Let $G:=\mathbb R/\mathbb Z$ (the torus group). Suppose that $f\colon G\to[-1,1]$ is a reasonably smooth function such that $\int_G f(z)dz=0$ and $\|\widehat f\|_\infty=1$, and that $g$ is the indicator function of an interval $I:=(-l,l)\subset G$. How small can the norm $\|f\ast g\|_\infty$ of a convolution be with these assumptions?
I am afraid this is a very basic question, but not exactly my cup of tea. I guess the identity $\widehat{f\ast g}=\widehat f\cdot\widehat g$ should be highly relevant, but I cannot see exactly how it helps.
Thank you in advance!
Note that for any function $h : G \to \mathbb{R}$, $$\|\hat{h}\|_\infty \leq \|h\|_1.$$ Apply this result to a special function $h$ and use the results you cited above to get a general result.