# Convergence of Cesaro sums on $L^p$

Let $$K_N=\frac{1}{N}\sum_{n=0}^{N-1}D_n(x)$$ be the Fejer kernel and let $$\sigma_N(f)=\frac{1}{N}\sum_{n=0}^{N-1}S_N(f)$$ where $$S_n(f)=\frac{1}{\pi}\int_{-\pi}^{\pi}f(\tau)D_n(t-\tau)d\tau=f*D_n.$$ With "*" we denote the convolution.

Then it is easy to prove that $$\sigma_N(f)=f*K_N$$.

My question is if $$f\in L^p[-\pi,\pi]$$ how to prove that $$\lim_{N\to \infty}\left\lVert \sigma_N(f)-f\right\rVert_p=0?$$

Hint: Fejer's Theorem says that if $$f$$ is a continuous periodic function then $$\|\sigma_N(f) -f\|_{\infty} \to 0$$. This also implies that $$\|\sigma_N(f) -f\|_{p} \to 0$$. Now use the fact that continuous periodic functions are dense in $$L^{p}$$. You will also need Young's inequality $$\|f*g\|_p \leq \|f\|_p \|g\|_{1}$$ and $$\|g\|_{1}=1$$ when $$g$$ is the Fejer kernel.
• The last inequaliti comes from the fact that if $1\leq p < q$ implies $||g||_q < ||g||_p$, right? – Emo Feb 23 '20 at 11:59
• The inequality is called Young's inequality. See en.wikipedia.org/wiki/Young%27s_convolution_inequality [ Put $q=1, r=p$ in that inequality] @Emo – Kavi Rama Murthy Feb 23 '20 at 12:07
• @Emo $\|f\|_p \leq (2\pi)^{1/p} \|f\|_{\infty}$. – Kavi Rama Murthy Feb 23 '20 at 12:49