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 (σn) 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}^nc_ke^{ikx},

where

c_n=\frac{1}{2\pi}\int_{-\pi}^\pi f(t)e^{-int}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.

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%2B0)%2Bf(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).

References