General uniform approximation theory by multivariate singular integral operators
Volume 103 / 2012
Annales Polonici Mathematici 103 (2012), 15-25
MSC: Primary 26A15, 41A17, 41A25, 41A35; Secondary 26D15, 41A36.
DOI: 10.4064/ap103-1-2
Abstract
We study the uniform approximation properties of general multivariate singular integral operators on $\mathbb{R}^{N}$, $N\geq1$. We establish their convergence to the unit operator with rates. The estimates are pointwise and uniform. The established inequalities involve the multivariate higher order modulus of smoothness. We list the multivariate Picard, Gauss–Weierstrass, Poisson–Cauchy and trigonometric singular integral operators to which this theory can be applied directly.
