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

Abstract

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 ω _r (f,δ). It reads

$$E_{n-1}(f)\leq c_{r}\omega_{r}\biggl(f,\frac{2\pi}{n}\biggr),$$

where c _r is some constant that depends only on r. It has been known that c _r admits the estimate c _r <r ^ar 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 c _r , far from increasing with r, does in fact decay exponentially fast. We also show that the same upper bound is valid for the constant c _r,p in the Stechkin inequality for L _p -metrics with p∈[1,∞), and for small r we present upper estimates which are sufficiently close to 1⋅γ ^* _r .