An extension of Vietoris's inequalities

We establish a best possible extension of a famous Theorem of Vietoris about the positivity of a general class of cosine sums. Our result refines and sharpens several earlier generalizations of this Theorem, and settles some open questions regarding the possibility of further improvement of it. Some new inequalities for trigonometric sums are given. We show that our results have applications within the context of positive sums of Gegenbauer polynomials and quadrature methods. We also obtain some existing estimates for the location of zeros of certain trigonometric polynomials under a weakened condition on their coefficients.