Non-Fourier-Lebesgue trigonometric series with nonnegative partial sums

It is proved that a trigonometric cosine series of the form _n^Emphasis>=0/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.Translated fromMatematicheskie Zametki, Vol. 59, No. 1, pp. 24–41, January, 1996.