Non-Fourier-Lebesgue trigonometric series with nonnegative partial sums

It is proved that a trigonometric cosine series of the form $\sum_{n=0}^\infty a_n\cos(nx)$ with nonnegative coefficients can be constructed in such a way that all of its partial sums are positive on the real axis. It converges to zero almost everywhere and is not a Fourier-Lebesgue series. Some other properties of trigonometric series with nonnegative partial sums are also studied.