On the Bernstein-Nikolsky inequality
Let \(1 \leq p < \infty\) and \(\sigma = (\sigma_ 1, \sigma_ 2, \dots, \sigma_ n)\), \(\sigma_ i > 0\), \(i=1,2, \dots,n\). Write \(M_{\sigma,p}\) for the space of entire functions of exponential type \(\sigma\) that belong to \(L^ p (\mathbb{R}^ n)\). The authors extend the Bernstein-Nikol'skij inequality by showing that \(\lim_{| \alpha | \to \infty} \sigma^{ - \alpha} \| D^ \alpha f \|_ p = 0\) for \(f \in M_{\sigma,p}\), where \(\alpha = (\alpha_ 1, \alpha_ 2, \dots, \alpha_ n)\) with \(\alpha_ i \geq 0\) for \(i = 1,2, \dots, n\). They also obtain Bernstein-Nikol'skij-type results for the directional derivatives of entire functions of exponential type.
scientific article; zbMATH DE number 13499