A Review on Some Linear Positive Operators Defined on Triangles

Abstract

We consider results regarding Bernstein and Cheney–Sharma-type operators that interpolate functions defined on triangles with straight and curved sides and we introduce a new Cheney–Sharma-type operator for the triangle with one curved side, highlighting the symmetry between the methods. We present some properties of the operators, their products and Boolean sums and some results regarding the remainders of the corresponding approximation formulas, using modulus of continuity and Peano’s theorem. Additionally, we consider some numerical examples to show the approximation properties of the given operators.

Keywords

• Bernstein operator
• Cheney–Sharma operator
• product and Boolean sum operators
• modulus of continuity
• degree of exactness
• error evaluation