Uniform approximation by smooth Picard multivariate singular integral operators revisited

In this article we reexamine the uniform approximation properties of smooth Picard multivariate singular integral operators over R N, N ≥ 1. We establish their convergence to the unit operator with rates. The estimates are pointwise and uniform. The established inequalities involve the multivariate first modulus of continuity. Our approach is based on a new multivariate trigonometric Taylor formula. At first we present in detail the general theory of uniform approximation by general smooth multivariate singular integral operators, which then is applied to the Picard operators case.