Journal of Approximation Theory 162 (2010) 576598 [616072]

Journal of Approximation Theory 162 (2010) 576–598
www.elsevier.com/locate/jat
Multivariate Bernstein–Durrmeyer operators with
arbitrary weight functions
Elena E. Berdysheva, Kurt Jetter
Institut f ¨ur Angewandte Mathematik und Statistik, Universit ¨at Hohenheim, D-70593 Stuttgart, Germany
Received 12 March 2009; received in revised form 23 October 2009; accepted 16 November 2009
Available online 22 November 2009
Communicated by Martin Buhmann
Abstract
In this paper we introduce a class of Bernstein–Durrmeyer operators with respect to an arbitrary
measureon the d-dimensional simplex, and a class of more general polynomial integral operators
with a kernel function involving the Bernstein basis polynomials. These operators generalize the well-
known Bernstein–Durrmeyer operators with respect to Jacobi weights. We investigate properties of the
new operators. In particular, we study the associated reproducing kernel Hilbert space and show that the
Bernstein basis functions are orthogonal in the corresponding inner product. We discuss spectral properties
of the operators. We make first steps in understanding convergence of the operators.
c
2009 Elsevier Inc. All rights reserved.
Keywords: Bernstein basis polynomials; Bernstein–Durrmeyer operator; Jacobi weight; Reproducing kernel Hilbert
space; Korovkin type theorem
1. Introduction
The motivation of this paper comes from paper [ 11], in which D.-X. Zhou and the second
author have applied the univariate Bernstein–Durrmeyer operator to bias-variance estimates
as they are common in learning theory. Bernstein–Durrmeyer operators are well known in
approximation theory, and their properties have been studied in great detail for special weight
functions, namely, for Jacobi weights. In this paper we study these operators under more general
assumptions and from a somewhat different point of view.
Corresponding author.
E-mail address: [anonimizat] (E.E. Berdysheva).
0021-9045/$ – see front matter c
2009 Elsevier Inc. All rights reserved.
doi:10.1016/j.jat.2009.11.005

E.E. Berdysheva, K. Jetter / Journal of Approximation Theory 162 (2010) 576–598 577
Let us start with notation. The standard simplex in Rdis the set
SdVDn
xD.x1;:::; xd/2RdV0x1;:::; xd1;0x1C


C xd1o
:
Instead of Cartesian coordinates, we will often refer to barycentric coordinates, which we denote
by a boldface symbol
xD.x0;x1;:::; xd/; x0VD1x1


xd:
Throughout the paper we will use more or less standard multi-index notation such as xD
x0
0x1
1


xd
dorjjD0C


Cdforx;2RdC1. Note that injjthe sum of the components
(and not the sum of their absolute values) is taken. In particular, for x2Sd, the components of x
are all non-negative, and they add up to 1.
LetPd
ndenote the space of d-variate algebraic polynomials of (total) degree at most n. The
d-variate Bernstein basis polynomials of degree nare defined by
B.x/VDn

xVDnW
0W1W::: dWx0
0x1
1


xd
d;jjDn;
where the components of D.0;1;:::; d/are non-negative integers. All together there are
nCd
d
DdimPd
nlinear independent polynomials of degree nthat constitute a basis of Pd
n. This
basis is often preferred to the monomial basis in numerical calculations, since it is known to be
better conditioned. It is also used in computer aided geometric design for the design of curves
and surfaces, since a polynomial represented in this basis can be fast and stably evaluated in a
recursive way by the famous de Casteljau algorithm.
In approximation theory, the Bernstein polynomial basis is used in various types of positive
polynomial approximation schemes. One of them is the famous Bernstein polynomial operator
of degree n,
BnVC.Sd/!Pd
n; f7!X
jjDnf
n
B;
which maps continuous functions on the simplex onto Pd
n. This positive linear operator
reproduces constant functions and linear polynomials, i.e., we have
X
jjDnB.x/D1; (1.1)
and
X
jjDni
nB.x/Dxi;iD0;:::; d: (1.2)
In this paper we study Bernstein–Durrmeyer polynomial operators and more general
polynomial integral operators for general weight functions. Let be a non-negative and bounded
(regular) Borel measure on Sdsatisfying the positivity assumption :
Assumption P. Ifpis a polynomial which is non-negative on Sdand is not identically zero, thenR
Sdpd> 0.
Without loss of generality, we will also assume that the measure is normalized to be a
probability measure:

578 E.E. Berdysheva, K. Jetter / Journal of Approximation Theory 162 (2010) 576–598
Assumption N. The measure satisfiesR
SddD1:
The spaces Lq
.Sd/, 1q<1, are defined in a usual way as spaces of (equivalence classes)
of real-valued functions f, for whichjfjqis integrable with respect to , with the norm
kfkLq
.Sd/VDZ
Sdjf.x/jqd.x/1=q
;1q<1:
The space L1
.Sd/is the space of -essentially bounded functions with the norm
kfkL1.Sd/VDess sup
x2Sdjf.x/j:
We shall also consider the space C.Sd/of continuous, real-valued functions on the simplex, with
the norm
kfkC.Sd/VDmax
x2Sdjf.x/j:
Let1denote the constant function equal to 1, i.e.,
1Vx7!1.x/1;x2Sd:
Due to Assumption N ,k1kLq
.Sd/D1 for all q. Note that C.Sd/L1
.Sd/andLq
.Sd/L1
.Sd/
for all 1q1 . On the space L2
.Sd/we consider the (semi-)inner product
hfjgiVDZ
Sdf.x/g.x/d.x/: (1.3)
Assumption P guarantees thathpjpi>0 for any polynomial pwhich is not identically zero.
Definition 1.1. The-weighted Bernstein–Durrmeyer operator of degree n on the simplex Sdis
the operator
Mn;VL1
.Sd/!Pd
n; f7!X
jjDnhfjBi
h1jBiB: (1.4)
As far as we know, this operator in this generality has never been studied in the literature.
For applications in learning theory, however, it is important to consider this general setting,
although the measure will refer to an unknown probability distribution. This generalization
was shortly mentioned by Berens and Xu [ 5] but not further investigated. On the other hand,
the Bernstein–Durrmeyer operator (1.4) for the unweighted case (i.e., d .x/Ddxup to
normalization) and, more general, for Jacobi weights (see (1.6) ) is well known and very
well studied. It was introduced in the one-dimensional unweighted case by Durrmeyer and,
independently, by Lupas ¸ and first studied by Derriennic [ 7]. It was extended to Jacobi weights
by P ˘alt˘anea [ 12] and studied by Berens and Xu [ 5]. The multidimensional case was dealt,
e.g., in [ 8,9]. For details and further references, see the papers mentioned above and [ 3].
In particular, Ditzian’s paper [ 9] gives a good overview over properties of the multivariate
Bernstein–Durrmeyer operator with Jacobi weights. In this special case, the weighted integrals of

E.E. Berdysheva, K. Jetter / Journal of Approximation Theory 162 (2010) 576–598 579
the Bernstein basis polynomials are explicitly known as values of the multivariate beta function
B./VDZ
SdxedxDdQ
iD0.i/
.jj/; (1.5)
where2RdC1
CandeD.1;1;:::; 1/. The normalized Jacobi weight is the measure
d.x/D1
B.Ce/xdx (1.6)
withD.0;1;:::; d/2RdC1satisfying the integrability condition >e. In this case,
we will denote the inner product (1.3) by
hfjgiD1
B.Ce/Z
Sdf.x/g.x/xdx: (1.7)
Due to (1.5) ,
h1jBiD1
B.Ce/Z
SdB.x/xdxDn
B.CCe/
B.Ce/; (1.8)
and the Jacobi-weighted Bernstein–Durrmeyer operator is given by
Mn;.f/DX
jjDnc;.f/B;c;.f/D1
B.CCe/Z
Sdf.x/xCdx: (1.9)
A more general operator than the Bernstein–Durrmeyer operator with Jacobi weights
was considered by P ˘alt˘anea. In his book [ 13, Section 5.2] he studied (univariate)
Bernstein–Durrmeyer operators of the form (1.4) with the weight (up to normalization)
d.x/Dx.1x/h.x/dx;x2T0;1U; (1.10)
where h2CT0;1U,h.t/ >0 for all t2T0;1U,; >1. Such weights are sometimes called
generalized Jacobi weights .
2. The kernel functions
Definition 2.1. For a given sequence !D.!/jjDnof non-negative numbers, the !-weighted
Bernstein basis kernel of degree n is defined as the function
Tn;!.x;y/DX
jjDn!B.x/B.y/;x;y2Sd:
It gives rise to a family of -weighted integral operators

Ln;!;.f/

.x/DZ
SdTn;!.x;y/f.y/d.y/: (2.1)
It is evident that these linear operators are positive, i.e.,
f0H) Ln;!;.f/0;

580 E.E. Berdysheva, K. Jetter / Journal of Approximation Theory 162 (2010) 576–598
and symmetric, i.e.,

Ln;!;.f/jgiDhfjLn;!;.g/
;f;g2L2
.Sd/: (2.2)
The-weighted Bernstein–Durrmeyer operator refers to the special case where the !-weights
are the canonical weights chosen according to the formula
1
!Dh1jBiDZ
SdB.x/d.x/>0;jjDnI (2.3)
the integral is positive due to Assumption P . This is the choice of weights which enforces the
operator to reproduce constant functions, that is,
Mn;.1/DX
jjDnBD1; (2.4)
see(1.1) . The action of the operator on polynomial spaces will be considered in the following
sections. In this section, we deal with an interpretation of the !-weighted Bernstein basis kernels
as reproducing kernels in an associated polynomial inner product space.
2.1. Properties of the kernel functions
Lemma 2.2. The kernel Tn;!is non-negative, symmetric, and continuous on SdSd, with bound
max
x;y2SdTn;!.x;y/max
jjDn!DV
n: (2.5)
Moreover, the kernel is a positive (semi-)definite function on Sd.
Proof. The first statements are clear since
0Tn;!.x;y/
nX
jjDnB.x/D
n
forx;y2Sd. The positive definiteness follows from
NX
i;jD1cicjTn;!.xi;xj/DX
jjDn!(NX
iD1ciB.xi/)2
0
for all points x1;:::; xN2Sdand all real numbers .ci/N
iD1.
Kernels of this type are called Mercer Kernels . Their properties are used in a standard
construction of an associated reproducing kernel Hilbert space (RKHS) that will be considered
below, cf. [ 6, Section 3].
Remark. Estimate (2.5) is sharp under the following condition. Let
VdVDfeiViD0;:::; dg
denote the set of vertices of the simplex Sd, in barycentric notation. So, all the coordinates of ei
are zero except at position i, where the entry is 1. Since
B.ei/D;neiforjjDnandiD0;:::; d;

E.E. Berdysheva, K. Jetter / Journal of Approximation Theory 162 (2010) 576–598 581
we find
max
x;y2SdTn;!.x;y/max
iD0;:::;dTn;!.ei;ei/Dmax
iD0;:::;d!nei:
This tells that the estimate is sharp whenever max jjDn!is attained at 2nVd. An example
will be given in Section 3.2.
2.2. The reproducing kernel Hilbert space
Consider the span of the functions
KxVDTn;!.x;
/;x2Sd;
and the inner product
*NX
iD1ciKxiMX
jD1djKyj+
VDNX
iD1MX
jD1ciTn;!.xi;yj/dj: (2.6)
The span is Pd
n, and we arrive at the space of algebraic polynomials of degree at most n, imposed
with an inner product structure such that the kernel Tn;!is a reproducing kernel. Note that since
in our case the span is finite dimensional, we do not have to perform the completion process
which is usual in such constructions.
Definition 2.3. ByHn;!we denote the space Pd
nequipped with the inner product (2.6) —which
we denote byh
j
iHn;!—and with the corresponding norm kfkHn;!VDphfjfiHn;!.
The reproducing kernel property refers to the simple identity
f.x/DhfjKxiHn;!DhfjTn;!.x;
/iHn;!
forf2Pd
nandx2Sd.
Theorem 2.4. The space Hn;!is compactly embedded into C .Sd/, with bound
kfkC.Sd/p

nkfkHn;!:
Proof. The embedding property is clear from the fact that the space Hn;!has finite dimension.
More interesting is the explicit expression for the bound. For reader’s convenience, we repeat the
proof given in [ 6]. We have
jf.x/jDhfjKxiHn;!kfkHn;!
kKxkHn;!DkfkHn;!
p
Tn;!.x;x/
forx2Sd, and the result follows from Lemma 2.2 .
2.3. Orthogonality of the Bernstein basis polynomials
The next result is our main result in this section. It shows that the new structure of the
polynomial space results in an orthogonality property for the Bernstein basis functions.

582 E.E. Berdysheva, K. Jetter / Journal of Approximation Theory 162 (2010) 576–598
Theorem 2.5. If!>0for allwithjjDn, then the normalized Bernstein basis polynomials
eBVDp!B;jjDn;
are orthonormal in Hn;!, i.e.,
heBjeBiHn;!D; forjjDjjDn:
Proof. We consider the collocation matrix
MDeB.x
/

jjDj
jDn
for the discrete simplicial set of nodes x
D1
n
, with
2NdC1
0andj
jDn. Here,refers
to columns and
refers to rows of the matrix, for some chosen ordering of the multi-indices
(e.g., lexicographic). The matrix Mis regular, since
MD
B.x
/

jjDj
jDnD!;
where the right-hand factor is the diagonal matrix with diagonal entriesp!,jjDn, and the
left-hand factor is the collocation matrix using the Bernstein basis functions of degree nand the
discrete simplicial nodes of order nas interpolation points. It is well known that this collocation
matrix is regular.
Denote
eB.y/VD0
BB@:::
eB.y/
:::1
CCA
jjDnand C.y/VD0
BB@:::
Kx
.y/
:::1
CCA
j
jDn:
The basis transformation formula C.y/DMeB.y/yields the identity

Tn;!.x
;x
0/
j
jDj
0jDnD



C.x
0/



j
0jDn
DM



eB.x
0/



j
0jDnDM MT:
SinceeB.y/DM1C.y/, we conclude (with MTVD.M1/T):

heBjeBiHn;!
jjDjjDnDheBjeBTiHn;!
DM1hCjCTiHn;!MT
DM1
Tn;!.x
;x
0/
j
jDj
0jDnMT
DM1MMTMTDI:
This shows that the Gramian of the normalized Bernstein basis polynomials with respect to the
inner product in Hn;!is the identity matrix I.
In terms of the (not normalized) Bernstein basis polynomials, the orthogonality property reads
as follows:

BjB
Hn;!D;1
!forjjDjjDn: (2.7)

E.E. Berdysheva, K. Jetter / Journal of Approximation Theory 162 (2010) 576–598 583
Therefore,
hpjqiHn;!DX
jjDncd
!
forpDP
jjDncBandqDP
jjDndB.
This shows an interesting connection to the inner product used in Theorem 1 and Corollary
1 of [ 2], see also [ 10] for the unweighted case. Denote Di jVD@
@xj@
@xifor 1i<jd,
D0jVD@
@xjfor 1jd, and DkVDQ
0i<jdDki j
i j, where ki j2N0andkD.ki j/0i<jd.
Further, denote XkVDQ
0i<jd.xixj/ki j. It was proved in [ 10] that
X
jkjn.njkj/W
nWkWXkDkB.x/DkB.x/D;B.x/
forjjDjjDnandx2Sd. Integrating this identity with respect to the measure , we obtain
X
jkjn.njkj/W
nWkWZ
SdXkDkB.x/DkB.x/d.x/D;h1jBi:
Comparing this with (2.7) , we arrive at the following statement.
Corollary 2.6. Let!be the canonical weights as defined in (2.3) . Then for all polynomials
p;q2Pd
nwe have
hpjqiHn;!DX
jkjn.njkj/W
nWkWZ
SdXkDkp.x/Dkq.x/d.x/:
If d.x/Dw.x/dxwith a smooth weight function wwhich satisfies certain conditions on the
boundary of the simplex Sd, one can integrate the last identity by parts, as we did in Corollary 1
of [2] for the case of Jacobi weights. We omit the details. This identification of the inner product
shows thath
j
iHn;!can be extended to spaces of smooth functions using appropriate differential
operators. This gives a hint to a possible construction of a class of ‘Sobolev type’ spaces on the
simplex.
3. The integral operators
Given the!-weighted Bernstein basis kernel of Definition 2.1 , we consider the corresponding
integral operator Ln;!;defined in (2.1) as a linear operator mapping functions f2Lq
.Sd/onto
polynomials p2Pd
n. Since the kernel is positive and continuous, we have
kLn;!;kL1.Sd/!L1.Sd/
D sup
f2L1.Sd/;kfkL1.Sd/D1Z
SdZ
SdTn;!;.x;y/f.y/d.y/d.x/
D sup
f2L1.Sd/;kfkL1.Sd/D1Z
SdZ
SdTn;!;.x;y/jf.y/jd.y/d.x/

584 E.E. Berdysheva, K. Jetter / Journal of Approximation Theory 162 (2010) 576–598
D sup
f2L1.Sd/;kfkL1.Sd/D1Z
Sd X
jjDn!Z
SdB.x/d.x/B.y/!
jf.y/jd.y/
n;!;VDmax
jjDn!Z
SdB.x/d.x/: (3.1)
On the other hand,
kLn;!;kL1.Sd/!L1.Sd/D sup
f2L1.Sd/;kfkL1.Sd/D1Z
SdTn;!;.x;y/f.y/d.y/
Z
SdX
jjDn!B.x/B.y/d.y/
n;!;:
Using the Riesz–Thorin interpolation theorem and taking into account (2.4) , we find
Lemma 3.1. For general!-weights and 1q 1 , the operators (2.1) are bounded as
operators from Lq
.Sd/into Lq
.Sd/, and
kLn;!;kLq
.Sd/!Lq
.Sd/n;!;:
If!are the canonical weights as in (2.3) , we have
kMn;kLq
.Sd/!Lq
.Sd/D1: (3.2)
3.1. Spectral properties of the integral operator
In this section we study spectral properties of the operator
Ln;!;VL2
.Sd/!Hn;!: (3.3)
Here both the domain and the range of the operator carry an inner product structure with inner
products (1.3) and(2.6) , respectively. Most results of this section follow from those of papers
[6,14], where the general theory of compact integral operators was applied to similar problems
in a more general context. However, since our operators have finite rank, one can proceed more
directly using elementary linear algebra, and we are going to do so. In particular, we will give
full proofs.
In this section, we assume that all weights !,jjD n, are strictly positive. We use the
notationeBfor the vector of the normalized Bernstein basis polynomials of degree n, as in the
proof of Theorem 2.5 . The kernel of the integral operator can be written as
Tn;!.x;y/DeBT.x/eB.y/:
From this we obtain the following representation of the operator:
Ln;!;.f/DeBTheBjfi: (3.4)
The right-hand factor here is a column vector where the inner product is taken componentwise.
The action of the operator on L2
.Sd/can now be easily understood. The orthogonal complement
.Pd
n/?ofPd
nwith respect to the -product (1.3) is in the kernel of the operator, and the action
onPd
nwill be described by specializing fto a polynomial pDeBTc, where cis a vector of

E.E. Berdysheva, K. Jetter / Journal of Approximation Theory 162 (2010) 576–598 585
coefficients. Note that, due to the positivity of the weights !, the components of eBform a basis
ofPd
n. Now
Ln;!;.eBTc/DeBTG c; (3.5)
where
GD
heBjeBi

jjDjjDn(3.6)
is the Gramian of the normalized Bernstein basis polynomials with respect to the -product. The
spectral decomposition of the Gramian can be used to characterize the spectral properties of the
integral operator. Here, the following well-known result will be useful.
Lemma 3.2. For p 1;:::; pN2L2
.Sd/, letGk, kD1;:::; N, denote the Gramian of the system
fp1;:::; pkg. Then
detGkC1DE2
k;.pkC1/
detGk;kD1;:::; N1;
where
E2
k;.pkC1/VDmin
p2PkkpkC1pk2
L2.Sd/;PkVDspanfp1;:::; pkg:
In particular, iffp1;:::; pNgis a linearly independent system of polynomials, then all these
determinants are strictly positive.
Proof. For the sake of completeness, we give a proof. By the normal equations, the best
approximation pDPk
`D1a`;kp`2Pkto the function pkC1is characterized by the fact
that pkC1pis orthogonal to Pkwith respect to the inner product (1.3) . If we subtract the
a`;k-multiple of the `th column, and then the same multiple of the `th row,`D1;:::; k,
from the last column (the last row, respectively) of GkC1, the non-diagonal entries in the last
column (and the last row) will vanish, except for the diagonal entry which becomes kpkC1Pk
`D1a`;kp`k2
L2.Sd/DE2
k;.pkC1/. From this the recurrence formula for the determinants is
clear.
The additional statement follows from the positivity assumption. We have det G1D
kp1k2
L2.Sd/>0 and E2
k;>0,kD1;:::; N1.
The Gramian (3.6) is regular, hence positive definite, if the weights !are strictly positive.
We choose a set of eigenpairs for G,
.
;v
/;j
jDn;
with (real) eigenvalues

>0;j
jDn;
and eigenvectors v
2RN,ND
nCd
d
, which we assume to be orthonormalized, i.e.,
vT

v
0D
;
0;j
jDj
0jDn:
The polynomials
p
DeBTv
;j
jDn; (3.7)
then provide a system of eigenfunctions for our integral operator.

586 E.E. Berdysheva, K. Jetter / Journal of Approximation Theory 162 (2010) 576–598
Theorem 3.3. Let!>0for allwithjjDn. Then the polynomials (3.7) are eigenfunctions
of the integral operator (2.1) satisfying
Ln;!;.p
/D
p
;j
jDn;
and
hp
jp
0iD

;
0;j
jDj
0jDn:
Moreover, the restriction of the operator to Pd
nis an isomorphism.
The map
eB
$ep
VD1p
p
;j
jDn;
provides an isometry between the two inner product spaces
Pd
n;h
j
iHn;!

and
Pd
n;h
j
i

,
since
;DheBjeBiHn;!Dhepjepi;jjDjjDn:
Next we prove one further property of the integral operator.
Corollary 3.4. If!>0for allwithjjDn, then
hLn;!;.f/jpiHn;!Dhfjpi
for f2L2
.Sd/and p2Pd
n.
Proof. With pDeBTcDP
jjDnceB, formula (3.4) yields

Ln;!;.f/jp
Hn;!D
eBTheBjfijeBTc
Hn;!
DP
jjDnP
jjDnheBjfiheBjeBiHn;!c
DP
jjDncheBjfiDhpjfi;
due to the orthogonality of the normalized Bernstein basis polynomials. 
Next we discuss the norm of operator Ln;!;as an operator from L2
.Sd/intoHn;!.
Theorem 3.5. If!>0for allwithjjDn, then the operator (3.3) has the norm
kLn;!;kL2.Sd/!Hn;!Dp
.G/;
where.G/VDmaxj
jDn
is the spectral radius of the Gramian (3.6) .
Proof. We have, with pDeBTc2Pd
n, where c2RN,ND
nCd
d
,
kLn;!;k2
L2.Sd/!Hn;!Dmax
06Dp2PdnkLn;!;.p/k2
Hn;!
kpk2
L2.Sd/Dmax
06Dc2RNcTG2c
cTGc
using (3.5) and(3.6) . The latter Rayleigh quotient is bounded by the spectral radius of the
Gramian, and this bound is sharp. 

E.E. Berdysheva, K. Jetter / Journal of Approximation Theory 162 (2010) 576–598 587
In the case of the canonical weights we have .G/D1, since the corresponding -weighted
Bernstein–Durrmeyer operator is contractive and reproduces constant functions; see (3.2) and
(2.4) , respectively. Thus, we obtain
Corollary 3.6. For the-weighted Bernstein–Durrmeyer operator we have
kMn;kL2.Sd/!Hn;!D1:
In the general setting we arrive at the difficult problem of finding or estimating the spectrum
ofG. However, an easy estimate can be derived as follows. Gis a positive definite matrix, with
trace
trace GDX
jjDn!Z
SdB2
.x/d.x/DZ
SdTn;!.x;x/d.x/:
ByLemma 2.2 we thus obtain
Corollary 3.7. The operator norm in Theorem 3.5is bounded by
kLn;!;kL2.Sd/!Hn;!p

n;
with
nas in (2.5) .
Remark. The considerations of this section can be also carried out in the case where some
of the weights !vanish. In Theorem 3.3 , one has to decompose L2
.Sd/into the space
spanfeBVjjD ngand its orthogonal complement, which is the kernel of the operator. In
Theorem 3.5 , we replace NbyN0Ddim spanfeBVjjD ngandGbyG0, the Gramian of
non-vanishing eB’s. In this case, we have .G/D.G0/.
3.2. The Jacobi-weighted Bernstein–Durrmeyer operator
In this subsection we consider the Bernstein–Durrmeyer operator (1.9) with respect to the
Jacobi weight (1.6) . Due to the special structure of the weights !, the operator Mn;has
properties that are not valid in the general setting.
Recall that Mn;is a positive polynomial operator on L1
.Sd/, with norm
kMn;kLq
.Sd/!Lq
.Sd/D1;1q1;
which reproduces constant functions.
3.2.1. Sharpness of estimate (2.5)
Estimate (2.5) for the kernel of the operator Mn;is sharp whenever the exponents iin(1.6)
are all non-negative, except possibly one. This follows from the following
Lemma 3.8. If!are the canonical weights as in (2.3) with respect to the Jacobi weight (1.6) ,
and if the exponents iin(1.6) are all non-negative, except at most one, then maxjjDn!is
attained at2nVd.
Proof. According to (1.8) ,
!1
Dh1jBiD1
B.Ce/nW
.nCjjCdC1/dY
iD0.iCiC1/
iW:

588 E.E. Berdysheva, K. Jetter / Journal of Approximation Theory 162 (2010) 576–598
Denote

VDdY
iD0.iCiC1/
iWD
ndP
iD1iC0C1

ndP
iD1i
WdY
iD1.iCiC1/
iW:
We need to show that, under the assumptions of the lemma, min jjDn
is attained at 2nVd.
First we prove that min jjDn
is attained on the boundary of the simplex nSd. We switch to
the Cartesian coordinates D.1;:::; d/. Let j2f1;:::; dg. For the forward difference of

in the direction ejwe have

j
VD
Cej

D0
BBB@jCjC1
jC1ndP
iD1iC0
ndP
iD1i1
CCCAdY
iD1.iCiC1/
iW
ndP
iD1iC0

ndP
iD1i1
W
D
j
ndX
iD1i!
0.jC1/!dY
iD1.iCiC1/
iW
ndP
iD1iC0
.jC1/
ndP
iD1i
W:
We see that each
j
is a linear function of times a positive term. Thus,
changes the
monotonicity at most once in each variable. We first consider the case when jjD0. Then there
is an index jwith 1jdsuch thatj00. But then
j
has constant sign. Thus,
is
monotone with respect to jand cannot have a local minimum. Consequently, the minimum is
attained on the boundary of nSd.
Now supposejj6D0. Consider the system of linear equations
j
ndX
iD1i!
0.jC1/D0;jD1;:::; d:
Ifjj6D0, this system has the unique solution D.1;:::;d/withjDj
jj.nCd/1,
jD1;:::; d. This point provides a local minimum only if the coefficients of jin
j
, which
are equal toj0, are positive for every jD1;:::; d, i.e., if
jC0<0;jD1;:::; d: (3.8)
The minimum is attained strictly inside the simplex only if j>0,jD1;:::; d, andPd
iD1j<n1, which is equivalent to
j
jj.nCd/>1;jD0;:::; d: (3.9)
It is not difficult to see that the latter is only possible if all the exponents j,jD0;:::; dare
negative. Indeed, if we suppose that jj>0, we get from (3.9) thatj>jj
nCd>0,jD0;:::; d,
which contradicts (3.8) . Thus,jj<0, but then we get from (3.9) thatj<jj
nCd<0,
jD0;:::; d, as desired. At this point we have shown that min jjDn
is attained at the boundary

E.E. Berdysheva, K. Jetter / Journal of Approximation Theory 162 (2010) 576–598 589
ofnSdwhenever at least one of the exponents i, 0id, is non-negative. On the other hand,
if the exponents i, 0id, are all negative, it follows from (3.9) that minjjDn
will be
attained strictly inside the simplex nSdforn>max jD0;:::;djj
id, so that the assumption about
negativity of the exponents cannot be relaxed for all n.
Now let us consider a face of dimension d1, on which the minimum is attained. Without loss
of generality, we may assume that this is the face with the equation dD0. Repeating the above
considerations for the simplex nSd1with the barycentric coordinates eD.0;:::; d1/and
the weighteD.0;:::; d1/, we see that the minimum is attained on the boundary of nSd1
whenever at least one of the exponents 0;:::; d1is non-negative. Repeating the process, we
obtain the statement of the lemma. 
Remark. The assumptions of the lemma cannot be relaxed. As an example, take dD2,nD2,
0D1D 1=2, and2D1=2. Then
.0;0;2/D15
163
2,
.1;0;1/D
.0;1;1/D3
83
2,
.2;0;0/D

.0;2;0/D3
163
2, and
.1;1;0/D1
83
2. Thus, minjjDn
is attained at D.1;1;0/622V2.
3.2.2. Orthogonal polynomials as eigenfunctions
The Bernstein–Durrmeyer operator Mn;has remarkable spectral properties. Because of their
importance, we are going to discuss them in detail. All results of this section, except for the
remark below, are well known, e.g., [ 7,8,5,9].
An important property of the Bernstein–Durrmeyer operator with respect to Jacobi weights
is that this operator is degree preserving . This means that, for a monomial '.x/VDxwith
jjDmn, we have
Mn;.'/DC'Cpm1; (3.10)
where pm12Pd
m1andC6D0 is a constant. In particular,
Mn;.Pd
m/Pd
m;mD0;1;:::; n: (3.11)
One can see this, e.g., as follows. A direct computation shows that the monomials '.x/,
jjDmn, are mapped by Mn;onto the polynomials
Mn;.'/DX
jjDnc;B
with
c;DB.CCCe/
B.CCe/D.nCjjCdC1/
.nCmCjjCdC1/dY
iD0.iCiCiC1/
.iCiC1/:
Thus,
c;Dq./;jjDn;jjDmn;
where qis a polynomial in of coordinate degree . It follows by a well-known property
of Bernstein–B ´ezier coefficients of a polynomial, cf. Proposition 2.1 in [ 15] and its proof, that
Mn;.'/is a polynomial of coordinate degree , which proves (3.10) .
Let
E0;VDPd
0I Em;VDPd
m\.Pd
m1/?;m>0;

590 E.E. Berdysheva, K. Jetter / Journal of Approximation Theory 162 (2010) 576–598
denote the spaces of polynomials orthogonal with respect to the inner product (1.7) . It follows
from the property (3.11) and the symmetry property (2.2) that the kernel of Mn;is.En;/?, and
that the spaces Em;,mD0;:::; n, are eigenspaces of the operator Mn;. The corresponding
eigenvalues are (e.g., [ 9])
n;m;Dn
m


nCjjCdCm
m;0mn: (3.12)
The detailed knowledge of spectral properties of the operators was extensively used for
studying approximation properties of Mn;like, e.g., in [ 5] or [ 3, Theorem 7]. It allows us also
to elaborate on results of Section 3.1. Thus, the spectrum of the Gramian (3.6) is given by the
values (3.12) , wheren;m;has multiplicity 1 for mD0 and multiplicity
dimEm;DmCd
m
m1Cd
m1
DmCd1
m
formD1;:::; n.
Remark. Neither the degree preservation property nor the fact that the spaces of orthogonal
polynomials are eigenspaces of the operator are valid for the Bernstein–Durrmeyer operator in
the general setting (1.4) . As an example, consider S2with the measure
d.x/D.x1Cx2/dx: (3.13)
Then, for example, for 'e1.x/Dx1we have
M2;.'e1/D1
72×2
1C1
72×2
2C7
18×11
18x2C1
4:
The eigenvalues and the eigenfunction of M2;are:
1D1 p1D1
2D61Cp
2041
252p2D.x1x2/
7.x1Cx2/51p
2041
3D1
3p3D4x1C4x23
4D1
14p4Dx2
1Cx2
24x1x2
5D61p
2041
252p5D.x1x2/
7.x1Cx2/51Cp
2041
6D1
21p6D15.x1Cx2/220.x1Cx2/C6:
A direct calculation shows that the polynomials p2and p5are not orthogonal polynomials,
i.e., they are not orthogonal to all polynomials of lower degree.
In the case of Jacobi weights, it follows from the spectral properties of the Bernstein–
Durrmeyer operator that if pis an eigenfunction of Mn;then pis an eigenfunction of MnC1;
as well. This property fails to hold in the general setting (1.4) , too. As an example, consider again
the measure defined in (3.13) . Then the polynomial pDx1x2is an eigenfunction of M1;but
not of M2;.

E.E. Berdysheva, K. Jetter / Journal of Approximation Theory 162 (2010) 576–598 591
4. Convergence
4.1. Convergence for test functions and Korovkin type theorems
We come back to the general case of the operator Ln;!;under Assumptions P andN. The
-weighted Bernstein–Durrmeyer operator (1.4) reproduces constant functions, see (2.4) . The
integral operators (2.1) in general do not reproduce constant functions. We have
1Ln;!;.1/DX
jjDn
1!Z
SdB.x/d.x/
B;
and thus the condition
limn!1max
jjDn!Z
SdB.x/d.x/1D0
implies the convergence
limn!1k1Ln;!;.1/kC.Sd/D0:
Next we give a sufficient condition for convergence in C.Sd/for linear functions. Let ei2Vd.
We consider the linear test functions
'ei.x/DxeiDxi; iD1;:::; d;
1x1


xd;iD0:(4.1)
ForiD0;:::; dandjjDndefine
b.i/
;!;VD!Z
SdxeiB.x/d.x/: (4.2)
We will denote these coefficients by b.i/
;in the case of the -weighted Bernstein–Durrmeyer
operator, and by b.i/
;in the case of the Bernstein–Durrmeyer operator with the Jacobi weight
(1.6) .
Lemma 4.1. Let0id. The condition
limn!1max
jjDnb.i/
;!;i
nD0 (4.3)
implies
limn!1k'eiLn;!;.'ei/kC.Sd/D0: (4.4)
The proof is clear since, by (1.2) ,
'eiLn;!;.'ei/DX
jjDni
nb.i/
;!;
B;iD0;:::; d: (4.5)
Remark. In the case of the canonical weights, the coefficients b.i/
;are given by
b.i/
;DR
SdxeiB.x/d.x/R
SdB.x/d.x/:

592 E.E. Berdysheva, K. Jetter / Journal of Approximation Theory 162 (2010) 576–598
This allows an interesting interpretation of these coefficients as the expectation of the random
variable Xiwith respect to the probability measureB.x/d.x/R
SdB.x/d.x/supported on the simplex. Our
assumption relates these measures to the Dirac measures supported at points1
n
,j
jDn.
Condition (4.3) is satisfied, for example, in the case of the Jacobi-weighted Bernstein–
Durrmeyer operators (1.9) . Indeed,
b.i/
;DR
SdxeiB.x/xdxR
SdB.x/xdxDB.CCeiCe/
B.CCe/DiCiC1
nCjjCdC1; (4.6)
and thus
b.i/
;i
nDO1
n
uniformly in: (4.7)
We will give a generalization in the next section. On the other hand, it is not difficult to construct
an example of an integral operator, even with the canonical weights, for which convergence (4.4)
fails. For simplicity, we consider the one-dimensional case, i.e., dD1,S1DT0;1U. Take a
number a2.0;1/, and consider the measure d .x/Dw.x/dxwith
w.x/D8
<
:1
a;0xa;
0;a<x1:
The corresponding Bernstein–Durrmeyer operator has the form

Mn;.f/

.y/DnX
kD0Ra
0f.x/xk.1x/nkdxRa
0xk.1x/nkdxn
k
yk.1y/nk;y2T0;1U:
Obviously, for the linear function 'e1.x/Dxwe have
Mn;.'e1/

.y/a,y2T0;1U, so that
convergence (4.4) cannot hold.
The example above shows, in particular, that C.Sd/is not the right space for studying
convergence of the -weighted Bernstein–Durrmeyer operators, and one should restrict the
consideration to the support of the measure . Numerical experiments show that convergence
holds at least inside the support of for a large class of measures. An interesting question would
be to determine conditions on the measure which guarantee convergence
limn!1kfMn;.f/kL1.Sd/D0
for each f2L1
.Sd/.
For the Bernstein–Durrmeyer operator (1.4) , a simple test for the convergence in L1
.Sd/is
given by the following Korovkin type theorem.
Theorem 4.2. If condition (4.3) is fulfilled for each i D0;1;:::; d, then
limn!1kfMn;.f/kL1.Sd/D0
for each f2L1
.Sd/.
Proof. Condition (4.3) imply convergence for the linear monomials 'ei,iD1;:::; d, and for the
linear function 'e0D1Pd
iD1'ei. Thus, convergence holds for the constant function 1as well.

E.E. Berdysheva, K. Jetter / Journal of Approximation Theory 162 (2010) 576–598 593
In addition, the operators Mn;are contractions; see (3.2) . Convergence for each f2L1
.Sd/
follows from a Korovkin type theorem of Berens and Lorentz for linear contractions on the space
L1; see Section 5 of [ 4].
To establish convergence in C.Sd/, one has to consider, in addition, quadratic functions
'2ei.x/Dx2eiDx2
i,iD1;:::; d. Taking into account that
'2eiDX
jjDni.i1/
n.n1/B;
we obtain the condition
limn!1max
jjDn!Z
Sdx2eiB.x/d.x/i.i1/
n.n1/D0: (4.8)
Now we can formulate the result for the general integral operators (2.1) .
Theorem 4.3. If condition (4.3) is fulfilled for each i D0;1;:::; d, and condition (4.8) is
fulfilled for each iD1;:::; d, then
limn!1kfLn;!;.f/kC.Sd/D0
for each f2C.Sd/.
4.2. A class of Jacobi-like measures
In this section we restrict our considerations to the Bernstein–Durrmeyer operators (1.4) . We
describe a class of measures with the property that Theorem 4.3 is valid for the corresponding
-weighted Bernstein–Durrmeyer operator. This class generalizes the Jacobi weights and the
class (1.10) considered by P ˘alt˘anea.
Theorem 4.4. Suppose that the measure has the form
d.x/Dw.x/dx;
and suppose that there are Jacobi exponents  >eand constants 0<a;A<1such
that
ax
B.Ce/w.x/Ax
B.Ce/;x2Sd: (4.9)
Ifjjjj<1, then (4.3) is fulfilled for each i D0;1;:::; d,(4.8) is fulfilled for each
iD1;:::; d, and, moreover,
k'eiMn;.'ei/kC.Sd/DO
n1.jjjj/
2
;n!1; (4.10)
k'2eiMn;.'2ei/kC.Sd/DO
n1.jjjj/
2
;n!1: (4.11)
In particular,
limn!1kfMn;.f/kC.Sd/D0 (4.12)
for each f2C.Sd/.

594 E.E. Berdysheva, K. Jetter / Journal of Approximation Theory 162 (2010) 576–598
Proof. First we prove convergence for the linear test functions (4.1) . The coefficients (4.2) in our
case have the form
b.i/
;DR
SdxCeid.x/R
Sdxd.x/:
We will compare them with the coefficients b.i/
;for the Jacobi-weighted Bernstein-Durrmeyer
operator with the weightx
B.Ce/, which were calculated in (4.6) . We have
b.i/
;b.i/
;DR
Sd
xiiCiC1
nCjjCdC1
xd.x/
R
Sdxd.x/;
and, using the Cauchy–Schwarz inequality and (4.9) ,

b.i/
;b.i/
;2
R
Sd
xiiCiC1
nCjjCdC12
xd.x/
R
Sdxd.x/
A
aB.Ce/
B.Ce/R
Sd
xiiCiC1
nCjjCdC12
xCdx
R
SdxCdx
DA
aB.Ce/
B.Ce/B.CCe/
B.CCe/R
Sd
xiiCiC1
nCjjCdC12
xCdx
R
SdxCdx:(4.13)
We will estimate the order of (4.13) asn!1 . For the factorB.CCe/
B.CCe/we have
B.CCe/
B.CCe/D.nCjjCdC1/
.nCjjCdC1/dY
iD0.iCiC1/
.iCiC1/
njjjjdY
iD01
.iC1/ii;
the latter follows from the formula lim n!1nba.nCa/
.nCb/D1, e.g. (6.1.46) in [ 1]. Since
Qd
iD0.iC1/ii1 (and this estimate cannot be improved for all ,and), we finally
get
B.CCe/
B.CCe/DO
njjjj
;n!1:
Now let us consider the last factor in (4.13) . We have
R
Sd
xiiCiC1
nCjjCdC12
xCdx
R
SdxCdx
DR
SdxCC2eidxR
SdxCdx2iCiC1
nCjjCdC1R
SdxCCeidxR
SdxCdxCiCiC1
nCjjCdC12
D.iCiC2/. iCiC1/
.nCjjCdC2/.nCjjCdC1/iCiC1
nCjjCdC12

E.E. Berdysheva, K. Jetter / Journal of Approximation Theory 162 (2010) 576–598 595
D.iCiC1/.niCjjiCd/
.nCjjCdC2/.nCjjCdC1/2.nCiC1/.nCjjiCd/
.nCjjCdC2/.nCjjCdC1/2
DO1
n
;n!1;
uniformly in. Consequently,

b.i/
;b.i/
;2
DO
n1C.jjjj/
;n!1;
uniformly in, and, taking into account (4.7) , we obtain
max
jjDnb.i/
;i
nDO
n1.jjjj/
2
;n!1:
Thus, ifjjjj<1, then (4.3) is fulfilled for each iD0;1;:::; d. The convergence rate in
(4.10) follows from (4.5) . For the quadratic test functions, a similar consideration shows that (4.8)
is fulfilled for each iD1;:::; dand that the convergence rate in (4.11) holds. Thus, convergence
(4.12) holds by Theorem 4.3 .
An example of a non-Jacobi measure which satisfies the assumptions of Theorem 4.4 is
d.x/Dw.x/dxwithw.x/Dxln1
x,x2T0;1U. Indeed,
x.1x/w.x/px.1x/;0x1:
In particular,k'e1Mn;.'e1/kC.Sd/DO
n1
4
andk'2e1Mn;'2e1kC.Sd/DO
n1
4
,
n!1 .
Note that our class (4.9) is more general than P ˘alt˘anea’s class (1.10) . For example, the function
considered in the previous paragraph does not belong to the class (1.10) . Also, our proof is
different from P ˘alt˘anea’s one in [ 13], since he essentially used the continuity of the function hin
definition (1.10) .
4.3. K-functional estimates
In the last section of the paper, we give K-functional estimates for the L1
-approximation
by the operators (1.4) and(2.1) . These estimates generalize one of the estimates of Theorem 2
in [11].
LetC1.Sd/be the subspace of functions g2C.Sd/with continuous partial derivatives @ig,
iD1;:::; d, and semi-norm
krgkC.Sd/VDmax
iD1;:::;dk@igkC.Sd/;
and let
kgkC1.Sd/VDmaxfkgkC.Sd/;krgkC.Sd/g:
We will give estimates in terms of the K-functionals
K.fIt/VD inf
g2C1.Sd/n
kfgkL1.Sd/CtkrgkC.Sd/o
and
eK.fIt/VD inf
g2C1.Sd/n
kfgkL1.Sd/CtkgkC1.Sd/o
:

596 E.E. Berdysheva, K. Jetter / Journal of Approximation Theory 162 (2010) 576–598
Theorem 4.5. For the integral operator Ln;!;, we have the error estimate
kfLn;!;.f/kL1.Sd/
1Cn;!;
eK
fIe
n;!;
1Cn;!;!
;f2L1
.Sd/;
withn;!;as in (3.1) ,
e
n;!;VD

1Ln;!;.1/

L1.Sd/CdX
iD1

Ln;!;.j'eixi1j/

.x/

L1.Sd/;
and the test functions 'eias in (4.1) .
For the Bernstein–Durrmeyer operator Mn;, we have the error estimate
kfMn;.f/kL1.Sd/2K
fI
n;
2
;f2L1
.Sd/;
with

n;VDdX
iD1

Mn;.j'eixi1j/

.x/

L1.Sd/:
In the second term of the definition of e
n;!; , the operator Ln;!; is first applied to the
function y7!j'ei.y/xi1.y/j, and then the integral is taken with respect to the variable x.
n;
is defined similarly.
Proof. The proof is standard. Since
fLn;!;.f/D.fg/Ln;!;.fg/C.gLn;!;.g//
forg2C1.Sd/and
k.fg/Ln;!;.fg/kL1.Sd/
1Cn;!;

kfgkL1.Sd/; (4.14)
we have to estimate the norm of gLn;!;.g/. For g2C1.Sd/andx;y2Sdwe have
g.y/g.x/DdX
iD1@ig./. yixi/
with2Sda convex combination of xandy. Therefore,
jg.y/g.x/jkr gkC.Sd/dX
iD1jyixij
and
jgg.x/1jkr gkC.Sd/dX
iD1j'eixi1j;x2Sd: (4.15)
Fixx2Sd. Since the operator Ln;!;is positive, we have

gLn;!;.g/

.x/g.x/Ln;!;.g.x/1/.x/C
Ln;!;.g.x/1g/

.x/
jg.x/j
1Ln;!;.1/

.x/C
Ln;!;.jg.x/1gj/

.x/

E.E. Berdysheva, K. Jetter / Journal of Approximation Theory 162 (2010) 576–598 597
kgkC.Sd/
1Ln;!;.1/

.x/Ckr gkC.Sd/dX
iD1
Ln;!;.j'eixi1j/

.x/;
where in the last step we have used estimate (4.15) . Integrating with respect to the variable x
yields

gLn;!;.g/

L1.Sd/kgkC1.Sd/e
n;: (4.16)
We arrive at the estimate
kfLn;!;.f/kL1.Sd/.1Cn;!;/(
kfgkL1.Sd/Ce
n;
1Cn;!;kgkC1.Sd/)
for arbitrary g2C1.Sd/, and taking the infimum over all possible g2C1.Sd/gives the desired
bound.
The proof for Mn;repeats the proof for Ln;!;with the following changes. Since n;!;D1
in this case, estimate (4.14) becomes
k.fg/Mn;.fg/kL1.Sd/2kfgkL1.Sd/:
Since Mn;reproduces constant functions, estimate (4.16) can be replaced by

gMn;.g/

L1.Sd/kr gkC.Sd/
n;:
Acknowledgments
We thank Hubert Berens, Paul Nevai, and Joachim St ¨ockler for helpful and inspiring
discussions. We thank Carl de Boor and the anonymous referees for their remarks which helped
to improve the presentation of the paper.
References
[1]M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions, Dover Publ., Inc., New York, 1970.
[2]E. Berdysheva, K. Jetter, J. St ¨ockler, New polynomial preserving operators on simplices: Direct results, J. Approx.
Theory 131 (2004) 59–73.
[3]E. Berdysheva, K. Jetter, J. St ¨ockler, Durrmeyer operators and their natural quasi-interpolants, in: K. Jetter, et al.
(Eds.), Topics in Multivariate Approximation and Interpolation, Elsevier, Amsterdam, 2006, pp. 1–21.
[4]H. Berens, G.G. Lorentz, Sequences of contractions in L1-spaces, J. Func. Anal. 15 (1974) 155–165.
[5]H. Berens, Y . Xu, On Bernstein–Durrmeyer polynomials with Jacobi weights, in: C.K. Chui (Ed.), Approximation
Theory and Functional Analysis, Academic Press, Boston, 1991, pp. 25–46.
[6]F. Cucker, S. Smale, On the mathematical foundations of learning, Bull. Amer. Math. Soc. 39 (2001) 1–49.
[7]M.-M. Derriennic, Sur l’approximation de fonctions int ´egrables surT0;1Upar des polyn ˆomes de Bernstein modifies,
J. Approx. Theory 31 (1981) 325–343.
[8]M.-M. Derriennic, On multivariate approximation by Bernstein-type polynomials, J. Approx. Theory 45 (1985)
155–166.
[9]Z. Ditzian, Multidimensional Jacobi-type Bernstein–Durrmeyer operators, Acta Sci. Math. (Szeged) 60 (1995)
225–243.
[10] K. Jetter, J. St ¨ockler, An identity for multivariate Bernstein polynomials, Comput. Aided Geom. Design 20 (2003)
563–577.
[11] K. Jetter, D.-X. Zhou, Approximation with polynomial kernels and SVM classifiers, Adv. Comput. Math. 25 (2006)
323–344.
[12] R. P˘alt˘anea, Sur un op ´erateur polynomial defini sur l’ensemble des fonctions int ´egrables, “Babes ¸-Bolyai” Univ.
Fac. Math. Res. Semin. 2 (1983) 101–106.

598 E.E. Berdysheva, K. Jetter / Journal of Approximation Theory 162 (2010) 576–598
[13] R. P˘alt˘anea, Approximation Theory Using Positive Linear Operators, Birkh ¨auser-Verlag, Boston, 2004.
[14] S. Smale, D.-X. Zhou, Learning theory estimates via integral operators and their approximations, Constr. Approx.
26 (2007) 153–172.
[15] S. Waldron, On the Bernstein–B ´ezier form of Jacobi polynomials on a simplex, J. Approx. Theory 140 (2006)
86–99.