Kantorovich-Bernstein polynomials

Abstract

A gap between saturation and direct-converse theorems for Kantorovich-Bernstein polynomials will be closed for a steady rate of convergence. The present theorems unify the above-mentioned results. Furthermore, it is shown that for steady rates our converse results are an improvement on both weak-type converse theorems and strong-weak-type converse theorems for the KantorovichBernstein polynomials.

The lower estimate for linear positive operators, I

“…In the algebraic case a key role in this development (to prove saturation results) is played by an important paper by Lorentz [11]. Modifying the technique introduced by Berens and Lorentz [1], the second author pointed out in [12] that for many operators the inversb and saturation results can be proved in a common way (see [7], [8], [12], and [13] for details). Of course, the aim of these investigations is in fact to give a best possible lower estimate in some sense, while the upper estimate in many cases is known (Jackson-type theorem).…”

The Second Ditzian-Totik modulus revisited: refined estimates for positive linear operators

“…The continuing work of the authors mentioned explicitely in the above eventually culminated in, among other articles, a book by Ditzian and Totik [9], joint work of Ditzian and Zhou (see, e.g., [10]), an important article by Ditzian and Ivanov [8], a paper by Totik [36], and a significant contribution of Knoop and Zhou [23], [24]. A somewhat partial, but nonetheless streamlined, account of what had been achieved up to 1993, say, is given in the book of DeVore and Lorentz [5].…”

Weak Type Inequalities for Best Simultaneous Approximation in Banach Spaces

“…In this section we generalize the admissible order functions used in approximation theorems to an Abelian group of functions 8, and we introduce a relation O in 8 which gives an easy tool to compare the different orders of elements of 8. Several attempts to extend power functions can be found in papers on approximation theory, see e.g., P. L. Butzer and K. Scherer [9], P. L. Butzer et al [4], Z. Ditzian and X. Zhou [14] as well as in S. Jansche and R. L. Stens [18] and E. van Wickeren [31]. A rigorous application of the theory of regularly varying functions seems to have been first carried out in [17], and in [16], where one finds a detailed elaboration and proofs of the material of this section.…”