On the convergence of spectral methods involving non-compact operators

05/23/2023

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by   Thomas Trogdon, et al.

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Motivated by Fredholm theory, we develop a framework to establish the convergence of spectral methods for operator equations ℒ u = f. The framework posits the existence of a left-Fredholm regulator for 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.