# Fejér's theorem

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

Explicitly,

where

and

with *F*_{n} being the *n*th order Fejér kernel.

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 *L*^{1}(-π,π). If the left and right limits *f*(*x*_{0}±0) of *f*(*x*) exist at *x*_{0}, or if both limits are infinite of the same sign, then

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 σ_{n} is replaced with (C, α) mean of the Fourier series (Zygmund 1968, Theorem III.5.1).

## References

- Zygmund, Antoni (1968),
*Trigonometric series*(2nd ed.), Cambridge University Press (published 1988), ISBN 978-0-521-35885-9.