Fejér's theorem

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In mathematics, Fejér's theorem, named for 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, we have

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.

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