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:$c\_n=frac\{1\}\{2pi\}int\_\{-pi\}^pi\; f(t)e^\{-int\}dt,$and:$sigma\_n(x)=frac\{1\}\{n\}sum\_\{k=0\}^\{n-1\}s\_k(x)=frac\{1\}\{2pi\}int\_\{-pi\}^pi\; f(x-t)F\_n(t)dt,$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 harv|Zygmund|1968|loc=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

:$sigma\_n(x\_0)\; o\; frac\{1\}\{2\}left(f(x\_0+0)+f(x\_0-0)\; ight).$

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 harv|Zygmund|1968|loc=Theorem III.5.1.

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