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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.
41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiń≠-type inequalities)
41A50 Best approximation, Chebyshev systems
42A10 Trigonometric approximation