On the boundedness below of trigonometric polynomials of best approximation

V. S. Kolesnikov

Matematičeskie zametki, Tome 79 (2006) no. 6, pp. 870-878

Résumé

As A. S. Belov proved, the partial sums of an even $2\pi$-periodic function f expanded in a Fourier series with convex coefficients $\{a_n\}_{n=0}^\infty$, are uniformly bounded below if the conditions $a_n = O(n^{-1})$, $n\to\infty$, are satisfied; moreover, this assertion is no longer valid if the exponent $-1$ in this condition is replaced by a greater one.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 $L_{2\pi}^1$.