Approximations and expansions Definition

Szász–Mirakyan operator

Builds on Poisson distribution Positive linear operator

Definition

Let $f \colon [0, \infty) \to \R$ satisfy $\abs{f(t)} \le C e^{A t}$ for some constants $C, A \ge 0$ and all $t \ge 0$ . For an integer $n \ge 1$ , the Szász–Mirakyan operator $S_n$ maps $f$ to the function $S_n(f)(x) = e^{-nx} \sum_{k=0}^{\infty} \frac{(nx)^k}{k!} \, f\left(\frac{k}{n}\right), \qquad x \ge 0.$

Remarks

The growth bound makes the series converge for all $x \ge 0$ . Each $S_n$ is a positive linear operator. The weights $e^{-nx} (nx)^k / k!$ are the probabilities of the Poisson distribution with mean $nx$ .

Example: if $N$ is Poisson with mean $nx$ , then $E[N] = nx$ and $E[N^2] = nx + n^2 x^2$ . So $S_n(1) = 1$ , $S_n(t)(x) = x$ and $S_n(t^2)(x) = x^2 + x/n$ .

Non-example: $f(t) = e^{t^2}$ is not in the domain, since $e^{k^2/n^2} (nx)^k / k!$ does not tend to $0$ as $k \to \infty$ when $x > 0$ .

From the three moments and Korovkin's theorem, $S_n f \to f$ uniformly on each interval $[0, b]$ for every bounded continuous $f$ .

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Tag	`XKAWV4CCCB`
URI	`m2://XKAWV4CCCB`
Kind	definition
Writing	W0 stub
MSC	41A36, 41A35, 41A25
Defines	Szász–Mirakyan operator
Uses	Poisson distribution, Positive linear operator

Szász–Mirakyan operator

Definition

Remarks

Reading path

Further reading

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