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In 1944 Favard [5, pp. 229, 239] introduced a discretely defined operator which is a discrete analogue of the familiar Gauss-Weierstrass singular convolution integral. In the present paper we consider a slight generalization F_n,σ_n of the Favard operator and its Durrmeyer variant  and study the local rate of convergence when applied to locally smooth functions. The main result consists of the complete asymptotic expansions for the sequences  and  as n tends to infinity. Furthermore, these asymptotic expansions are valid also with respect to simultaneous approximation. Finally, we define left quasi-interpolants for the Favard operator and its Durrmeyer variant in the sense of Sablonniere.

Mathematics Subject Classification (2010): 41A36, 41A60, 41A28.

Keywords: Approximation by positive operators, asymptotic expansions, simultaneous approximation.