On Trigonometric Interpolation and Its Applications

TL;DR

This work introduces a FFT-accelerated trigonometric interpolation method with improved convergence properties, including uniform rates for derivatives, and extends applicability to non-periodic functions through a smooth cut-off extension. Theoretical results are supported by coefficient decay bounds and convergence proofs, aided by Abel transform techniques. Numerically, the method demonstrates high accuracy and stability, outperforming Trapezoid/Simple Simpson rules for integrals and standard Runge-Kutta methods for ODEs, while exhibiting a local error property that mitigates error propagation. The approach holds significant practical impact for efficient numerical integration and ODE solving across periodic and non-periodic domains, with potential for broad extensions.

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.

Figures (8)

• Figure 1: The graphs of $h_M$ over $[s-\delta, s]$ with $(s,e, \delta)=(-1,1, 1)$ and $M=2^8$ for three cases: $r=0.1$ (red), $r=0.5$ (blue) and $r=1$ (green).
• Figure 2: The graphs of the $fh$'s extension $(hf)_{ext}$ over a period $[2s-e-3\delta, e+\delta]$ with $(s,e,\delta)=(2,3,1)$. The function is even after parallel shift left by $s-\delta=1$. Note that $fh=f=(x-2.5)^2$ over $[s,e]$ as expected.
• Figure 3: The graphs of $f(x)$ vs $\hat{f}_M(x)$ over $[e-\delta,s+\delta]$ with $(s,e, p, q, \delta)=(2,3,7,8,1)$. The figure is plotted by $2^{12}$ sample points.
• Figure 4: Plots of the normalized difference of consecutive error at grid points with $\lambda=1/2^6$ with $[s,e]=[2,3]$
• Figure 5: The solutions of ODE (\ref{['ODE_1']})-(\ref{['ODE_2']}) by Algorithm \ref{['alg_ode']} with three initial values at $s$: $(0, 1, 2)$ with $q=8$.

Theorems & Definitions (13)

• Theorem 1.1
• Theorem 2.1
• Remark 1
• Theorem 2.2
• Theorem 2.3
• Lemma 3.1
• proof
• proof
• Lemma 3.2
• proof