# Fejér's theorem

In mathematics, Fejér's theorem, named after Hungarian mathematician Lipót Fejér, states that if f:R → C is a continuous function with period 2π, then the sequencen) of Cesàro means of the sequence (sn) of partial sums of the Fourier series of f converges uniformly to f on [-π,π].

Explicitly,

$s_{n}(x)=\sum _{k=-n}^{n}c_{k}e^{ikx},$ where

$c_{k}={\frac {1}{2\pi }}\int _{-\pi }^{\pi }f(t)e^{-ikt}dt,$ and

$\sigma _{n}(x)={\frac {1}{n}}\sum _{k=0}^{n-1}s_{k}(x)={\frac {1}{2\pi }}\int _{-\pi }^{\pi }f(x-t)F_{n}(t)dt,$ with Fn being the nth order Fejér kernel.

Then

$\lim _{N\to \infty }(\sigma _{N}f)(x)=f(x)$ with uniform convergence.

A more general form of the theorem applies to functions which are not necessarily continuous (Zygmund 1968, Theorem III.3.4). Suppose that f is in L1(-π,π). If the left and right limits f(x0±0) of f(x) exist at x0, or if both limits are infinite of the same sign, then

$\sigma _{n}(x_{0})\to {\frac {1}{2}}\left(f(x_{0}+0)+f(x_{0}-0)\right).$ Existence or divergence to infinity of the Cesàro mean is also implied. By a theorem of Marcel Riesz, Fejér's theorem holds precisely as stated if the (C, 1) mean σn is replaced with (C, α) mean of the Fourier series (Zygmund 1968, Theorem III.5.1).