<abstract xmlns="http://www.w3.org/1999/xhtml"><p>This research and survey article deals exclusively with the study of the approximation of generalized multivariate Gauss-Weierstrass singular integrals to the identity-unit operator. Here we study quantitatively most of their approximation properties. The multivariate generalized Gauss-Weierstrass operators are not in general positive linear operators. In particular we study the rate of convergence of these operators to the unit operator, as well as the related simultaneous approximation. These are given via Jackson type inequalities and by the use of multivariate high order modulus of smoothness of the high order partial derivatives of the involved function. Also we study the global smoothness preservation properties of these operators. These multivariate inequalities are nearly sharp and in one case the inequality is attained, that is sharp. Furthermore we give asymptotic expansions of Voronovskaya type for the error of multivariate approximation. The above properties are studied with respect to <italic>L<sub>p</sub></italic>norm, 1 ≤ <italic>p</italic> ≤ <italic>∞</italic>.</p></abstract>

This research and survey article deals exclusively with the study of the approximation of generalized multivariate Gauss-Weierstrass singular integrals to the identity-unit operator. Here we study quantitatively most of their approximation properties. The multivariate generalized Gauss-Weierstrass operators are not in general positive linear operators. In particular we study the rate of convergence of these operators to the unit operator, as well as the related simultaneous approximation. These are given via Jackson type inequalities and by the use of multivariate high order modulus of smoothness of the high order partial derivatives of the involved function. Also we study the global smoothness preservation properties of these operators. These multivariate inequalities are nearly sharp and in one case the inequality is attained, that is sharp. Furthermore we give asymptotic expansions of Voronovskaya type for the error of multivariate approximation. The above properties are studied with respect to Lpnorm, 1 ≤ p ≤ ∞.

Complete Approximations by Multivariate Generalized Gauss-Weierstrass Singular Integrals

Moroccan Journal of Pure and Applied Analysis

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

This research and survey article deals exclusively with the study of the approximation of generalized multivariate Gauss-Weierstrass singular integrals to the...

Complete Approximations by Multivariate Generalized Gauss-Weierstrass Singular Integrals

Published Online: Dec 14, 2020

Page range: 134 - 172

Received: Oct 10, 2020

Accepted: Nov 25, 2020

DOI: https://doi.org/10.2478/mjpaa-2021-0012

Keywords
Approximations by multivariate Gauss-Weierstrass singular integral, Multivariate moduli of smoothness

© 2020 George A. Anastassiou, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Complete Approximations by Multivariate Generalized Gauss-Weierstrass Singular Integrals

Published Online: Dec 14, 2020

Page range: 134 - 172

Received: Oct 10, 2020

Accepted: Nov 25, 2020

DOI: https://doi.org/10.2478/mjpaa-2021-0012

KeywordsApproximations by multivariate Gauss-Weierstrass singular integral, Multivariate moduli of smoothness

© 2020 George A. Anastassiou, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
Approximations by multivariate Gauss-Weierstrass singular integral, Multivariate moduli of smoothness