Let $\delta > 0$ be small and $\chi\_{[-\delta, \delta]}(t)$ be a characteristic function on the interval $[-\delta, \delta]$. I am interested in a trigonometric polynomial $S$ such that
$$
\chi\_{[-\delta, \delta]}(t) \leq S(t)
$$
such that and $S(t) = 0$ if $|t| > 2 \delta$ with some control on the coefficients of $S$. I apologize that the question is rather vague but I am looking for a place to start looking into something like this. 


I would greatly appreciate any references or comments! Thank you!



A "nice" trigonometric polynomial approximation of a characteristic function
Pick any $f$ that bounds above the characteristic function of $[-\delta, \delta]$. For example the sine kernel $(\frac{\sin x}{x})^2$ or a higher power conveniently rescaled to fit your requirements. Then make your upper bound $f(t)$ periodic by considering $F(t) = \sum\_{n} f(t + n)$. Then make sure that your $f$ has the property that $\widehat{f}$ is compactly supported (this is guaranteed by Paley Wiener if $f$ is analytic and has no more than exponential growth in the complex plane). Then apply Poisson summation to see that $F(t) = \sum\_{n} \widehat{f}(n) e^{2\pi i t n}$ is in fact your desired trigonometric polynomial. 


