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

This paper deals with the continuity of the sharp constant K( T,X) with respect to the set T in the Jackson-Stechkin inequality , 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( $\mathbb{T}^d $, ℂ) is taken for X and the subspace of functions g ∈ L( $\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. [ABSTRACT FROM AUTHOR]