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 (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, we have

[s_n(x)=\sum_^nc_ne^,]

where

[c_n=\frac\int_^\pi f(t)e^dt,]

and

[\sigma_n(x)=\frac\sum_^s_k(x)=\frac\int_^\pi f(x-t)F_n(t)dt,]

with F_n being the nth order Fejér kernel.

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