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On the exact constant in the Jackson-Stechkin inequality for the uniform metric. (English) Zbl 1181.41017
In the well-known Jackson-Stechkin inequality the value of the best approximation $ E_{n-1}(f)$ of a $2\pi -$periodic function by trigonometric polynomials of degree $\leq n-1$ is estimated by $r-$th modolus of smoothness $\omega _r$ of $f$. This inequality has the form $$E_{n-1}(f)\leq c_r w_r(f;\frac{2\pi}{n}),$$ where $c_r$ is some constant that depends only on $r$. The main result is that $$ (1-\frac{1}{r+1})\gamma _r^*\leq c_r<5\gamma_r^*,\ \text{ where }\gamma _r^*= \frac{1}{{r\choose [\frac{r}{2}]}}\asymp \frac{r^{1/ 2}}{2^r}.$$ Moreover, the same upper bound is valid for the constant $c_{r,p}$ in the Stechkin inequality for $L_p-$metrics with $p\in [1,\infty ).$

MSC:
41A17Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
41A44Best constants (approximations and expansions)
42A10Trigonometric approximation
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References:
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