Motivated by Fredholm theory, we develop a framework to establish the convergence of spectral methods for operator equations $\mathcal L u = f$. The framework posits the existence of a left-Fredholm regulator for $\mathcal L$ and the existence of a sufficiently good approximation of this regulator. Importantly, the numerical method itself need not make use of this extra approximant. We apply the framework to finite-section and collocation-based numerical methods for solving differential equations with periodic boundary conditions and to solving Riemann--Hilbert problems on the unit circle. We also obtain improved results concerning the approximation of eigenvalues of differential operators with periodic coefficients.

翻译：受弗雷德霍姆理论的启发，我们建立了一个框架来证明算子方程 $\mathcal L u = f$ 的谱方法收敛性。该框架假设 $\mathcal L$ 存在一个左弗雷德霍姆调节器，并且该调节器具有足够好的近似。重要的是，数值方法本身无需使用这一额外近似。我们将该框架应用于有限截断和基于配置的方法，以求解具有周期边界条件的微分方程，以及求解单位圆上的黎曼-希尔伯特问题。我们还得到了关于具有周期系数的微分算子特征值近似问题的改进结果。