Approximation by simple partial fractions on the semi-axis

Abstract

This paper investigates the simple partial fractions (that is, the logarithmic derivatives of polynomials) all of whose poles lie within the angular domain , for any . It is shown that they are contained in a proper half-space of the space for any (in particular, they are not dense in this space) and conversely, they are dense in for any , where . The distances from the poles of a simple partial fraction  to the semi-axis are estimated in terms of the degree of the fraction  and its norm in . The approximation properties of sets of simple partial fractions of degree at most  are investigated, as well as properties of the least deviations from these sets for the functions . Bibliography: 14 titles.