On the continuity of the sharp constant in the Jackson-Stechkin inequality in the space L
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Authors: | V S Balaganskii |
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Institution: | 19765. Institute of Mathematics and Mechanics, Ural Branch, Russian Academy of Sciences, Ekaterinburg, Russia
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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 $ , ?) is taken for X and the subspace of functions g ∈ L 2( $\mathbb{T}^d $ , ?), 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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