I'm trying to answer this problem: Consider a real function f, bandlimited by frequency $\omega$, which satisfy 
$$\int\_{-\infty}^\infty f(x)^2dx=c.$$
(For pure mathematicians: "bandlimited" means that it Fourier transform is supported 
on $[-\omega,\omega]$.)


Is the derivative of this function limited in absolute value? That is, is there an expression $A(\omega,c)$ for which $|f'(x)|<A(\omega,c)$ for all x? 


Thank for the help!



Derivative of Band-limited functions
The answer is "yes". We have
$$f(x)=\int\_{-\omega}^\omega g(t)e^{2\pi itx}dt,$$
so
$$|f'(x)|^2\leq\left(2\pi\int\_{-\omega}^\omega |t||g(t)|dt\right)^2\leq\frac{8\pi^2\omega^3}{3}\| g\|\_2,$$
by Cauchy-Bounyakovski-Schwarz inequality. It remains to notice that $\| g\|\_2=\| f\|\_2$ according to Parseval . Equality when $g(t)=t$.


