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Aldaz, J. M.
Preferred name
Aldaz, J. M.
Official Name
Aldaz, J. M.
Research Output
Now showing 1 - 3 of 3
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Publication
Shape preserving properties of generalized Bernstein operators on extended Chebyshev spaces
2009-12, Aldaz, J. M., Kounchev, Ognyan, Render, Hermann
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
]
.
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Publication
On real-analytic recurrence relations for cardinal exponential B-splines
2007-10, Aldaz, J. M., Kounchev, Ognyan, Render, Hermann
Let LN+1 be a linear differential operator of order N + 1 with constant coefficients
and real eigenvalues λ 1, ..., λ N+1, let E( N+1) be the space of all C∞-solutions of
LN+1 on the real line.We show that for N 2 and n = 2, ...,N, there is a recurrence
relation from suitable subspaces εn to εn+1 involving real-analytic functions, and
with εN+1 = E(Λ N+1) if and only if contiguous eigenvalues are equally spaced.
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Publication
Bernstein operators for exponential polynomials
2009-04, Aldaz, J. M., Kounchev, Ognyan, Render, Hermann
Let
L
be a linear differential operator with constant coefficients of order
n
and complex eigenvalues
λ
0
,...,λ
n
. Assume that the set
U
n
of all solutions of the
equation
Lf
= 0 is closed under complex conjugation. If the length of the interval [
a,b
]
is smaller than
Ï€/M
n
, where
M
n
:= max
{|
Im
λ
j
|
:
j
= 0
,...,n
}
, then there exists a basis
p
n,k
,
k
= 0
,...n
, of the space
U
n
with the property that each
p
n,k
has a zero of order
k
at
a
and a zero of order
n
−
k
at
b,
and each
p
n,k
is positive on the open interval (
a,b
)
.
Under the additional assumption that
λ
0
and
λ
1
are real and distinct, our first main
result states that there exist points
a
=
t
0