On Trigonometric Interpolation and Its Applications

TL;DR

This paper introduces a new trigonometric interpolation algorithm that improves efficiency and accuracy, especially for derivatives and non-periodic functions, with applications to integral estimation and solving ODEs, outperforming traditional methods.

Contribution

It proposes a novel trigonometric interpolation method with enhanced convergence and applicability to non-periodic functions, leveraging FFT for better efficiency and error control.

Findings

Achieves uniform convergence rates for functions and derivatives.

Outperforms Trapezoid/Simpson methods in integral estimation.

Outperforms Runge-Kutta in solving ODEs.

Abstract

In this paper, we propose a new trigonometric interpolation algorithm and establish relevant convergent properties. The method adjusts an existing trigonometric interpolation algorithm such that it can better leverage Fast Fourier Transform (FFT) to enhance efficiency. The algorithm can be formulated in a way such that certain cancellation effects can be effectively leveraged for error analysis, which enables us not only to obtain the desired uniform convergent rate of the approximation to a function, but desired uniform convergent rates for its derivatives as well. We further enhance the algorithm so it can be applied to non-periodic functions defined on bounded intervals. Numerical testing results confirm decent accurate performance of the algorithm. For its application, we demonstrate how it can be applied to estimate integrals and solve linear/non-linear ordinary differential equation (ODE). The test results show that it significantly outperforms Trapezoid/Simpson method on integral and standard Runge-Kutta algorithm on ODE. In addition, we show some numerical evidences that estimation error of the algorithm likely exhibits ``local property", i.e. error at a point tends not to propagate, which avoids significant compounding error at some other place, as a remarkable advantage compared to polynomial-based approximations.