Szász-Mirakyan operator

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In functional analysis, a discipline within mathematics, the Szász-Mirakjan[1] operators are generalizations of Bernstein polynomials to infinite intervals. They are defined by

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

where x\in[0,\infty)\subset\mathbb{R} and n\in\mathbb{N}.[2][3]

Contents

[edit] Basic results

In 1964, Cheney and Sharma showed that if f is convex and non-linear, the sequence (\mathcal{S}_n(f))_{n\in\mathbb{N}} decreases with n (\mathcal{S}_n(f)\geq f).[4] 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).

[edit] 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\rightarrow\infty.[2]

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

[edit] 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.[5]

[edit] References

[edit] Footnotes

  1. ^ Also spelled Mirakyan and Mirakian
  2. ^ a b Szász, Otto (1950). "Generalizations of S. Bernstein's polynomials to the infinite interval". Journal of Research of the National Bureau of Standards 45 (3): 239–245. 
  3. ^ Walczak, Zbigniew (2003). "On modified Szasz-Mirakyan operators". Novi Sad Journal of Mathematics 33 (1): 93–107. 
  4. ^ Cheney, Edward W.; A. Sharma (1964). "Bernstein power series". Canadian Journal of Mathematics 16 (2): 241–252. 
  5. ^ 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. 
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