Nonnegative Trigonometric Sums

We obtain estimates for PŽkŽ N ŽckŽ2 and PŽkŽ N ŽckŽ4, when PŽkŽ N ckeitkx 0, t0 D 0; and c0 D 1. It is shown that N X kD􀀀N ŽckŽ4 M=b C 1 and N X kD􀀀N ŽckŽ2 q2N C 1‘M=b C 1‘; where M D max ŽtkŽ and b D min”ŽtjŽ x j 6D 0•. When the cks are known to be nonnegative, the inequality N X kD􀀀N ŽckŽ2 M=b C 1 is established