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 Fourier 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：受弗雷德霍姆理论启发，我们建立了一个框架来论证算子方程$\mathcal L u = f$的谱方法的收敛性。该框架假设$\mathcal L$存在一个左弗雷德霍姆调节子，且该调节子存在足够好的近似。重要的是，数值方法本身无需利用这一额外近似。我们将该框架应用于以下两类数值方法：求解周期边界条件微分方程的傅里叶有限段法与配置法，以及求解单位圆上黎曼-希尔伯特问题的方法。此外，我们还获得了关于周期系数微分算子特征值逼近的改进结果。