Fast Algorithms for the computation of Fourier Extensions of arbitrary length

TL;DR

This paper introduces two efficient algorithms with ${ m O}(N ext{log}^2 N)$ complexity for computing Fourier extensions of arbitrary length, leveraging Prolate Spheroidal Wave theory to improve over previous methods.

Contribution

It develops two novel fast algorithms for Fourier extension computation for any domain size, generalizing prior work limited to specific cases, and exploits connections to Prolate Spheroidal Wave functions.

Findings

Algorithms achieve ${ m O}(N ext{log}^2 N)$ complexity.

Both algorithms are effective for general $T$ values.

The methods outperform previous ${ m O}(N^3)$ approaches.

Abstract

Fourier series of smooth, non-periodic functions on $[-1,1]$ are known to exhibit the Gibbs phenomenon, and exhibit overall slow convergence. One way of overcoming these problems is by using a Fourier series on a larger domain, say $[-T,T]$ with $T>1$, a technique called Fourier extension or Fourier continuation. When constructed as the discrete least squares minimizer in equidistant points, the Fourier extension has been shown shown to converge geometrically in the truncation parameter $N$. A fast ${\mathcal O}(N \log^2 N)$ algorithm has been described to compute Fourier extensions for the case where $T=2$, compared to ${\mathcal O}(N^3)$ for solving the dense discrete least squares problem. We present two ${\mathcal O}(N\log^2 N )$ algorithms for the computation of these approximations for the case of general $T$, made possible by exploiting the connection between Fourier extensions and Prolate Spheroidal Wave theory. The first algorithm is based on the explicit computation of so-called periodic discrete prolate spheroidal sequences, while the second algorithm is purely algebraic and only implicitly based on the theory.