checking if a function is positive using Fourier coefficients

Question

Given a function $$f(x) = \sum_{k=0}^N a_k\ \sin(k\pi x)$$ defined over the region $S = [0, 1]$, is there some way to check if $f(x) \geq 0$ for all $x \in S$ using the coefficients $\{ a_k;\ k \leq N \}$? In particular, I was hoping there'd some inequality involving the coefficients that'll enable us to check this quickly.