Language:   Search:   Contact
Zentralblatt MATH has released its new interface!
For an improved author identification, see the new author database of ZBMATH.

Query:
Fill in the form and click »Search«...
Format:
Display: entries per page entries
Zbl 1228.41019
Cárdenas-Morales, D.; Garrancho, P.; Raşa, I.
Bernstein-type operators which preserve polynomials.
(English)
[J] Comput. Math. Appl. 62, No. 1, 158-163 (2011). ISSN 0898-1221

Summary: We present the sequence of linear Bernstein-type operators defined for $f\in C[0,1]$ by $B_{n}(f\deg \tau^{ - 1})\deg \tau$, $B_{n}$ being the classical Bernstein operators and $\tau$ being any function that is continuously differentiable $\infty$ times on $[0,1]$, such that $\tau (0)=0$, $\tau (1)=1$ and $\tau'(x)>0$ for $x\in [0,1]$. We investigate its shape preserving and convergence properties, as well as its asymptotic behavior and saturation. Moreover, these operators and others of King type are compared with each other and with $B_{n}$. We present as an interesting byproduct sequences of positive linear operators of polynomial type with nice geometric shape preserving properties, which converge to the identity, which in a certain sense improve $B_{n}$ in approximating a number of increasing functions, and which, apart from the constant functions, fix suitable polynomials of a prescribed degree. The notion of convexity with respect to $\tau$ plays an important role.
MSC 2000:
*41A36 Approximation by positive operators

Keywords: Bernstein-type operator; positive linear operator; Voronovskaja-type formula; generalized convexity

Highlights
Master Server