Approximation properties of recursively defined Bernstein-type operators
The authors consider polynomial type operators having the form \(A_n(f;x)= \sum^{k=n}_{k=0} \alpha_{n,k} x^k(1-x)^{n-k} f({k\over n})\) with \(n\in\mathbb{N} =\{1,2,\dots\}\), \(f\in C([0,1], \mathbb{R})\), \(x\in [0,1]\), \(\alpha_{n+1,k} =\alpha_{n,k} +\alpha_{n,k-1}\) for \(k\in \overline {1,n}\) and \(\alpha_{n,0} =\lambda_n\), \(\alpha_{n,n} =\rho_n\), where \((\lambda_n)_{n\in \mathbb{N}} \), \((\rho_n)_{n\in\mathbb{N}}\) are fixed sequences of real numbers. The authors study the uniform convergence of these operators together with some quantitative and regularity properties. Finally, in some particular cases, they investigate the behavior of the iterates.
scientific article; zbMATH DE number 980673