Shape preserving properties of generalized Bernstein operators on extended Chebyshev spaces

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
:
C
[
a,b
]
→
U
n
with strictly increasing nodes, fixing
f
0
,f
1
∈
U
n
.
If
U
n
⊂
U
n
+1
and
U
n
+1
has a non-negative Bernstein basis, then there exists a Bernstein
operator
B
n
+1
:
C
[
a,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
B
n
f
≥
B
n
+1
f
≥
f
for all (
f
0
,f
1
)-convex functions
f
∈
C
[
a,b
]
.
Furthermore,
B
n
f
is (
f
0
,f
1
)-convex for all
(
f
0
,f
1
)-convex functions
f
∈
C
[
a,b
]
.