Approximation error in regularized SVD-based Fourier continuations

We present an analysis of the convergence of recently developed Fourier continuation techniques that incorporates the required truncation of the Singular Value Decomposition (SVD). Through the analysis, the convergence of SVD-based continuations are related to the convergence of any Fourier approximation of similar form, demonstrating the efficiency and accuracy of the numerical method. The analysis determines that the Fourier continuation approximation error can be bounded by a key value that depends only on the parameters of the Fourier continuation and on the points over which it is applied. For any given distribution of points, a finite number of calculations can be performed to obtain this important value. Our numerical computations on evenly spaced points show that as the number of points increases, this quantity converges to a fixed value, allowing for broad conclusions on the convergence of Fourier continuations calculated with truncated SVDs. We conclude that Fourier continuations can obtain super-algebraic or even exponential convergence on evenly spaced points for non-periodic functions until the convergence is limited by a parameter normally chosen near the machine precision accuracy threshold.