Approximation by Linear Fractional Transformations of Simple Partial Fractions and Their Differences

Abstract

Let f be a function and ρ be a simple partial fraction of degree at most n. Under linear-fractional transformations, the difference f − ρ becomes the difference of another function and a certain simple partial fraction of degree at most n with a quadratic weight. We study applications of this important property. We prove a theorem on uniqueness of interpolating simple partial fraction, generalizing known results, and obtain estimates for the best uniform approximation of certain functions on the real semi-axis ℝ+. For continuous functions of rather common type we first obtain estimates of the best approximation by differences of simple partial fractions on ℝ+. For odd functions we obtain such estimates on the whole axis ℝ.