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On the continuity of the sharp constant in the Jackson-Stechkin inequality in the space L 2

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Abstract

This paper deals with the continuity of the sharp constant K(T,X) with respect to the set T in the Jackson-Stechkin inequality

$E(f,L) \leqslant K(T,X)\omega (f,T,X),$

, where E(f,L) is the best approximation of the function f ∈ X by elements of the subspace L ⊂ X, and ω is a modulus of continuity, in the case where the space L 2(\(\mathbb{T}^d \), ℂ) is taken for X and the subspace of functions g ∈ L 2(\(\mathbb{T}^d \), ℂ), for L. In particular, it is proved that the sharp constant in the Jackson-Stechkin inequality is continuous in the case where L is the space of trigonometric polynomials of nth order and the modulus of continuity ω is the classical modulus of continuity of rth order.

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  1. Institute of Mathematics and Mechanics, Ural Branch, Russian Academy of Sciences, Ekaterinburg, Russia

    V. S. Balaganskii

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  1. V. S. Balaganskii
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Original Russian Text © V. S. Balaganskii, 2013, published in Matematicheskie Zametki, 2013, Vol. 93, No. 1, pp. 13–44.

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Balaganskii, V.S. On the continuity of the sharp constant in the Jackson-Stechkin inequality in the space L 2 . Math Notes 93, 12–28 (2013). https://doi.org/10.1134/S0001434613010021

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