Representation of the Fourier transform as a weighted sum of the complex error functions

TL;DR

This paper presents a novel method to represent the Fourier transform as a weighted sum of complex error functions, simplifying computations and enabling non-periodic wavelet approximations for improved algorithmic efficiency.

Contribution

The paper introduces a new sampling-based approach that expresses the Fourier transform as a sum of complex error functions, offering a simplified and non-periodic wavelet approximation.

Findings

Fourier transform expressed as a weighted sum of complex error functions

Simplification using the properties of the complex error function

Potential for improved algorithmic implementation with non-periodic wavelet approximation

Abstract

In this paper we show that a methodology based on a sampling with the Gaussian function of kind $h\,{e^{ - {{\left( {t/c} \right)}^2}}}/\left( {{c}\sqrt \pi } \right)$, where ${c}$ and $h$ are some constants, leads to the Fourier transform that can be represented as a weighted sum of the complex error functions. Due to remarkable property of the complex error function, the Fourier transform based on the weighted sum can be significantly simplified and expressed in terms of a damping harmonic series. In contrast to the conventional discrete Fourier transform, this methodology results in a non-periodic wavelet approximation. Consequently, the proposed approach may be useful and convenient in algorithmic implementation.