# Sampling and Frequency Warping

## TL;DR

This paper introduces a frequency warping technique to transform non band-limited functions into band-limited ones, enabling Fourier decomposition and new basis functions, with theoretical and numerical analysis of reconstruction and convergence.

## Contribution

It proposes a novel frequency warping method that creates new orthonormal bases for Sobolev spaces, enhancing sampling and reconstruction of non band-limited functions.

## Key findings

- Effective frequency warping transforms functions into band-limited form
- New orthonormal bases are expressed via Laguerre functions for integer alpha
- Numerical experiments demonstrate improved reconstruction and convergence

## Abstract

Optimal sampling of non band-limited functions is an issue of great importance that has attracted considerable attention. We propose to tackle this problem through the use of a frequency warping: First, by a nonlinear shrinking of frequencies, the function is transformed into a band-limited one. One may then perform a decomposition in Fourier series. This process gives rise to new orthonormal bases of the Sobolev spaces H^alpha. When alpha is an integer, these orthonormal bases can be expressed in terms of Laguerre functions. We study the reconstruction and speed of convergence properties of the warping-based sampling. Besides theoretical considerations, numerical experiments are performed.