On the boundedness below of trigonometric polynomials of best approximation

Abstract

As A. S. Belov proved, the partial sums of an even 2π-periodic function f expanded in a Fourier series with convex coefficients {α n } n=0 ∞ , are uniformly bounded below if the conditions a n = O(n −1), n → ∞, are satisfied; moreover, this assertion is no longer valid if the exponent −1 in this condition is replaced by a greater one. In this paper, we obtain analogs of these results for trigonometric polynomials of best approximation to the function f in the metric of L 2π 1 .