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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
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:<math>\left[\mathcal{S}_n(f)\right](x)</math>=<math>e^{-nx}\sum_{k=0}^\infty{\frac{(nx)^k}{k!}f\left(\frac{k}{n}\right)}</math>
where <math>x\in[0,\infty)\subset\mathbb{R}</math> and <math>n\in\mathbb{N}</math>.<ref name="Szász">{{cite journal| last=Szász | first=Otto | year=1950 | title=Generalizations of S. Bernstein's polynomials to the infinite interval | journal=Journal of Research of the National Bureau of Standards | volume=45 | issue=3 | pages=239–245 | url=http://nvl.nist.gov/pub/nistpubs/jres/045/3/V45.N03.A09.pdf}}</ref><ref name="Walczak">{{cite journal| last=Walczak | first=Zbigniew | year=2003 | title=On modified Szasz–Mirakyan operators | journal=Novi Sad Journal of Mathematics | volume=33 | issue=1 | pages=93–107 | url=http://www.emis.de/journals/NSJOM/33_1/rad-08.pdf}}</ref>
 
==Basic results==
In 1964, Cheney and Sharma showed that if <math>f</math> is convex and non-linear, the sequence <math>(\mathcal{S}_n(f))_{n\in\mathbb{N}}</math> decreases with <math>n</math> (<math>\mathcal{S}_n(f)\geq f</math>).<ref name="Cheney">{{cite journal|last=Cheney|first=Edward W.|coauthors=A. Sharma|year=1964|title=Bernstein power series|journal=Canadian Journal of Mathematics|volume=16|issue=[http://books.google.com/books?id=RSNqggY5Q5cC&dq 2]|pages=241–252}}</ref> They also showed that if <math>f</math> is a polynomial of degree <math>\leq m</math>, then so is <math>\mathcal{S}_n(f)</math> for all <math>n</math>.
 
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 <math>f</math> is [[Continuous function|continuous]] on <math>(0,\infty)</math>, having a finite limit at infinity, then <math>\mathcal{S}_n(f)</math> [[Uniform convergence|converges uniformly]] to <math>f</math> as <math>n\rightarrow\infty</math>.<ref name="Szász"/>
This is analogous to [[Bernstein polynomial#Approximating continuous functions|a theorem stating that Bernstein polynomials approximate continuous functions on [0,1]]].
 
==Generalizations==
A [[Leonid Kantorovich|Kantorovich]]-type generalization is sometimes discussed in the literature. These generalizations are also called the [[Szász–Mirakjan–Kantorovich operator]]s.
 
In 1976, C. P. May showed that the [[Baskakov operators]] can reduce to the Szász–Mirakyan operators.<ref name="May">{{cite journal|last=May|first=C. P.|year=1976|title=Saturation and inverse theorems for combinations of a class of exponential-type operators|journal=Canadian Journal of Mathematics|volume=28|issue=6|pages=1224–1250|url=http://books.google.com/books?hl=en&lr=&id=irg7sNuSTT8C&oi=fnd&pg=PA1224&ots=cdhSVISxAs&sig=OW7T5zOvK9PSaEsmT9_40BG0Pdc}}</ref>
 
==References==
*{{cite book|last=Altomare|first=Francesco|coauthors=Michele Campiti|year=1994|title=Korovkin-Type Approximation Theory and Its Applications|publisher=Walter de Gruyter|isbn=3-11-014178-7}}
*{{cite journal| last=Favard | first=Jean | authorlink=Jean Favard | year=1944 | title=Sur les multiplicateurs d'interpolation | language=French | journal=Journal de Mathematiques Pures et Appliquees | volume=23 | issue=9 | pages=219–247}} (See also: [[Favard operators]])
*{{cite journal|last=Horová|first=Ivana|year=1968|title=Linear positive operators of convex functions|journal=Mathematica (Cluj)|volume=10|issue=33|pages=275–283|zbl=0186.11101}}
*{{cite journal| last=Kac | first=Mark | authorlink=Mark Kac | year=1938 | title=Une remarque sur les polynomes de M. S. Bernstein | language=French | journal=Studia Mathematica | volume=7 | pages=49–51 | zbl=0018.20704 | url=http://matwbn.icm.edu.pl/ksiazki/sm/sm7/sm715.pdf}}
*{{cite journal| last=Kac | first=M. | authorlink=Mark Kac | year=1939 | title=Reconnaissance de priorité relative à ma note 'Une remarque sur les polynomes de M. S. Bernstein' | language=French | journal=Studia Mathematica | volume=8 | pages=170 | jfm=65.0248.03 | url=http://matwbn.icm.edu.pl/ksiazki/sm/sm8/sm8111.pdf}}
*{{cite journal| last=Mirakjan | first=G. M. | year=1941 | trans_title=Approximation of continuous functions with the aid of polynomials  of the form <math>e^{-nx}\sum_{k=0}^{m_n}{C_{k,n}x^k}</math> | title=Approximation des fonctions continues au moyen de polynômes de la forme <math>e^{-nx}\sum_{k=0}^{m_n}{C_{k,n}x^k}</math> | language=French | journal=[[Comptes rendus de l'Académie des sciences de l'URSS]] | volume=31 | pages=201–205 | jfm=67.0216.03}}
*{{cite journal|last=Wood|first=B.|year=1969|month=July|title=Generalized Szasz operators for the approximation in the complex domain|journal=SIAM Journal on Applied Mathematics|volume=17|pages=790–801| zbl=0182.08801|issue=4|doi=10.1137/0117071|jstor=2099320}}
 
===Footnotes===
<references/>
 
{{DEFAULTSORT:Szasz-Mirakyan Operator}}
[[Category:Approximation theory]]

Latest revision as of 17:05, 12 February 2014

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