SCHWEITZER 1980 Problem 8

Let $f(x)$ be a nonnegative, integrable function on $(0,2\pi)$ whose Fourier series is $f(x)=a_0+\sum_{k=1}^{\infty} a_k \cos (n_k x)$ , where none of the positive integers $n_k$ divides another. Prove that $|a_k| \leq a_0$ .

G. Halasz