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