ACTA issues

Positivity of a class of cosine sums

Gavin Brown, Qinghe Yin

Acta Sci. Math. (Szeged) 67:1-2(2001), 221-247
2782/2009

 Abstract. Suppose that $a_0\ge a_1\ge\cdots \ge a_n>0$ and $(2k+1)a_{2k-1}\ge(2k+2)a_{2k}$ for $k\ge1$. Then for $0< x< \pi$ one has $$\sum_{k=0}^na_k\cos kx>0.$$ This is a further generalisation of a result of the first named author and Hewitt, which generalises a theorem of Vietoris. AMS Subject Classification (1991): 42A32 Received March 2, 1999. (Registered under 2782/2009.)