Phillips-type \(q\)-Bernstein operators on triangles (Q2023022)

Summary: The purpose of the paper is to introduce a new analogue of Phillips-type Bernstein operators \((\mathcal{B}_{m,q}^u f)(u,v)\) and \((\mathcal{B}_{n,q}^v f) (u,v)\), their products \((\mathcal{P}_{mn,q}f) (u,v)\) and \((\mathcal{Q}_{nm,q}f) (u,v)\), their Boolean sums \((\mathcal{S}_{mn,q}f) (u,v)\) and \((\mathcal{T}_{nm,q}f) (u,v)\) on triangle \(\mathcal{T}_h\), which interpolate a given function on the edges, respectively, at the vertices of triangle using quantum analogue. Based on Peano's theorem and using modulus of continuity, the remainders of the approximation formula of corresponding operators are evaluated. Graphical representations are added to demonstrate consistency to theoretical findings. It has been shown that parameter \(q\) provides flexibility for approximation and reduces to its classical case for \(q=1\).