Shape preserving properties of generalized Bernstein operators on Extended Chebyshev spaces |
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Authors: | J M Aldaz O Kounchev and H Render |
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Institution: | (1) Laboratoire de Modélisation et Calcul (LMC-IMAG), Université Joseph Fourier, BP 53, F-38041 Grenoble Cedex, France |
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Abstract: | We study the existence and shape preserving properties of a generalized Bernstein operator B
n
fixing a strictly positive function f
0, and a second function f
1 such that f
1/f
0 is strictly increasing, within the framework of extended Chebyshev spaces U
n
. The first main result gives an inductive criterion for existence: suppose there exists a Bernstein operator B
n
: Ca, b] → U
n
with strictly increasing nodes, fixing f0, f1 ? Un{f_{0}, f_{1} \in U_{n}} . If Un ì Un + 1{U_{n} \subset U_{n + 1}} and U
n+1 has a non-negative Bernstein basis, then there exists a Bernstein operator B
n+1 : Ca, b] → U
n+1 with strictly increasing nodes, fixing f
0 and f
1. In particular, if f
0, f
1, . . . , f
n
is a basis of U
n
such that the linear span of f
0, . . . , f
k
is an extended Chebyshev space over a, b] for each k = 0, . . . , n, then there exists a Bernstein operator B
n
with increasing nodes fixing f
0 and f
1. The second main result says that under the above assumptions the following inequalities hold
Bn f 3 Bn+1 f 3 fB_{n} f \geq B_{n+1} f \geq f |
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Keywords: | |
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