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On Bernstein-Nikol’skij inequalities that are best with respect to choice of harmonics. (English. Russian original) Zbl 0797.41014
Russ. Acad. Sci., Dokl., Math. 45, No. 2, 281-285 (1992); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 323, No. 2, 211-215 (1992).
For the space $${\mathcal T}_{\mathcal N}= \{x(t)= \sum_{| k|\leq{\mathcal N}} x_ k e^{i_ k t}\}$$ of trigonometric polynomials of degree at most $${\mathcal N}$$ the inequality $$\| x^{(r)}\|_ p\ll {\mathcal N}^{r+(1/q-1/p)_ +} \| x\|_ q$$, $$x\in {\mathcal T}_{\mathcal N}$$, $$1\leq p,q\leq\infty$$, is well known as the Bernstein-Nikol’skij inequality. In this paper the smallest constant with respect to the choice of $${\mathcal N}$$ harmonics in the Bernstein-Nikol’skij inequality for trigonometric polynomials in several variables is considered. In some cases an optimal collections of harmonics is given. The investigations of the author’s work [Thesis (Mowcow, 1978)] and N. S. Nikol’skaya [Sibir. Mat. Zh. 15, 395-412 (1975; Zbl 0307.42003)] are continued.
##### MSC:
 41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities) 41A50 Best approximation, Chebyshev systems 42A10 Trigonometric approximation
##### Keywords:
Bernstein-Nikol’skij inequality