Does a Fourier series always converge to its function?

Not pointwise in general; continuous functions can have Fourier series that diverge at points, but the series always converges in the mean-square sense for square-integrable functions, and summability methods recover uniform convergence for continuous ones.

What is the Gibbs phenomenon?

Near a jump discontinuity the partial sums of a Fourier series overshoot the function by a fixed proportion that does not vanish as more terms are added, an artifact of pointwise convergence at jumps.

Fourier Series

A Fourier series expands a periodic function as a sum of sines and cosines, decomposing it into its fundamental frequencies and raising the central question of when the series reconstructs the function.

Definition

A Fourier series is the representation of a periodic function as an infinite combination of sines and cosines, or complex exponentials, whose coefficients are determined by integrating the function against those basic oscillations.

Scope

This topic covers the Fourier coefficients of a periodic function, the partial sums and their Dirichlet kernel, pointwise and uniform convergence criteria, the Gibbs phenomenon at jumps, convergence in the mean and Parseval's identity, summability methods such as Cesaro and Abel means with the Fejer kernel, and completeness of the trigonometric system in square-integrable functions.

Core questions

How are the Fourier coefficients of a periodic function computed?
When does the Fourier series converge back to the function, and in what sense?
Why do summability methods restore convergence where partial sums fail?
Why does the trigonometric system form a complete orthonormal basis of square-integrable functions?

Key theories

Mean-square convergence and Parseval's identity: The Fourier series of a square-integrable periodic function converges to it in the mean-square sense, and the sum of the squared coefficients equals the squared norm of the function, expressing the trigonometric system as a complete orthonormal basis.
Fejer's theorem: The Cesaro means of the partial sums of the Fourier series of a continuous periodic function converge uniformly to the function, recovering convergence through averaging even when the partial sums themselves do not converge.

Clinical relevance

Fourier series are the foundation of spectral analysis of periodic signals, used in acoustics, vibration analysis, electrical engineering, and the solution of the heat and wave equations by separation of variables, where decomposing a state into frequency modes makes the equations solvable.

History

Fourier introduced trigonometric expansions in his 1822 theory of heat, claiming a generality that provoked decades of scrutiny. Dirichlet gave the first rigorous convergence theorem in 1829, and Fejer's 1900 summability result clarified convergence for continuous functions.

Key figures

Joseph Fourier
Lejeune Dirichlet
Lipot Fejer

Seminal works

stein2003fourier
katznelson2004

Frequently asked questions

Does a Fourier series always converge to its function?: Not pointwise in general; continuous functions can have Fourier series that diverge at points, but the series always converges in the mean-square sense for square-integrable functions, and summability methods recover uniform convergence for continuous ones.
What is the Gibbs phenomenon?: Near a jump discontinuity the partial sums of a Fourier series overshoot the function by a fixed proportion that does not vanish as more terms are added, an artifact of pointwise convergence at jumps.