平稳序列Kaczmarz重构的稳定性
Stability of the Kaczmarz Reconstruction for Stationary Sequences
摘要：
Kaczmarz算法是从内积$\langle f , φ_{n} \rangle $重建未知向量$f$的迭代法。在假设$\{ φ_{n} \}$为平稳序列的条件下，考虑了加性噪声对重建的影响。与其他重建方法（如帧重建）不同，Kaczmarz重建在存在噪声的情况下是不稳定的。然而，我们表明，重建可以通过放松Kaczmarz算法稳定，这种放松对应于阿贝尔求和时，被视为单位盘上的重建。此外，我们表明，对于某些噪声配置文件，如那些躺在$H^{\infty}(\mathbb{D})$或某些子空间的$H^{2}(\mathbb{D})$，放松版本的Kaczmarz算法可以完全消除腐败的噪声在内积。使用平稳序列的谱表示，我们表明，我们的放松版本的Kaczmarz算法也稳定重建的傅里叶级数展开在$L^2(μ)$时，$μ$是奇异的。
英文摘要：
The Kaczmarz algorithm is an iterative method to reconstruct an unknown vector $f$ from inner products $\langle f , φ_{n} \rangle $. We consider the problem of how additive noise affects the reconstruction under the assumption that $\{ φ_{n} \}$ 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 $L^2(μ)$ when $μ$ is singular.
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