Zero-Mean Cosine Polynomials which are Non-Negative for as Long as Possible

For a given integer n, all zero-mean cosine polynomials of orderat most n which are non-negative on 0,(n/(n+1))] are found,and it is shown that this is the longest interval 0,] on whichsuch cosine polynomials exist. Also, the longest interval 0,]on which there is a non-negative zero-mean cosine polynomialwith non-negative coefficients is found. As an immediate consequence of these results, the correspondingproblems of the longest intervals ,] on which there are non-positivecosine polynomials of degree n are solved. For both of these problems, all extremal polynomials are found.Applications of these polynomials to Diophantine approximationare suggested.