for Stationary Sequences

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.

Appendix

Proof of Proposition 2

The first statement follows from the second, but an alternative proof can be found in [10, Theorem 3.10]. The second statement follows from Fatou’s construction [23, Section II.A]. $\square$

Proof of Lemma 3

For the inner function $b^{K}$, let $\mu ^{K}$ denote the corresponding singular measure. By the Herglotz Representation, $\mu<< \mu ^{K}$. Since by [28] every $f \in H^2 \ominus b^{K} H^2$ has convergent Fourier series in $L^2(\mu ^{K})$, it follows that the Fourier series also converges in $L^2(\mu )$. $\square$