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

${\displaystyle \left[{\mathcal {S}}_{n}(f)\right](x)}$=${\displaystyle e^{-nx}\sum _{k=0}^{\infty }{{\frac {(nx)^{k}}{k!}}f\left({\frac {k}{n}}\right)}}$

where ${\displaystyle x\in [0,\infty )\subset \mathbb {R} }$ and ${\displaystyle n\in \mathbb {N} }$.[1][2]

Basic results

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 ${\displaystyle n}$ (${\displaystyle {\mathcal {S}}_{n}(f)\geq f}$).[3] 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 ${\displaystyle n}$.

A converse of the first property was shown by Horová in 1968 (Altomare & Campiti 1994:350).

Theorem on convergence

In Szász's original paper, he proved the following:

If ${\displaystyle f}$ is continuous on ${\displaystyle (0,\infty )}$, having a finite limit at infinity, then ${\displaystyle {\mathcal {S}}_{n}(f)}$ converges uniformly to ${\displaystyle f}$ as ${\displaystyle n\rightarrow \infty }$.[1]

This is analogous to a theorem stating that Bernstein polynomials approximate continuous functions on [0,1].

Generalizations

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.[4]

References

• Altomare, Francesco; Michele Campiti (1994). Korovkin-Type Approximation Theory and Its Applications. Walter de Gruyter. ISBN 3-11-014178-7.
• Favard, Jean (1944). "Sur les multiplicateurs d'interpolation". Journal de Mathématiques Pures et Appliquées (in French). 23 (9): 219–247. (See also: Favard operators)
• Horová, Ivana (1968). "Linear positive operators of convex functions". Mathematica (Cluj). 10 (33): 275–283. Zbl 0186.11101.
• Kac, Mark (1938). "Une remarque sur les polynomes de M. S. Bernstein" (PDF). Studia Mathematica (in French). 7: 49–51. Zbl 0018.20704.
• Kac, M. (1939). "Reconnaissance de priorité relative à ma note 'Une remarque sur les polynomes de M. S. Bernstein'" (PDF). Studia Mathematica (in French). 8: 170. JFM 65.0248.03.
• Mirakjan, G. M. (1941). "Approximation des fonctions continues au moyen de polynômes de la forme ${\displaystyle e^{-nx}\sum _{k=0}^{m_{n}}{C_{k,n}x^{k}}}$" [Approximation of continuous functions with the aid of polynomials of the form ${\displaystyle e^{-nx}\sum _{k=0}^{m_{n}}{C_{k,n}x^{k}}}$]. Comptes rendus de l'Académie des sciences de l'URSS (in French). 31: 201–205. JFM 67.0216.03.
• Wood, B. (July 1969). "Generalized Szasz operators for the approximation in the complex domain". SIAM Journal on Applied Mathematics. 17 (4): 790–801. doi:10.1137/0117071. JSTOR 2099320. Zbl 0182.08801.

Footnotes

1. Szász, Otto (1950). "Generalizations of S. Bernstein's polynomials to the infinite interval" (PDF). Journal of Research of the National Bureau of Standards. 45 (3): 239–245. doi:10.6028/jres.045.024.
2. Walczak, Zbigniew (2003). "On modified Szasz–Mirakyan operators" (PDF). Novi Sad Journal of Mathematics. 33 (1): 93–107.
3. Cheney, Edward W.; A. Sharma (1964). "Bernstein power series". Canadian Journal of Mathematics. 16 (2): 241–252. doi:10.4153/cjm-1964-023-1.
4. May, C. P. (1976). "Saturation and inverse theorems for combinations of a class of exponential-type operators". Canadian Journal of Mathematics. 28 (6): 1224–1250. doi:10.4153/cjm-1976-123-8.