Computational and Analytical Study of Variations and Generalizations of the FC-Gram Approximation Algorithm

TL;DR

This paper introduces a generalized FC-Gram framework (GenFC) that enhances the approximation accuracy of the original FC-Gram algorithm by providing greater flexibility in constructing periodic extensions, supported by convergence analysis and numerical experiments.

Contribution

The paper develops a flexible GenFC framework that generalizes the FC-Gram algorithm, improving approximation accuracy and encompassing previous methods as special cases.

Findings

GenFC achieves better approximation accuracy than the original FC-Gram.

Convergence rate of the trigonometric interpolant is established as $ ext{O}(n^{- ext{min}(r+eta, d)})$.

Numerical experiments confirm the theoretical convergence rates and improved accuracy.

Abstract

The FC-Gram algorithm approximates non-periodic functions to high order by constructing a periodic extension with controlled boundary behavior and applying trigonometric interpolation. In this paper we introduce a generalized FC-Gram framework (GenFC), which provides greater flexibility in the construction of the blending continuation of Gram polynomials. This flexibility gives better control over the shape of the periodic extension and leads to improved approximation accuracy. We establish a convergence theorem showing that the trigonometric interpolant converges at the rate $\mathcal{O}(n^{-\min(r+\beta,\,d)})$ in the supremum norm on the original interval, where $r$ is the smoothness of the target function, $d$ the number of Gram polynomials, and $\beta \in [0,1]$ a Fourier-decay parameter. The framework and its analysis are developed so that the modified FC-Gram method of [J. Sci. Comput., 105(1):8, 2025] is recovered as a particular case. Numerical experiments confirm the predicted convergence rates and show that the added flexibility of the GenFC framework leads to improved approximation accuracy, with the gains carrying over to a Fourier continuation solver for two-point boundary value problems.