Fejér's theorem
In mathematics, Fejér's theorem, named after Hungarian mathematician Lipót Fejér, states the following: == Explanation of Fejér's Theorem's == Explicitly, we can write the Fourier series of f {\displaystyle f} as f ( x ) = ∑ n = − ∞ ∞ c n e i n x {\displaystyle f(x)=\sum _{n=-\infty }^{\infty }c_{n}\,e^{inx}} where the n {\displaystyle n} th partial sum of the Fourier series of f {\displaystyle f} may be written as s n ( f , x ) = ∑ k = − n n c k e i k x , {\displaystyle s_{n}(f,x)=\sum _{k=-n}^{n}c_{k}e^{ikx},} where the Fourier coefficients c k {\displaystyle c_{k}} are c k = 1 2 π ∫ − π π f ( t ) e − i k t d t . {\displaystyle c_{k}={\frac {1}{2\pi }}\int _{-\pi }^{\pi }f(t)e^{-ikt}dt.} Then, we can define σ n ( f , x ) = 1 n ∑ k = 0 n − 1 s k ( f , x ) = 1 2 π ∫ − π π f ( x − t ) F n ( t ) d t {\displaystyle \sigma _{n}(f,x)={\frac {1}{n}}\sum _{k=0}^{n-1}s_{k}(f,x)={\frac {1}{2\pi }}\int _{-\pi }^{\pi }f(x-t)F_{n}(t)dt} with F n {\displaystyle F_{n}} being the n {\displaystyle n} th order Fejér kernel.