On L-convergence of Bernstein–durrmeyer Operators with Respect to Arbitrary Measure
نویسندگان
چکیده
S d := {x = (x1, . . . , xd) ∈ R : 0 6 x1, . . . , xd 6 1, 0 6 x1 + · · ·+ xd 6 1} denote the standard simplex in R. We denote by ∂S the boundary of S. We will also use barycentric coordinates on the simplex which we denote by the boldface symbol x = (x0, x1, . . . , xd), x0 := 1−x1−· · ·−xd. We will use standard multiindex notation such as x := x0 0 x α1 1 · · ·xd d and α n := (α0 n , α1 n , · · · , αd n )
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We introduce a class of Bernstein-Durrmeyer operators with respect to an arbitrary measure on a multi-dimensional simplex. These operators generalize the well-known Bernstein-Durrmeyer operators with Jacobi weights. A motivation for this generalization comes from learning theory. In the talk, we discuss the question which properties of the measure are important for convergence of the operators....
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