Global smoothness preservation by multivariate singular integrals
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.
Bulletin of the Australian Mathematical Society
Anastassiou, G., & Gal, S. (2000). Global smoothness preservation by multivariate singular integrals. Bulletin of the Australian Mathematical Society, 61 (3), 489-506. https://doi.org/10.1017/s0004972700022516