Binomial-type coefficients and classical approximation processes (Q2702491)

\textit{F. Altomare} [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 16, No. 2, 259-279 (1989; Zbl 0706.47022)] initiated a systematic study of the connection between some approximation processes and the solution of suitable evolution problems, the key ingredients being Voronovskaya's formula and Trotter's theorem. Chapter 6 of the monograph [\textit{F. Altomare}, \textit{M. Campiti}, Korovkin-type approximation theory and its applications (1994; Zbl 0924.41001)] gives a first account of the results of this type. Since then, many authors considered these problems either using different classical approximation processes or introducing new sequences of positive operators satisfying a prescribed Voronovskaya formula. The author discusses a class of approximation processes which can be obtained from classical Bernstein-type operators when the binomial coefficients are substituted by more general ones satisfying a similar recursive formula. In the interval \([0,1]\), the process taken into consideration is of Bernstein-type for continuous functions and of Bernstein-Kantorovitch-type for \(L_p\)-integrable functions; on the half-line Bernstein-Chlodovski-type operators and Baskakov-type operators are considered. A final section is devoted to Bernstein-type operators on the hypercube and on the standard simplex of \(\mathbb{R}^n\). The class of evolution problems which can be investigated by using the above mentioned approximation processes includes some diffusion models of particular interest as gene frequency models in population genetics when selection, migration, mutation and other factors occur.NEWLINENEWLINEFor the entire collection see [Zbl 0954.65001].