# Estimating the derivative of Fejer's Kernel

Fejer's Kernel is given by $$F_{N}(t) = \frac{\sin^{2}(Nt/2)}{N\sin^{2}(t/2)}$$ and has the derivative $$\frac{\sin(Nt/2)\cos(Nt/2)}{\sin^{2}(t/2)} - \frac{\cos(t/2)\sin^{2}(Nt/2)}{N\sin^{3}(t/2)}.$$ For $|t|\leq \pi$, we have $|\sin(Nt/2)| \leq CN|t|$ and $|\sin(t/2)|\geq c|t|$ for some constants $c$ and $C$. How can we show that $$|F_{N}'(t)| \leq \frac{A}{|t|^{2}}?$$

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Hint: If $|\sin(t/2)|\geq c|t|$ then $|\sin^2(t/2)| \geq c^2 |t|^2$, and so $$\frac{1}{|\sin^2(t/2)|} \leq \frac{1}{c^2 |t|^2}.$$ –  Antonio Vargas Mar 31 '12 at 2:42