# Fourier series windowed by a bump function

## TL;DR

This paper investigates how windowing a function with a bump function affects Fourier series convergence, extending classical results to non-periodic functions and supporting findings with numerical experiments.

## Contribution

It extends Jackson's classical Fourier series convergence results to non-periodic functions using bump function windowing.

## Key findings

- Theoretical extension of Fourier series convergence results.
- Numerical experiments confirm the theoretical predictions.

## Abstract

We study the Fourier transform windowed by a bump function. We transfer Jackson's classical results on the convergence of the Fourier series of a periodic function to windowed series of a not necessarily periodic function. Numerical experiments illustrate the obtained theoretical results.