On the Exact Constant in the Jackson–Stechkin Inequality for the Uniform Metric

By Simon Foucart, Y. Kryakin, A. Shadrin. 2006

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The classical Jackson–Stechkin inequality estimates the value of the best uniform approximation of a 2π-periodic function f by trigonometric polynomials of degree ≤n−1 in terms of its r-th modulus of smoothness ωr(f,δ). It reads $$E_{n-1}(f)\leq c_{r}\omega_{r}\biggl(f,\frac{2\pi}{n}\biggr),$$ where cr is some constant that depends only on r. It has been known that cr admits the estimate cr<rar and, basically, nothing else has been proved.The main result of this paper is in establishing that $$\biggl(1-\frac{1}{r+1}\biggr)\gamma_{r}^{*}\leq c_{r}<5\gamma_{r}^{*}\quad \gamma_{r}^{*}=\frac{1}{{r\choose\lfloor\frac{r}{2}\rfloor}}\asymp\frac{r^{1/2}}{2^{r}},$$ 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⋅γr*.

Subfields: Applied Mathematics, Numerical Analysis, Statistics and Probability

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