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On the Bernstein-Nikolsky inequality. II. (English) Zbl 0848.46014
There are considered Orlicz spaces \(L_\Phi (\mathbb{R}^n)\) with the Luxemburg norm \(|\cdot |_\Phi\) and spaces \(M_{\sigma, \Phi}\) of all entire functions of exponential type \(\sigma\) which as functions of real \(x\in \mathbb{R}^n\) belong to \(L_\Phi (\mathbb{R}^n)\). The author considers a connection between the spaces \({\mathcal S}'\) and \(M_{\sigma, \Phi}\) (Theorems 1 and 2, respectively) and properties of the elements from \(M_{\sigma, \Phi}\) and \(L_\Phi (\mathbb{R}^n)\) (Theorems 3 and 4, respectively). Very important is Theorem 5 with a nice proof which says that if \(f\) is such that \(D^\alpha f(x)\in L_\Phi (\mathbb{R}^n)\) (where \(\alpha\) is a multi-index) then \[ \liminf_{|\alpha |\to \infty} (|\xi^{-\alpha} |\;|D^\alpha f|_\Phi )^{1/ |\alpha |}\geq 1. \] [For part I see M. Morimoto and the author, same J. 14,No. 1, 231-238 (1991; Zbl 0803.41017)].
Reviewer: A.Waszak (Poznań)

46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
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