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

作者：Komarov M A^*

来源：Izvestiya - Mathematics, 2017, 81(3): 568-591.

DOI：10.1070/IM8548

• simple partial fraction
• approximation
• alternance
• uniqueness
• disc
• odd function
• lemniscate

In the problem of approximating real functions f by simple partial fractions of order <= n on closed intervals K = [c -rho, c + rho] subset of R, 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 z(j)* is not an element of B(c, rho) = {z : vertical bar z -c vertical bar <= rho} on the poles z(j)* of the fraction rho*(n, f, K; x) of best approximation, we omit the restriction k = n on the order k of this fraction. In the case of approximation of odd functions on [-rho,rho], we obtain a similar criterion under much weaker restrictions on the position of the poles z(j)* : the disc B(0,rho) 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 R+ to the set of poles of all simple partial fractions of order <= n which are normalized with weight 2 root x on R+ (a weighted analogue of Gorin's problem on the semi-axis).

• 出版日期2017