Bernstein operators for exponential polynomials

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
π/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
<t
1
<...<t
n
=
b
and positive numbers
α
0
,..,α
n
, such that the operator
B
n
f
:=
n
X
k
=0
α
k
f
(
t
k
)
p
n,k
(
x
)
satisfies
B
n
e
λ
j
x
=
e
λ
j
x
, for
j
= 0
,
1
.
The second main result gives a sufficient condition
guaranteeing the uniform convergence of
B
n
f
to
f
for each
f
∈
C
[
a,b
].