A criterion for the best uniform approximation by simple partial fractions in terms of alternance. II

Abstract

In the problem of approximating real functions by simple partial fractions of order on closed intervals , we obtain a criterion for the best uniform approximation which is similar to Chebyshev's alternance theorem and considerably generalizes previous results: under the same condition on the poles of the fraction of best approximation, we omit the restriction on the order of this fraction. In the case of approximation of odd functions on , we obtain a similar criterion under much weaker restrictions on the position of the poles : the disc is replaced by the domain bounded by a lemniscate contained in this disc. We give some applications of this result. The main theorems are extended to the case of weighted approximation. We give a lower bound for the distance from to the set of poles of all simple partial fractions of order which are normalized with weight on (a weighted analogue of Gorin's problem on the semi-axis).