Approximation by Max-Product Favard–Szász–Mirakjan Operators

Abstract

This chapter deals with the approximation and the shape preserving properties of the max-product Favard–Szász–Mirakjan operators, denoted by F _n ^(M)(f) in the non-truncated case, by $\mathcal{T}_{n}^{(M)}(f)$ in the truncated case and attached to bounded functions f with only positive values. It is worth mentioning that this restriction can be dropped by attaching to bounded functions f of variable sign the new max-product type operators $\overline{F}_{n}^{(M)}(f)(x) = F_{n}^{(M)}(f - a)(x) + a$, $\overline{\mathcal{T}}_{n}^{(M)}(f)(x) = \mathcal{T}_{n}^{(M)}(f - a)(x) + a$, with a < inff. Indeed, by following the ideas in Theorem 2.9.1 (see also Subsection 1.1.3, Property C) it is easily seen that all the approximation and shape preserving properties proved for F _n ^(M)(f)(x) and $\mathcal{T}_{n}^{(M)}(f)(x)$ below in this chapter remain valid for the max-product operators $\overline{F}_{n}^{(M)}(f)(x)$ and $\overline{\mathcal{T}}_{n}^{(M)}(f)(x)$.