On the real CF-method for polynomial approximation and strong unicity constants

AbstractWe consider uniform polynomial approximation on [ −1, 1]. For the class of functions which are analytic in an ellipse with foci ± 1 and sum of semiaxes greater than 8.1722…, we prove several asymptotic results on the best approximation. We describe the CF-approximation method and prove that, for our class of functions, the CF-approximation is “not far away” from the best one. With the help of this result we show a Kadec type result on the alternants and prove a conjecture of Poreda on the strong uniqueness constants. Also we prove a lemma on the distance between the best approximation and a “good” approximating polynomial