A note on transformed Fourier systems for the approximation of non-periodic signals

TL;DR

This paper compares various non-periodic function approximation techniques, including rank-1 lattices and transformed Fourier systems, highlighting their similar approximation errors in the context of non-periodic signals.

Contribution

It introduces a parameterized transformed Fourier system that achieves comparable approximation errors to existing lattice-based methods for non-periodic functions.

Findings

Transformed Fourier systems can match the approximation accuracy of lattice-based methods.

Sampling sets like Chebyshev- and tent-transformed nodes are effective for non-periodic approximation.

The proposed method offers a flexible alternative with similar error performance.

Abstract

A variety of techniques have been developed for the approximation of non-periodic functions. In particular, there are approximation techniques based on rank-$1$ lattices and transformed rank-$1$ lattices, including methods that use sampling sets consisting of Chebyshev- and tent-transformed nodes. We compare these methods with a parameterized transformed Fourier system that yields similar $\ell_2$-approximation errors.