2 00 6 On the exact constant in Jackson - Stechkin inequality for the uniform metric

Abstract

The classical Jackson-Stechkin inequality estimates the value of the best uniform approximation of a periodic function f by trigonometric polynomials of degree ≤ n− 1 in terms of its r-th modulus of smoothness ωr(f, δ). It reads En−1(f) ≤ cr ωr ( f, 2π n ) , where cr is some constant that depends only on r. It was known that cr admits the estimate cr < r ar and, basically, nothing else could be said about it. The main result of this paper is in establishing that (1− 1 r+1 ) γ r ≤ cr < 5 γ r , γ r = 1 ( r ⌊ r 2 ⌋ ) ≍ r 1/2 2r , i.e., that the Stechkin constant cr, far from increasing with r, does in fact decay exponentially fast. We also show that the same upper bound is valid for the constant cr,p in the Stechkin inequality for Lp-metrics with p ∈ [1,∞), and for small r we present upper estimates which are sufficiently close to 1 · γ r .

Cite this paper

@inproceedings{Shadrin2006206, title={2 00 6 On the exact constant in Jackson - Stechkin inequality for the uniform metric}, author={Alexei Shadrin}, year={2006} }