# Fejér's theorem

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Fejér's theorem

In mathematics, Fejér's theorem, named for Hungarian mathematician Lipót Fejér, states that if "f":R &rarr; C is a continuous function with period 2&pi;, then the sequence (&sigma;"n") of Cesàro means of the sequence ("s""n") of partial sums of the Fourier series of "f" converges uniformly to "f" on [-&pi;,&pi;] .

Explicitly,:$s_n\left(x\right)=sum_\left\{k=-n\right\}^nc_ke^\left\{ikx\right\},$where:$c_n=frac\left\{1\right\}\left\{2pi\right\}int_\left\{-pi\right\}^pi f\left(t\right)e^\left\{-int\right\}dt,$and:$sigma_n\left(x\right)=frac\left\{1\right\}\left\{n\right\}sum_\left\{k=0\right\}^\left\{n-1\right\}s_k\left(x\right)=frac\left\{1\right\}\left\{2pi\right\}int_\left\{-pi\right\}^pi f\left(x-t\right)F_n\left(t\right)dt,$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(-&pi;,&pi;). 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

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

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 &sigma;"n" is replaced with (C, α) mean of the Fourier series harv|Zygmund|1968|loc=Theorem III.5.1.

References

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