- Fejér's theorem
In mathematics, Fejér's theorem, named for Hungarian
mathematician Lipót Fejér, states that if "f":R → C is a continuous functionwith period 2π, then the sequence(σ"n") of Cesàro means of the sequence ("s""n") of partial sums of the Fourier seriesof "f" converges uniformly to "f" on [-π,π] .
Explicitly,:where:and:with "F""n" being the "n"th order
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
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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