A Simple Proof of the Uniform Convergence of Fourier Series in Solutions to the Wave Equation

TL;DR

This paper provides a straightforward proof, using nonstandard analysis, that Fourier series of smooth solutions to the wave equation on a finite interval converge uniformly, enhancing understanding of wave phenomena.

Contribution

It introduces a simple, nonstandard analysis-based proof of uniform convergence for Fourier series solutions to the wave equation.

Findings

Fourier series of smooth wave solutions converge uniformly.

Nonstandard methods simplify convergence proofs.

Applicable to functions with vanishing boundary conditions.

Abstract

Using nonstandard methods, we show that the time dependent Fourier series of any smooth function F, solving the wave equation, on a finite closed interval, with vanishing boundary conditions, converges uniformly to F.