NONNEGATIVE TRIGONOMETRIC POLYNOMIALS DIMITAR K. DIMITROV AND CLINTON A. MERLO Abstract. An extremal problem for the coefficients of sine polynomials, which are nonnegativein [0,π], posed and discussed by Rogosinski and Szeg˝ o is un- der consideration. An analog of the Fej´ er-Riesz representation of nonnegative- general trigonometric and cosine polynomials is proved for nonnegativesine polynomials. Various extremal sine polynomials for the problem of Rogosin- ski and Szeg˝ o are obtained explicitly. Associated cosine polynomials kn(θ) are constructed in such a way, that {kn(θ)} are summability kernels. Thus, the Lp, pointwise and almost everywhere convergence of the corresponding convolutions is established. 1. Introduction Among the various reasons for the interest in the problem of constructing non- negativetrigonometric polynomials are: the Gibbs phenomenon [19, Chapter II, §9], univalent functions and polynomials [8], positive Jacobi polynomial sums [3], orthogonal polynomials on the unit circle [18], zero-free regions for the Riemann zeta-function [1, 2], just to mention a few. Our interest in this subject comes from the classical Approximation Theory. In this paper we construct some new positive summability kernels. Recall that the sequence {k n (θ)} of even, nonnegativecontinuous 2π-periodic functions is called an even positive kernel if k n (θ) are normalized by (1/2π)  π -π k n (θ)dθ = 1 and they converge to zero in any closed subset of (0, 2π). It is classically known that the convolutions of such kernels with 2π-periodic functions f ∈ L p [-π,π] converge to f in the L p -norm, for 1 ≤ p ≤∞. To the best of our knowledge, Fej´ er [6] was the first to construct such a kernel. He proved that (1.1) F n (θ)=1+2 n  k=1  1 - k n +1  cos kθ are nonnegative, and established the uniform convergence of the corresponding convolutions with continuous functions. These convolutions are nothing but the Ces`aro means of the Fourier series. Jackson [10, 11] used the kernel J n (θ)= F 2 n (θ) to prove his celebrated approximation theorem. The basic tool for constructing nonnegativetrigonometric polynomials T (θ) is Fej´ er and Riesz’ (see [7]) theorem which states that T (θ) is nonnegativeif and only if there exists an algebraic polynomial R(z), such that T (θ)= |R(e iθ )| 2 . However, 1991 Mathematics Subject Classification. Primary 42A05, 26D05. Key words and phrases. Nonnegative trigonometric polynomials, extremal polynomials, summability kernel, Fej´ er-Riesz type theorem, Lp convergence, pointwise and almost everywhere convergence. Research supported by the Brazilian Science Foundations CNPq under Grant 300645/95-3, FAPESP under Grants 97/06280-0 and 98/11977-3. 1