STABILITY OF BEST RATIONAL CHEBYSHEV-APPROXIMATION

In this paper we consider best Chebyshev approximation to continuous functions by generalized rational functions using an optimization theoretical approach introduced in [B. Brosowski and C. Guerreiro, On the characterization of a set of optimal points and some applications, in "Approximation and Optimization in Mathematical Physics" (B. Brosowski and E. Martensen, Eds.), pp. 141-174, Verlag Peter Lang, Frankfurt a.M./Bern, 1983]. This general approach includes, in a unified way, usual, weighted, one-sided, unsymmetric, and also more general rational Chebychev approximation problems with side-conditions. We derive various continuity conditions for the optimal value, for the feasible set, and the optimal set of the corresponding optimization problem. From these results we derive conditions for the upper semicontinuity of the metric projection, which include some of the results of Werner [On the rational Tschebyscheff operator, Math. Z. 86 (1964), 317-326] and Cheney and Loeb [On the continuity of rational approximation operators, Arch. Rational Mech. Anal. 21 (1966), 391-401]. © 1990.