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,

$s_n(x)=\sum_{k=-n}^nc_ke^{ikx},$

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 F_n 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

$\sigma_n(x_0) \to \frac{1}{2}\left(f(x_0+0)+f(x_0-0)\right).$

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 σ_n is replaced with (C, α) mean of the Fourier series (Zygmund 1968, Theorem III.5.1).

References

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