Operators of Durrmeyer Type with Respect to Arbitrary Measure

Abstract

AbstractIn this paper, we give an overview of operators of Durrmeyer type with respect to arbitrary measure. Our construction includes the Bernstein–Durrmeyer operator, the Szász–Mirakjan–Durrmeyer operator, and the Baskakov–Durrmeyer operator with respect to arbitrary measure. We are particularly interested in the convergence of the operators. We discuss the uniform and the pointwise convergence as well as convergence in the corresponding weighted \(L^p\)-spaces. A new result is the statement on the \(L^p\)-convergence of the Szász–Mirakjan–Durrmeyer operator and the Baskakov–Durrmeyer operator without additional restrictions on the measure.KeywordsBernstein–Durrmeyer operatorSzász–Mirakjan–Durrmeyer operatorBaskakov–Durrmeyer operatorUniform convergencePointwise convergence \(L^p\)-convergence2010 AMS Subject Classification:41A36