Baskakov operator

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 : [\mathcal{L}_n(f)](x) = \sum_{k=0}^\infty {(-1)^k \frac{x^k}{k!} \phi_n^{(k)}(x) f\left(\frac{k}{n}\right)} where x\in[0,b)\subset\mathbb{R} ( b can be \infty ), n\in\mathbb{N} , and (\phi_n)_{n\in\mathbb{N}} is a sequence of functions defined on [0,b] that have the following properties for all n,k\in\mathbb{N} : # \phi_n\in\mathcal{C}^\infty[0,b] . Alternatively, \phi_n has a Taylor series on [0,b) . # \phi_n(0) = 1 # \phi_n is completely monotone, i.e. (-1)^k\phi_n^{(k)}\geq 0 . #There is an integer c such that \phi_n^{(k+1)} = -n\phi_{n+c}^{(k)} whenever n>\max\{0,-c\} They are named after V. A. Baskakov, who studied their convergence to bounded, continuous functions.

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 [\mathcal{L}_n(f)](x) = \sum_{k=0}^\infty {(-1)^k \frac{x^k}{k!} \phi_n^{(k)}(x) f\left(\frac{k}{n}\right)} where x\in[0,b)\subset\mathbb{R} ( b can be \infty ), n\in\mathbb{N} , and (\phi_n)_{n\in\mathbb{N}} is a sequence of functions defined on [0,b] that have the following properties for all n,k\in\mathbb{N} : \phi_n\in\mathcal{C}^\infty[0,b] . Alternatively, \phi_n has a Taylor series on [0,b) . \phi_n(0) = 1 \phi_n is completely monotone, i.e. (-1)^k\phi_n^{(k)}\geq 0 . There is an integer c such that \phi_n^{(k+1)} = -n\phi_{n+c}^{(k)} 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

Footnotes

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