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.
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The author expresses gratitude to anonymous referee for valuable suggestions.
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Volosivets, S.S. (2019). Approximation by Trigonometric Polynomials in Stechkin Majorant Spaces. In: Abell, M., Iacob, E., Stokolos, A., Taylor, S., Tikhonov, S., Zhu, J. (eds) Topics in Classical and Modern Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-12277-5_21
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DOI: https://doi.org/10.1007/978-3-030-12277-5_21
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