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.
[1][2] In 1964, Cheney and Sharma showed that if
is convex and non-linear, the sequence
is a polynomial of degree
A converse of the first property was shown by Horová in 1968 (Altomare & Campiti 1994:350).
In Szász's original paper, he proved the following as Theorem 3 of his paper: This is analogous to a theorem stating that Bernstein polynomials approximate continuous functions on [0,1].
A Kantorovich-type generalization is sometimes discussed in the literature.
These generalizations are also called the Szász–Mirakjan–Kantorovich operators.
In 1976, C. P. May showed that the Baskakov operators can reduce to the Szász–Mirakyan operators.