I apologise if this is trivial or well known to be impossible:


Can one find a finite set of integers $2\leq a\_1<a\_2<\ldots<a\_m<\infty$
such that for the function defined as
$$
f\_{a\_1,\ldots,a\_m}(x)=\max\left\{\left|\frac{\sin (a\_1 x)}{\sin x}\right| ,\left|\frac{\sin (a\_2 x)}{\sin x}\right| , \cdots,\left|\frac{\sin (a\_m x)}{\sin x}\right| \right\}
$$
the minimum satisfies
$$
\min\{f\_{a\_1,\ldots,a\_m}(x):0\leq ~x~\leq 2\pi\}>1?
$$


**Edit** I can restrict to $x$ not equal to an odd multiple of $\pi/2$ as well.


More generally can the lower bound be made larger? Say larger than a small positive integer? Or a slowly growing function of $m$?



The minimum of the maximum of a sequence of sinc functions
It looks like this is impossible. Correct me if I am wrong.


For example, $f\_{a\_1,\cdots,a\_m}(\pi/2)\leq 1$, so the minimum cannot exceed 1.


