Approximation by simple fractions
In this paper, a number of problems concerning the uniform approximation of complex-valued continuous functions \(f(z\)) on compact subsets of the complex plane by simple fractions of the form \(\Theta_n(z) =\sum_{j=1}^n1/(z-z_0)\) are considered. It is shown that the best approximation of a function \(f\) by the fractions \(\Theta_n\) is of the same order of vanishing as the best approximations by polynomials of degree \(\leq n\). Let \(f(z) = \sum_{\nu=0}^\infty\alpha_\nu(f)z^\nu\) be a function regular in the unit disk \(\Delta\) with \(|\alpha_\nu(f)|\leq M\), \(\nu\in\mathbb N\). Then the simple Padé fractions \(\Theta_n\) of the function \(F(z) = f(z)/(2M)\) uniformly converge to \(F\) on any disk \(\Delta_\lambda=\{z : |z|\leq\lambda < 1\}\) and \(\|F-\Theta_n\|_{\Delta_\lambda}\leq\frac{1+n}{1-\lambda}\lambda^n\). This estimate is sharp. Denote by \(\rho(\Theta_n, K)\) the Euclidean distance between the set \(\{z_j\}\) of the poles of \(\Theta_n\) and a set \(K\in\overline{\mathbb C}\). One particular statement about the lower bound for the distances \(\rho(\Theta_m, K)\) depending on \(\|f - \Theta_m\|_\Delta K\) is proved. Suppose that \(P\) is a polynomial of degree \(n\geq 1\), \(Q_{nk}\) is a polynomial of degree \(nk\geq 1\), and for certain \(\varepsilon\in (0,1)\) and \(a\in(0, k')\) the estimate \(\|P-\Theta_{nk}\|_\Delta\leq a^kk^{-k}\) holds on \(\Delta\). Then for sufficiently large \(k\) we have \(\rho(\Theta_{nk}, \partial\Delta)\geq C k\), where \(C\) depends only on \(a,\varepsilon, \|P\|\).
approximation by simplest fractions of complex functions
scientific article; zbMATH DE number 1927775