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Bernstein-Nikol’skij inequalities for functions of several variables, that are the best as concerns the choice of harmonics. (English. Russian original) Zbl 0783.41012
Mosc. Univ. Math. Bull. 47, No. 6, 1-4 (1992); translation from Vestn. Mosk. Univ., Ser. I 1992, No. 6, 3-6 (1992).
The author considers terms of the following type $T_ N(r,p,q)=\inf_{K_ N} \sup_{x\in L(K_ N), x\neq 0} \| x^{(r)}\|_ p/\| x\|_ q,$ where $$K_ N$$ is any set of $$N$$-harmonic multi-indices in $$\mathbb{Z}_ 0^ N$$ and $$L(K_ N)$$ the linear span of $$\{e^{i(k,t)}\mid k\in K_ N\}$$. Thus, the quantities $$T_ N(r,p,q)$$ represent the best constants (for all possible choices of sets $$K_ N$$) for Bernstein-Nikol’skij inequalities for polynomials from the spaces $$L(K_ N)$$. The author proves a theorem that establishes growth estimates for $$T_ N(r,p,q)$$ in terms of $$N$$ different choices of the parameters $$r$$, $$p$$ and $$q$$.
Reviewer: E.Quak (Schwerte)
##### MSC:
 41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities) 42A10 Trigonometric approximation
##### Keywords:
Bernstein-Nikol’skij inequalities