 
Summary: ROCKY MOUNTAIN
JOURNAL OF MATHEMATICS
Volume 33, Number 3, Fall 2003
AN EXTREMAL NONNEGATIVE SINE POLYNOMIAL
ROBERTO ANDREANI AND DIMITAR K. DIMITROV
ABSTRACT. For any positive integer n, the sine polynomi
als that are nonnegative in [0, ] and which have the maximal
derivative at the origin are determined in an explicit form.
Associated cosine polynomials Kn() are constructed in such
a way that {Kn()} is a summability kernel. Thus, for each
p, 1 p and for any 2periodic function f Lp[, ],
the sequence of convolutions Kn f is proved to converge to
f in Lp[, ]. The pointwise and almost everywhere conver
gences are also consequences of our construction.
1. Introduction and statement of results. There are various
reasons for the interest in the problem of constructing nonnegative
trigonometric polynomials. Among them are the Gibbs phenomenon
[16, Section 9], univalent functions and polynomials [7], positive Jacobi
polynomial sums [1] and orthogonal polynomials on the unit circle [15].
Our study is motivated by a basic fact from the theory of Fourier
