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.
AbstractWe establish the global smoothness preservation of a function f by the sequence of linear positive operators. Our estimate is in terms of the second order Ditzian-Totik modulus of smoothness. Application is given to the Bernstein operator.
Global smoothness preservation by multivariate singular integrals
