This paper pioneers the application of reproducing kernel spaces in proving the stability of numerical methods for approximating linear differential equations. We analyze families of products of Hardy space Toeplitz operators that do not fall into a well-understood class of operators through explicit techniques, many of which are based upon exploiting the reproducing kernel Hilbert space structure of the Hardy space on the unit disk and properties of Toeplitz operators. This results in bounds that would not be feasible through more general functional analytic techniques. In particular, for these operators, we relate the power bound and a resolvent condition of Kreiss-Ritt type since it is well-known that such a relationship implies the stability of certain methods for the numerical solution of linear differential equations. Our methods apply mutatis mutandis to operators of the form $T_{g(z)}^{-1}T_{f(z)}T_{g(z)}$ where $f(z)$ is a polynomial in $z$ and $\overline{z}$ and $g(z)$ is a polynomial in $z$, which arise frequently in the numerical solution of the Cauchy problem for linear ordinary, partial, and delay differential equations used as models for processes in science and engineering.

翻译：本文开创性地应用再生核空间证明线性微分方程数值逼近方法的稳定性。我们通过显式技巧分析不属于已充分理解算子类别的Hardy空间Toeplitz算子乘积族，这些技巧主要基于单位圆盘上Hardy空间的再生核Hilbert空间结构及Toeplitz算子性质。由此导出的估计界无法通过更一般的泛函分析技术获得。具体而言，针对这类算子，我们建立了幂有界性与Kreiss-Ritt型预解条件之间的关联——已知此类关联蕴含特定线性微分方程数值求解方法的稳定性。我们的方法经适当修正可适用于形如 $T_{g(z)}^{-1}T_{f(z)}T_{g(z)}$ 的算子（其中 $f(z)$ 为 $z$ 与 $\overline{z}$ 的多项式，$g(z)$ 为 $z$ 的多项式），这些算子频繁出现在作为科学与工程过程模型的线性常微分方程、偏微分方程及延迟微分方程Cauchy问题的数值求解中。