Abstract
Introducing a higher order modulus of smoothness based on $q$-integers, in this paper first we obtain Jackson-type estimates in approximation by Jackson-type generalizations of the $q$-Picard and $q$-Gauss-Weierstrass singular integrals and give their global smoothness preservation property with respect to the uniform norm. Then, we study approximation and geometric properties of the complex variants for these $q$ -singular integrals attached to analytic functions in compact disks. Finally, we prove approximation properties of these $q$-singular integrals attached to vector-valued functions.
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