Pointwise Approximation By Monotone Sequences of Polynomials

Abstract

In a recent paper [2], Gal and Szabados obtained, for $f \in C_{\left[ { - 1,1} \right]}$, sequences {Pn} and{Qn} satisfying Qn(x) ≦ Qn+1(x) ≦ f(x) ≦ Pn+1 ≦ Pn(x)such that

$$||P_n (x) - Q_n (x)||\underline { \leqslant 8} \sum\limits_{k = [n/2] - 1}^\infty {k^{ - 1} E_k (f),{\text{ }}n\underline{ \geqslant 4} } $$

under the condition

$$\sum\limits_{k = 1}^\infty {k^{ - 1} K_k (f) < \infty } $$

. Xie and Zhou in [4] showed that one can construct such monotone polynomial sequences which do achieve the best uniform approximation rate for a continuous function, making no condition, in a quite delicate constructive way just by perturbation by constants of a subsequence of the best approximation polynomials. By considering that the pointwise estimate for such type of approximation might be potentially useful in some algebraic approximation cases, one should be interested to establish Jackson type rate. However, this problem is not easy. This paper will present an affirmative answer.