On positive linear operators
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Date
2019
Authors
Jannat, Farzaneh
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Abstract
Approximation theory received much attention in the last century, and an intriguing area in this field is positive linear operators. This thesis is a literature survey on positive linear operators.
We discuss some of the well-known examples of positive linear operators. Preserving $k$-monotonicity by positive linear operators is another interesting topic in this area. We are interested if $L_n(f;x)$ is $k$-monotone whenever function $f$ is $k$-monotone. For a sequence of positive linear operators $\{L_n(f;x)\}$, it is a natural question if this sequence converges to $f(x)$ and how fast is this convergence. We study the saturation of these operators. Usually, there is a relation between the rate of convergence and the smoothness of the function being approximated. If increasing smoothness of functions does not result in the increase of the degree of approximation, this phenomenon is called saturation. We discuss iteration of general positive linear operators and give numerical examples.
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Mathematics