Evolution equations associated with recursively defined Bernstein-type operators
The theory of evolution equations associated with the classical Bernstein operators \((B_n)\) is presented in Chapter 6 of the book [\textit{F. Altomare} and \textit{M. Campiti}, Korovkin-type approximation theory and its applications (1994)]. In a previous paper [J. Approximation theory 87, No. 3, 243-269 (1996; Zbl 0865.41023)], \textit{M. Campiti} and \textit{G. Metafune} introduced and studied a generalization of the classical Bernstein operators. Namely, given two sequences \((\lambda_n)\) and \((\rho_n)\) of real numbers, they considered the operators \[ A_nf(x)= \sum^n_{k=0} \alpha_{k,n} x^k(1-x)^{n-k} f(k/n), \quad f\in C[0,1],\;x\in [0,1], \] with \(\alpha_{n,k}\) determined by \(\alpha_{n+1,k} =\alpha_{n,k} +\alpha_{n,k-1}\), \(k=1, \dots, n\) and \(\alpha_{n,0} =\lambda_n\), \(\alpha_{n,n} =\rho_n\). The use of \((A_n)\) instead of \((B_n)\) enables the authors to approximate the solutions of a larger class of degenerate second order parabolic problems.
scientific article; zbMATH DE number 980674