Extremal Positive Trigonometric Polynomials

There are various reasons for the interest in the problem of constructing nonnegative trigonometric polynomials. Among them are: Cesàro means and Gibbs’ phenomenon of the the Fourier series, approximation theory, univalent functions and polynomials, positive Jacobi polynomial sums, orthogonal polynomials on the unit circle, zero-free regions for the Riemann zeta-function, just to mention a few. In this paper we summarize some of the recent results on nonnegative trigonometric polynomials. Needless to say, we shall not be able to cover all the results and applications. Because of that this short review represents our personal taste. We restrict ourselves to the results and problems we find interesting and challenging. One of the earliest examples of nonnegative trigonometric series is the Poisson kernel