An Extremal Nonnegative Sine Polynomial

Abstract

For any positive integer n, the sine polynomials that are nonnegative in [0, π] and which have the maximal derivative at the origin are determined in an explicit form. Associated cosine polynomials K n (θ) are constructed in such a way that {K n (θ)} is a summability kernel. Thus, for each p, 1 ≤ p ≤ ∞ and for any 2π-periodic function f ∈ L p [−π, π], the sequence of convolutions K n * f is proved to converge to f in L p [−π, π]. The pointwise and almost everywhere convergences are also consequences of our construction.

An extremal problem for polynomials

“…According to Problem 52 in [7, p. 79] or (1.3) in [8], a nonnegative trigonometric polynomial of the form…”

“…In the pioneer work [19], W. Rogosinski and G. Szegő considered and discussed possible ways to solve a large variety of extremal problems for such polynomials. A very particular case of their results reads as follows: for P (z) ∈ T N there holds the exact estimate (1) |a 2 | ≤ 2µ N , N is odd, 2η N , N is even, where µ N = cos 2π N +3 is the largest root of the equation U (N +1)/2 (x) = 0, while η N is the maximal root of U ′ N 2 +1 (x) − U ′ N 2 (x) = 0. Here, U j with j ∈ N 0 denote the Chebyshev polynomials 1 of the second kind and U ′ j their derivatives, defined by…”

Roots of Trigonometric Polynomials and the Erdős–turán Theorem

“…Nonnegative trigonometric polynomials and their properties have been actively studied for a very long time, we refer to a survey of Dimitrov [9], the papers [2,4,5,10,18,22] and references therein. One can take these trigonometric polynomials and interpret them as the real part of a polynomial in the complex plane evaluated on the boundary of the unit disk, this leads to the polynomials…”

“…Normalized to , it is given by where and n has to be sufficiently large depending on ρ. A slightly less well‐known 1912 example due to Young , again normalized to , is Nonnegative trigonometric polynomials and their properties have been actively studied for a very long time, we refer to a survey of Dimitrov , the papers , and references therein. One can take these trigonometric polynomials and interpret them as the real part of a polynomial in the complex plane evaluated on the boundary of the unit disk, this leads to the polynomials We note that the con...…”

Inequalities for two sine polynomials

“…91-92]. Variants and generalizations of Lukács' inequality are given in [1], [6], [8]- [11], [15], [16, p. 140]. …”