Inequalities of Bernstein type for derivatives of meromorphic functions and approximation by meromorphic functions on the real axis (Q1896826)

Let \(\gamma\) be a closed or open subset of \(\mathbb{R}\). For \(1 < p < \infty\) and \(\alpha > 0\), \(\varepsilon > 0\) the authors introduce linear subspaces \(K_{\alpha, p}\) and \(N_{\alpha, p, \varepsilon}\) \((= M_{\alpha,s} (\varepsilon)\) in the author's notation). The function of \(K_{\alpha, p}\) are characterized by the finiteness of a seminorm \(|\cdot |_{\alpha, p}\) defined in terms of the best approximation (in the \(L_p\) sense) \(a_{\varepsilon, h}\) by entire functions of order \(1 + \varepsilon\) and type \(2^k\). The functions in \(N_{\alpha, p}\) are characterized by the finiteness of a seminorm defined in terms of the best approximation (in the \(L_p\)-sense). \(E_m(I)\) by polynomials in intervals \(I\) of length \(\approx \delta\), \(I \subset \gamma\). By skillful use of the theory of approximation by rational functions the author shows the inlcusions \[ K_{\alpha, p \cap} L_p \supset N_{\alpha, p, 0} \quad N_{\alpha, p, \varepsilon \cap} L_p \supset K_{\alpha, p}. \] These inclusions imply inequalities of Bernstein type for meromorphic functions whithout real poles. The paper contains generalizations to classes of functions defined on a subset \(\gamma\) of curves other than the real axis.