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Mat. Zametki, 2016, Volume 99, Issue 2, Pages 215–238 (Mi mz10506)  

Inequalities between Best Polynomial Approximations and Some Smoothness Characteristics in the Space $L_2$ and Widths of Classes of Functions

S. B. Vakarchuka, V. I. Zabutnayab

a Alfred Nobel University Dnepropetrovsk
b Dnepropetrovsk National University

Abstract: We obtain exact constants in Jackson-type inequalities for smoothness characteristics $\Lambda_k(f)$, $k\in \mathbb{N}$, defined by averaging the $k$th-order finite differences of functions $f \in L_2$. On the basis of this, for differentiable functions in the classes $L^r_2$, $r\in \mathbb{N}$, we refine the constants in Jackson-type inequalities containing the $k$th-order modulus of continuity $\omega_k$. For classes of functions defined by their smoothness characteristics $\Lambda_k(f)$ and majorants $\Phi$ satisfying a number of conditions, we calculate the exact values of certain $n$-widths.

Keywords: best polynomial approximation, smoothness characteristics, Jackson-type inequality, modulus of continuity, Bernstein $n$-width of a function class, Rolle's theorem.

DOI: https://doi.org/10.4213/mzm10506

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English version:
Mathematical Notes, 2016, 99:2, 222–242

Bibliographic databases:

UDC: 517.5
Received: 29.04.2014

Citation: S. B. Vakarchuk, V. I. Zabutnaya, “Inequalities between Best Polynomial Approximations and Some Smoothness Characteristics in the Space $L_2$ and Widths of Classes of Functions”, Mat. Zametki, 99:2 (2016), 215–238; Math. Notes, 99:2 (2016), 222–242

Citation in format AMSBIB
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\paper Inequalities between Best Polynomial Approximations and Some Smoothness Characteristics in the Space~$L_2$ and Widths of Classes of Functions
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\pages 215--238
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