A New Class of Modified Bernstein Operators Original Research Article

The left Bernstein quasi-interpolant operator introduced by Sablonnière is a kind of modified Bernstein operator that has good stability and convergence rate properties. However, we recently found that it is not very convenient for practical applications. Fortunately, we showed in a previous paper that there exist many operators having stability and convergence rate properties similar to those of Sablonnièreʹs operator. In this paper, we introduce another specific class of such operators generated from the operator introduced by Stancu. We present detailed results about this class, some of which can be applied to numerical quadrature. Finally, we clarify its advantages and assert that it is more natural and more convenient, both theoretically and practically, than that of Sablonnière. Our paper, at the same time, provides several new results regarding Stancuʹs operator.