Is there any intuitive definition for the class of periodic functions that have only non-negative Fourier coefficients? Or at least (relatively weak) sufficient conditions for a periodic function to have non-negative Fourier coefficients?
I am particularly interested in describing a class of even functions with period $2\pi$ and zero mean value that satisfy the condition above. By writing the expression for the Fourier coefficients of such function $f$, it becomes obvious that it has non-negative Fourier coefficients if and only if
$$\int_0^\pi f(x) \cos(nx)\,dx \geq 0\quad\forall n \geq 1.$$
I would like to indentify sufficient conditions for $f$ to satisfy the inequality above. An example of a function that satisfies this inequality is the triangular wave (with maximum positive amplitude at $x=0$, with zero mean value and period $2\pi$).
Thank you in advance!