Iterations and fixed points for the Bernstein max-product operator (Q2871097)

The Bernstein max-product approximation operator, for the first time defined and formally studied by \textit{S. G. Gal} on Pages 325 and 326 in [Shape-preserving approximation by real and complex polynomials. Basel: Birkhäuser (2008; Zbl 1154.41002)], is given by the formula NEWLINE\[NEWLINEB_{n}^{(M)}(f)(x)=\frac{\bigvee _{k=0}^{n}p_{n,k}(x)f(\frac{k}{n})}{\bigvee_{0}^{n}p_{n,k}(x)}\,,NEWLINE\]NEWLINE NEWLINEwhere \(p_{n,k}(x)=\binom {n} {k}x^{k}(1-x)^{n-k}\) and \(\bigvee_{0}^{n}p_{n,k}(x)=\max_{k=\{0, 2, \dots, n\}}\{p_{n,k}(x)\}\). The fact that the Bernstein max-product operator does not loose the approximation properties makes this theory applicable and more advantaged [\textit{B. Bede} et al., Int. J. Math. Math. Sci. 2009, Article ID 590589, 26 p. (2009; Zbl 1188.41016)]. In the paper under review, the authors prove that the max-product Bernstein operator \(B_{n}^{(M)}:C_{+}[0, 1]\overrightarrow{}\longrightarrow C_{+}[0, 1]\) is non-expansive for each \(n\in {\mathbb{N}}\) with respect to the uniform norm in \(C_{+}[0, 1]\), where \(C_{+}[0, 1]\) is the set of all continuous functions from \([0, 1]\) into \(\mathbb{R}_{+}\). After proving two inequalities, namely Theorem 2.2 and Lemma 2.3, they obtain some estimates related to the Bernstein max-product approximation operator, and present a fixed point theorem. They also investigate Ishikawa iterations for \(B_{n}^{(M)}\).