Szász-Mirakyan operator

In functional analysis, a discipline within mathematics, the Szász-Mirakjan [Also spelled "Mirakyan" and "Mirakian"] operators are generalizations of Bernstein polynomials to infinite intervals. They are defined by:$left\; [mathcal\{S\}\_n(f)\; ight]\; (x)=e^\{-nx\}sum\_\{k=0\}^infty\{frac\{(nx)^k\}\{k!\}fleft(frac\{k\}\{n\}\; ight)\}$where $xin\; [0,infty)subsetmathbb\{R\}$ and $ninmathbb\{N\}$.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] 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]

Basic results

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

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 $f$ is continuous on $(0,infty)$, then $mathcal\{S\}\_n(f)$ converges uniformly to $f$ as $n\; ightarrowinfty$.This is analogous to [Bernstein polynomial#Approximating continuous functions|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-Mirakyan-Kantorovich operators.

In 1976, C. P. May showed that the Baskakov operators can reduce to the Szász-Mirakyan operators.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]

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=3110141787
*cite journal| last=Favard | first=Jean | authorlink=Jean Favard | year=1944 | title=Sur les multiplicateurs d'interpolation | journal=Journal de Mathematiques Pures et Appliquees | volume=23 | issue=9 | pages=219–247 fr icon (Also see 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|id=Zbl|0186.11101
*cite journal| last=Kac | first=Mark | authorlink=Mark Kac | year=1938 | title=Une remarque sur les polynomes de M. S. Bernstein | journal=Studia Mathematica | volume=7 | pages=49–51 | id=Zbl|0018.20704 | url=http://matwbn.icm.edu.pl/ksiazki/sm/sm7/sm715.pdf fr icon
*cite journal| last=Kac | first=Mark | authorlink=Mark Kac | year=1939 | title=Reconnaissance de priorité relative à ma note 'Une remarque sur les polynomes de M. S. Bernstein' | journal=Studia Mathematica | volume=8 | pages=170 | id=JFM|65.0248.03 | url=http://matwbn.icm.edu.pl/ksiazki/sm/sm8/sm8111.pdf fr icon
*cite journal| last=Mirakjan | first=G. M. | year=1941 | title=Approximation of continuous functions with the aid of polynomials of the form $e^\{-nx\}sum\_\{k=0\}^\{m\_n\}\{C\_\{k,n\}x^k\}$ ( _fr. Approximation des fonctions continues au moyen de polynômes de la forme $e^\{-nx\}sum\_\{k=0\}^\{m\_n\}\{C\_\{k,n\}x^k\}$) | journal=Proceedings of the USSR Academy of Sciences | volume=31 | pages=201-205 | id=JFM|67.0216.03 fr icon
*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|url=http://links.jstor.org/sici?sici=0036-1399%28196907%2917%3A4%3C790%3AGSOFTA%3E2.0.CO%3B2-F | id=Zbl|0182.08801|issue=4|doi=10.1137/0117071

Footnotes