Szász–Mirakyan operator

In functional analysis, a discipline within mathematics, the Szász–Mirakyan operators (also spelled "Mirakjan" and "Mirakian") are generalizations of Bernstein polynomials to infinite intervals, introduced by Otto Szász in 1950 and G. M. Mirakjan in 1941. They are defined by

where ${\displaystyle x\in [0,\infty )\subset \mathbb {R} }$ and ${\displaystyle n\in \mathbb {N} }$.

In 1964, Cheney and Sharma showed that if ${\displaystyle f}$ is convex and non-linear, the sequence ${\displaystyle ({\mathcal {S}}_{n}(f))_{n\in \mathbb {N} }}$ decreases with ${\displaystyle n}$ (${\displaystyle {\mathcal {S}}_{n}(f)\geq f}$). They also showed that if ${\displaystyle f}$ is a polynomial of degree ${\displaystyle \leq m}$, then so is ${\displaystyle {\mathcal {S}}_{n}(f)}$ for all ${\displaystyle n}$.

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