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Approximation by Trigonometric Polynomials in Stechkin Majorant Spaces

  • Sergey S. Volosivets
Chapter
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

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 En(f)p in \(L^p_{2\pi }\), 1 ≤ p ≤, satisfying the inequality En(f)p ≤ n, \(n\in \mathbb Z_+\), where C does not depend on n, εn 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 Sk(f) are partial Fourier sums of f and {ank : 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.

Keywords

Majorant space Best approximation Degree of approximation Jackson and Bernstein inequalities Linear means of Fourier series 

2010 Mathematics Subject Classification

Primary 42A10; Secondary 42A24 41A17 

Notes

Acknowledgements

The author expresses gratitude to anonymous referee for valuable suggestions.

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Sergey S. Volosivets
    • 1
  1. 1.Department of Mechanics and MathematicsSaratov State UniversitySaratovRussia

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