Convergence of Cesaro sum of Fourier series at lebesgue points.

Question

Let $f : \mathbb R \to \mathbb C$ be a $ 1 $ periodic function ; the $f$ is determined by its values on $[0,1]$ . Let $$ \sigma_N(f)(x):=\dfrac 1{N+1}\sum_{k=0}^NS_k(f)(x) , \quad\forall x \in [0,1] , \quad\forall N \in \mathbb N, $$ where $ S_n(f)(x):=\sum_{k=-n}^n \widehat f (k) e^{2\pi ikx} , \forall x \in [0,1] , \forall n \in \mathbb N $, where $$ \widehat f (k):=\int_{0}^1 f(t)e^{-2\pi ikt}dt , \forall k \in \mathbb Z $$ Simplifying we can also see that $$ \sigma_N(f)=\int_{-1/2}^{1/2}f(x-t)F_N(t)dt, $$ where $$ F_N(x):=\dfrac 1{N+1}\dfrac {\sin^2 (\pi(N+1)x)}{\sin^2(\pi x)} ,\quad \forall x \in [0,1] ,\quad\forall N \in \mathbb N. $$ I want to show that if $ x_0 $ is the lebesgue point of $ f $, then $ \sigma_Nf(x_0)\to f(x_0) $ when $ N\to\infty $. I have already known that if $ f $ is continuous at the point $ x_0 $, then $ \sigma_Nf(x_0)\to f(x_0) $ when $ N\to\infty $. However I cannot show the this for such case. Can you give me some hints?

Answer 1

Hint: Since $\int F_N=1$ you have $$f(x_0)-\int f(x_0-t)F_N(t) =\int(f(x_0)-f(x_0-t))F_N(t)\,dt,$$hence$$\left| f(x_0)-\int f(x_0-t)F_N(t)\right| \le\int|f(x_0)-f(x_0-t)|F_N(t)\,dt.$$

One way to connect this with the fact that $x_0$ is a Lebesgue point would be to integrate by parts. Or show that $F_N$ is dominated by a linear combination of characteristic functions of intervals cenetered at the origin...