EXAMPLES OF TRIGONOMETRIC SERIES WITH NONNEGATIVE PARTIAL-SUMS

Let {alpha(n)}(infinity)(n=1) be a monotone sequence of non-negative r eal numbers. In this paper the conditions [GRAPHICS] are proved to be necessary and sufficient for all partial sums of the trigonometric sin e series Sigma(n=1)(infinity) alpha(n) sin(nx) to be positive on the i nterval (0, pi). New conditions on the coefficients of a trigonometric cosine series ensuring that all its partial sums are positive on the real axis are presented.