My question comes from trying to understand a technical step in this paper by Bourgain.

Let $R,L$ be positive integers and let $f(x)=\sum_{|n|\leq RL}a_ne^{2\pi inx}$ be a trigonometric polynomial. Assume $f(x)>0$. Let $F_L(x)=\sum_{|n|\leq L}\frac{L-|n|}Le^{2\pi inx}$ be the Fejer kernel. Define (as usual) the convolution $$(f*F_L)(x)=\int_0^1f(t)F_L(x-t)dt=\sum_{|n|\leq L}a_n\frac{L-|n|}Le^{2\pi inx}$$ Does it follow (and how) that $$f(x)\leq10R(f*F_L)(x)$$?

In the paper we have a concrete $f(x)$, so maybe this is not true in general. There we have $$a_n=1-\cos\left(2\pi\frac{RL-|n|}N\right)$$ where $N$ is a large positive integer, (at least bigger than $4RL$).