Meaning of uniformly Cesàro summable

Question

There is a theorem like: The Fourier series of a continuous function $f(x)$ defined on $[-\pi,\pi]$ of period $2\pi$ is Uniformly Cesàro summable to $f(x)$. Now, I don't know the definition of Uniform Cesaro Summability. Can anyone explain this to me and give the proof of this theorem ?

Answer 1

If $(S_n)$ is the partial sum sequence of the Fourier series of $f$ then uniform Cesaro summability means $\sigma_n \to f$ uniformly, where $\sigma_n =\frac {S_0+S_1+...+S_n} {n+1}$. This result is called Fejer's Theorem and you can find it in many books as well as Wikipedia. See, for example, 6.1.1, p. 87 of Edward's Fourier Series. (Take $k=0$).

Answer 2

Let $S_n f$ be equal to n-nth partiał sum od Fourier series od tej function $f$, i.e, $$S_n f (x) = \frac{a_0}{2}+\sum_{k=1}^n (a_k\cos kx + b_k\sin kx )$$. Ten thę Cęsaro partiał sum arę equał to $$c_n f(x)=\frac{S_1 f(x) +...+S_n f(x)}{n}.$$

And Thę Theorem says that $$c_n f\to f$$ uniformły on $[-\pi ,\pi].$