I saw a statement that if $f$ is continuous,$2\pi$-periodic function which is $C^{1}$ piecewisely on $[-\pi,\pi]$, then its Fourier series converges uniformly to $f$ on $[-\pi,\pi]$. I was wondering how to prove it.
I know if $f$ is Lipschitz continuous,$2\pi$-periodic function on $[-\pi,\pi]$, then its Fourier series converges uniformly to $f$ on $[-\pi,\pi]$. Hence I tend to show piecewise $C^{1}$ can imply Lipschitz on the interval. But I am not sure how to handle the endpoints. Btw, by using "Lipschitz continuous on an interval", I mean "uniform Lipschitz", not "Lipschitz at each point".
Is my guess correct? If not, is there any other ways to prove the statement?
If piecewise $C^1$ means that there are finitely many points $x_1\dots x_n$ such that $f$ is $C^1$ on $[-\pi,+\pi]/\setminus\{x_1\dots x_n\}$ and all the left- and right-sided limits of $f'(x)$, $x\searrow x_i$, $x\nwarrow x_i$ exist, then one can prove that $f$ is Lipschitz continuous.
In fact the Lipschitz modulus can be bounded from above by $$ L := \sup_{x\in [-\pi,+\pi]/\setminus\{x_1\dots x_n\}} |f'(x)|. $$ To see this, take $y_1<y_2$ such that $y_1<x_i < \dots < x_k <y_2$. Then $$ |f(y_1)-f(y_2)| = | f(y_1) - f(x_i) + f(x_i) - \dots -f(x_k) + f(x_k) - f(y_2)| \le L ( |y_1-x_i| + |x_i-x_{i+1}| + \dots + |x_k-y_2|) = L |y_1-y_2|. $$ Here, I used the fact that $f$ is $C^1$ on each sub-interval. Hope this helps.