可 injectivity 和 weak*-to-weak 连续性足以保证 $\ell^1$-正则化在收敛率中的表现
Injectivity and weak*-to-weak continuity suffice for convergence rates in $\ell^1$-regularization
AI总结：
本文证明了在线性病态方程中，若精确解稀疏且算子可 injective 且 weak*-to-weak 连续，则 $\ell^1$-正则化的收敛率为 $O(δ)$。
摘要：
我们证明了对于线性病态方程，若精确解稀疏且算子可 injective 且 weak*-to-weak 连续，则 $\ell^1$-正则化的收敛率始终为 $O(δ)$。在相同假设下，非稀疏解的情况下的收敛率也得到了证明。这些结果基于文献中用于证明收敛率的某些源型条件自动满足的事实。
英文摘要：
We show that the convergence rate of $\ell^1$-regularization for linear ill-posed equations is always $O(δ)$ if the exact solution is sparse and if the considered operator is injective and weak*-to-weak continuous. Under the same assumptions convergence rates in case of non-sparse solutions are proven. The results base on the fact that certain source-type conditions used in the literature for proving convergence rates are automatically satisfied.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
