Baskakov operator explained

In functional analysis, a branch of mathematics, the Baskakov operators are generalizations of Bernstein polynomials, Szász–Mirakyan operators, and Lupas operators. They are defined by

[ l{L} _{n(f)](x)} = \sum ^{inf ty} _{k =0} {( -1 ) ^{k} (x ^{k})/(k!) \phi ^{(k)} _{n} (x) f\left( (k)/(n) \right)} where x\in[ 0 ,b)\subset R (b can be inf ty ), n\in N , and (\phi _{n)} _{n\inN} is a sequence of functions defined on [ 0 ,b] that have the following properties for all n,k\in N :\phi ^{inf ty[ 0 ,b]} _{n\in l{C}} . Alternatively, \phi _{n} has a Taylor series on [ 0 ,b) .\phi _{n(0)} = 1 \phi _{n} is completely monotone, i.e. ( -1 ) ^{(k)} _{n} \geq 0 .

1. There is an integer

c such that \phi ^{(k +1 )} _{n} = - n\phi ^{(k)} _{n + c} whenever n> max \{ 0 , - c\} They are named after V. A. Baskakov, who studied their convergence to bounded, continuous functions.

Basic results

The Baskakov operators are linear and positive.

References

• Baskakov . V. A. . 1957 . ru:Пример последовательности линейных положительных операторов в пространстве непрерывных функций . An example of a sequence of linear positive operators in the space of continuous functions . . Russian . 113 . 249–251.