Fejér's theorem

Fejér's Theorem—Let ${\displaystyle f:\mathbb {R} \to \mathbb {C} }$ be a continuous function with period ${\displaystyle 2\pi }$, let ${\displaystyle s_{n}(f)}$ be the nth partial sum of the Fourier series of ${\displaystyle f}$, and let ${\displaystyle \sigma _{n}(f)}$ be the sequence of Cesàro means of the sequence ${\displaystyle s_{n}(f)}$, that is the sequence of arithmetic means of ${\displaystyle s_{0}(f),...,s_{n}(f)}$. Then the sequence ${\displaystyle \sigma _{n}(f)}$ converges uniformly to ${\displaystyle f}$ on ${\displaystyle \mathbb {R} }$ as n tends to infinity.