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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Aral et al. (2008) studied this question.
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