Best Local Approximation by Simplest Fractions

In this paper, we present two theorems on best local approximation by simplest fractions, i.e., by logarithmic derivatives of algebraic polynomials with complex coefficients. In Theorem 1, we obtain an analog of Bernstein's well-known theorem on the description of $n$-times continuously differentiable functions on the closed interval $\Delta\subset\mathbb R$ in terms of local approximations in the uniform metric by algebraic polynomials. Theorem 2 describes the simplest Padé fraction as the limit of the sequence of simplest fractions of best uniform approximation and is an analog of Walsh's well-known result on the classical Padé fractions.