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Abstract. The aim of this paper is to study the convergence and approximation properties of the bivariate operators and GBS operators of Durrmeyer-type. IntroductionLet N be the set of positive integers andfor any x ∈ [0, 1], where p m,k (x) are the fundamental polynomials of Bernstein, defined as followsfor any x ∈ [0, 1] and any k ∈ {0, 1, . . . , m}. These operators were introduced in 1967 by J.L. Durrmeyer in [10] and were studied in 1981 by M.M. Derriennic in [9]. The purpose of this paper is to give a representation for the bivariate operators and GBS operators of Durrmeyer-type, to establish a convergence theorem for these operators. We also give an approximation theorem for these operators in terms of the first modulus of smoothness and of the mixed modulus of smoothness.

The aim of this paper is to study the convergence of the sequence of generalized Boolean sum (GBS) operators (UBm)m≥1 for B-continuous functions f ∈ Cb(∆4).

In this paper, we obtain the locus of generalized Torricelli-Fermat points of a triangle.