Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Overiteration of $d$-variate tensor product Bernstein operators: a quantitative result

Ana-Maria Acu, Heiner Gonska

Abstract

Extending an earlier estimate for the degree of approximation of overiterated univariate Bernstein operators towards the same operator of degree one, it is shown that an analogous result holds in the $d$-variate case. The method employed can be carried over to many other cases and is not restricted to Bernstein-type or similar methods.

Overiteration of $d$-variate tensor product Bernstein operators: a quantitative result

Abstract

Extending an earlier estimate for the degree of approximation of overiterated univariate Bernstein operators towards the same operator of degree one, it is shown that an analogous result holds in the -variate case. The method employed can be carried over to many other cases and is not restricted to Bernstein-type or similar methods.
Paper Structure (9 sections, 1 theorem, 43 equations, 1 figure)

This paper contains 9 sections, 1 theorem, 43 equations, 1 figure.

Table of Contents

  1. Introduction and historical remarks
  2. The non-quantitative approach of Agratini and Rus revisited
  3. The Zhuk extension in the bi- and $d$-variate cases
  4. Zhuk construction-univariate case
  5. Construction of the bivariate Zhuk extension
  6. Zhuk extension, $d$-variate case
  7. An estimate for $d$-variate tensor product Bernstein operators
  8. Optimality
  9. Concluding remark

Key Result

Theorem 2.1

(Jachymski 10) For fixed $l_1,\dots,l_d\in {\mathbb N}=\{1,2,\dots\}$ one has Here $B_{l_1}\otimes\dots \otimes B_{l_d}$ is the $d$-variate tensor product operator on $C([0,1]^d)$ and

Figures (1)

  • Figure 1:

Theorems & Definitions (1)

  • Theorem 2.1