The behavior of the derivatives of the algebraic polynomials of best approximation

Abstract

In a recent paper M. Hasson [3] has investigated the behavior of the derivatives of the polynomials of best approximation of a function f on (1, 11. These investigations have led to norm estimates on the derivatives of the polynomials and on the distance between these derivatives and the respective derivatives of the function. Separate results were obtained when the norms were taken over the whole interval and when they were taken on a subinterval (where the estimates are significantly better). This calls for pointwise estimates in the spirit of the results of Timan [6] and Trigub [7]. In Section 2 we obtain pointwise estimates on the distance between the derivatives of the polynomials of best approximation off and the respective derivatives of J These results extend and unify those of Hasson [3]. In Section 3 we consider higher-order derivatives and obtain estimates on the growth of the sequence of derivatives of the polynomials of best approximation. Some special cases of these results are due to Hasson [3].