Global Smoothness Preservation and Simultaneous Approximation by Multivariate General Singular Integrals

2011 ◽  
pp. 33-45
Author(s):  
George A. Anastassiou
Keyword(s):  
Simultaneous Approximation ◽  
Singular Integrals ◽  
Global Smoothness Preservation ◽  
Global Smoothness

scholarly journals Global smoothness preservation by multivariate singular integrals

2000 ◽  
Vol 61 (3) ◽  
pp. 489-506
Author(s):  
George A. Anastassiou ◽  
Sorin G. Gal
Keyword(s):  
Singular Integrals ◽  
Moduli Of Smoothness ◽  
Jackson Type ◽  
Global Smoothness Preservation ◽  
Univariate Case ◽  
Global Smoothness ◽  
Preservation Property

By using various kinds of moduli of smoothness, it is established that the multivariate variants of the well-known singular integrals of Picard, Poisson-Cauchy, Gauss-Weierstrass and their Jackson-type generalisations satisfy the “global smoothness preservation” property. The results are extensions of those proved by the authors for the univariate case.

scholarly journals Complete Approximations by Multivariate Generalized Gauss-Weierstrass Singular Integrals

2021 ◽  
Vol 7 (1) ◽  
pp. 134-172
Author(s):  
George A. Anastassiou
Keyword(s):  
Singular Integrals ◽  
Modulus Of Smoothness ◽  
High Order ◽  
Linear Operators ◽  
Multivariate Approximation ◽  
Approximation Properties ◽  
Global Smoothness Preservation ◽  
Global Smoothness ◽  
Unit Operator ◽  
Derivatives Of

AbstractThis 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 ≤ ∞.

Miscellaneous Progress in Global Smoothness Preservation

2000 ◽  
pp. 473-484 ◽  
Author(s):  
George A. Anastassiou ◽  
Sorin G. Gal
Keyword(s):  
Global Smoothness Preservation ◽  
Global Smoothness
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