Given the following function \begin{align*} g: \mathbb{R}&\rightarrow\mathbb{C}\\ x&\mapsto \begin{cases} x &\mbox{, $-\pi<x<\pi$}\\ 0&\mbox{, $x=-\pi$}\\ g(x+2\pi)=g(x)&\mbox{for all $x\in\mathbb{R}$,} \end{cases} \end{align*}
The Fourier series of the function in the point 3. diverges at 0 but I don't the conditions,also I don't knwo how to prove the points 1 and 2
1.Show that for all $n\in\mathbb{Z}^{+}$ and $t\in\mathbb{R}$ \begin{align*} |S_{n}(g,t)|\leq \pi+2, \end{align*} where $S_{n}$ defines the Nth partial sum of the Fourier Series.
2.It is defined for each $n\in\mathbb{Z}^{+}$ \begin{align*} \varphi_{n}:\mathbb{R}&\rightarrow\mathbb{C}\\ x&\mapsto\frac{1}{\pi+2}e^{int}S_{n}(g,x) \end{align*} Show that there exists $K>0$ such that for all $n\in\mathbb{Z}^{+}$ \begin{align*} |S_{n}(\varphi_{n},0)|>K \ln{n}. \end{align*}
- What condition must satisfy the positive integers $n_{j}$ and $\lambda_{j}$ so that the Fourier series of the function
\begin{align*} f:\mathbb{R}&\rightarrow\mathbb{C}\\ x&\mapsto\sum_{j=1}^{\infty}\frac{1}{2^{j}}e^{i\lambda_{j}x}\varphi_{n_{j}}(x) \end{align*} diverge for $t=0$ and converges for all $t\in[-\pi,0)\cup(0,\pi]$.