Approximation by Trigonometric Polynomials in Stechkin Majorant Spaces

Abstract

In this paper we consider Stechkin majorant spaces $\mathcal E_p(\varepsilon )$ such that $f\in \mathcal E_p(\varepsilon )$ has best trigonometric approximations E _n(f)_p in $L^p_{2\pi }$, 1 ≤ p ≤∞, satisfying the inequality E _n(f)_p ≤ Cε _n, $n\in \mathbb Z_+$, where C does not depend on n, ε _n ↓ 0. We prove that the trigonometric system is a basis in these spaces. The general estimates of best approximation in $\mathcal E_p(\varepsilon )$ including Jackson and Bernstein inequalities are established. For $\tau _n(f)(x)=\sum ^n_{k=0}a_{nk}S_k(f)(x)$, where S _k(f) are partial Fourier sums of f and {a _nk : n ≥ 0, 0 ≤ k ≤ n} satisfies certain condition of generalized monotonicity type, some bounds for the degree of approximation $\|f-\tau _n(f)\|{ }_{\mathcal E_p(\varepsilon )}$ are obtained. The sharpness of such results is proved under some restrictions. Also some applications of obtained results to the approximation in Hölder–Lipschitz spaces are given.