Fejer's kernel derivative

Question

Fejer's Kernel is given by $$F_{N}(t) = \frac{\sin^{2}(Nt/2)}{N\sin^{2}(t/2)}$$ and has the derivative $$\frac{\sin(Nt/2)\cos(Nt/2)}{\sin^{2}(t/2)} - \frac{\cos(t/2)\sin^{2}(Nt/2)}{N\sin^{3}(t/2)}.$$ Estimate the derivative s $F_N'(x)=-2NF_{N-1}(t)$$sin(nt)$$ $.Can we show $|F'_N(t)|\leq CN$

Answer 1

I have not checked your calculations, but if they are correct then there is no constant C such that $|F_N'(t)|=2N|F_{N-1} (t)| |\sin (Nt)| \leq CN$. Just take $t=\frac {\pi} {2N}$.

Answer 2

Since $F_N(0)=N$ and $F_N\left(\frac{2\pi}{N}\right)=0$, by Lagrange's theorem $|F_N'(\xi)|=\frac{N^2}{2\pi}$ for some $\xi\in\left(0,\frac{2\pi}{N}\right)$.
The best bound we can prove for $F_N'$ is a quadratic one, not a linear one. And from $$ F_N(t)=\sum_{|j|\leq N}\left(1-\frac{|j|}{N}\right) e^{ijt}$$ the bound $\left|F_N'(t)\right|\leq \frac{1}{3}N^2$ is straightforward.