J. Math. Anal. Appl. 424 (2015) 13741379 [616068]

J. Math. Anal. Appl. 424 (2015) 1374–1379
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Journal of Mathematical Analysis and Applications
www.elsevier.com/locate/jmaa
Uniform and pointwise convergence of Bernstein–Durrmeyer
operators with respect to monotone and submodular set functions
Sorin G. Gala,∗, Bogdan D. Oprisb
aDepartment of Mathematics and Computer Science, University of Oradea, Universitatii Street No. 1,
410087, Oradea, Romania
bBabes–Bolyai University, Faculty of Mathematics and Computer Science, M. Kogalniceanu Street No. 1,
400084 Cluj-Napoca, Romania
a r t i c l e i n f o a b s t r a c t
Article history:
Received 20 September 2014
A vailable online 9 December 2014
Submitted by P. Nevai
Keywords:
Monotone and submodular set
functionChoquet integral
Bernstein–Durrmeyer operator
Uniform convergence
Pointwise convergenceWe consider the multivariate Bernstein–Durrmeyer operator M n,μ in terms of the
Choquet integral with respect to a monotone and submodular set function μon
the standard d-dimensional simplex. This operator is nonlinear and generalizes
the Bernstein–Durrmeyer linear operator with respect to a nonnegative, bounded
Borel measure (including the Lebesgue measure). We prove uniform and pointwise
convergence of M n,μ(f)(x)t o f(x)a s n →∞ , generalizing thus the results obtained
in the recent papers [1]and [2].
© 2014 Published by Elsevier Inc.
1. Introduction
Starting from the paper [3], in other three recent papers [1,2,7] , uniform, pointwise and Lpconvergence
(respectively) of Mn,μ(f)(x)t o f(x)( a s n →∞ ) were obtained, where Mn,μ(f)(x) denotes the multivariate
Bernstein–Durrmeyer linear operator with respect to a nonnegative, bounded Borel measure μ, defined on
the standard simplex
Sd=/braceleftbig
(x1, …, x d); 0≤x1, …, x d≤1,0≤x1+…+xd≤1/bracerightbig
,
by
Mn,μ(f)(x)=/summationdisplay
|α|=n/integraltext
Sdf(t)Bα(t)dμ(t)/integraltext
SdBα(t)dμ(t)·Bα(x): =/summationdisplay
|α|=nc(α, μ)·Bα(x),x ∈Sd,n∈N, (1)
*Corresponding author.
E-mail addresses: [anonimizat] (S.G. Gal), [anonimizat] (B.D. Opris).
http://dx.doi.org/10.1016/j.jmaa.2014.12.0120022-247X/© 2014 Published by Elsevier Inc.

S.G. Gal, B.D. Opris / J. Math. Anal. Appl. 424 (2015) 1374–1379 1375
where fis supposed to be μ-integrable on Sd. Also, in formula (1), we have denoted α=(α0, α1, …, αn),
with αj≥0f o r all j=0, …, n, |α| =α0+α1+… +αn=nand
Bα(x)=n!
α0!·α1!·…·αn!(1−x1−x2−…−xd)α0·xα1
1·…·xαd
d
:=n!
α0!·α1!·…·αn!·Pα(x).
The goal of the present note is to show that the results in [1]and [2]on pointwise and uniform convergence
remain valid in the more general setting when μis a monotone, bounded and submodular set function on Sd
and the integrals appearing in the expression of the coefficients c(α, μ)i n formula (1)are Choquet integrals
with respect to μ.
2. Preliminaries
In this section we present concepts and results used in the next main section.
Definition 2.1. Let (Ω, C)b e a measurable space, i.e. Ωis a nonempty set and Cis a σ-algebra of subsets
inΩ.
(i) (See, e.g., [8, p. 63].) The set function μ :C→ [0, +∞]i s called a monotone set function (or capacity)
if μ(∅) =0a n d A, B∈C, with A ⊂B, implies μ(A) ≤μ(B). Also, μis called submodular if
μ(A∪B)+μ(A∩B)≤μ(A)+μ(B ),for all A, B ∈C.
If μ(Ω) =1 , then μis called normalized.
(ii) (See [4], or [8, p. 233].) Let μbe a normalized, monotone set function defined on C. Recall that
f:Ω→Ris called C-measurable if for any B, Borel subset in R, we have f−1(B) ∈C.
If f:Ω→Ris C-measurable, then for any A ∈C, the Choquet integral is defined by
(C)/integraldisplay
Afdμ=+∞/integraldisplay
0μ/parenleftbig
Fβ(f)∩A/parenrightbig
dβ+0/integraldisplay
−∞/bracketleftbig
μ/parenleftbig
Fβ(f)∩A/parenrightbig
−μ(A)/bracketrightbig
dβ,
where Fβ(f) ={ω∈Ω; f(ω) ≥β}. If (C) /integraltext
Afdμ exists in R, then fis called Choquet integrable on A.
Note that if f≥0o n A, then the term integral /integraltext0
−∞in the above formula becomes equal to zero.
When μis the Lebesgue measure (i.e. countably additive), then the Choquet integral (C) /integraltext
Afdμreduces
to the Lebesgue integral.
In what follows, we list some known properties we need for the main section.
Remark 2.2. Let us suppose that μ :C→ [0, +∞]i s a monotone set function. Then, the following properties
hold:
(i) (C) /integraltext
Ais positively homogeneous, i.e. for all a ≥0w e have (C) /integraltext
Aafdμ =a ·(C) /integraltext
Afdμ(for f≥0
see, e.g., [8, Theorem 11.2 (5), p. 228] and for fof arbitrary sign, see, e.g., [5, p. 64, Proposition 5.1 (ii)]).
(ii) In general, (C) /integraltext
A(f+g)dμ /negationslash=(C) /integraltext
Afdμ +(C) /integraltext
Agdμ. However, we have
(C)/integraldisplay
A(f+c)dμ=(C)/integraldisplay
Afdμ+c·μ(A),
for all c ∈Rand fof arbitrary sign (see, e.g., [8, pp. 232–233] , or [5, p. 65]).

1376 S.G. Gal, B.D. Opris / J. Math. Anal. Appl. 424 (2015) 1374–1379
If μis submodular too, then the Choquet integral is sublinear, that is
(C)/integraldisplay
A(f+g)dμ≤(C)/integraldisplay
Afdμ+(C)/integraldisplay
Agdμ,
for all f, gof arbitrary sign and lower bounded (see, e.g., [5, p. 75, Theorem 6.3]).
(iii) If f≤gon Athen (C) /integraltext
Afdμ ≤(C) /integraltext
Agdμ(see, e.g., [8], p. 228, Theorem 11.2 (3) for f, g≥0a n d
p. 232 for f, gof arbitrary sign).
(iv) Let f≥0. By the definition of the Choquet integral, it is immediate that if A ⊂Bthen
(C)/integraldisplay
Afdμ≤(C)/integraldisplay
Bfdμ
and if, in addition, μis finitely subadditive, then
(C)/integraldisplay
A∪Bfdμ≤(C)/integraldisplay
Afdμ+(C)/integraldisplay
Bfdμ.
(v) By the definition of the Choquet integral, it is immediate that
(C)/integraldisplay
A1·dμ(t)=μ(A).
(vi) Simple concrete examples of monotone and submodular set functions μcan be obtained from a
probability measure Mon a σ-algebra on Ω(i.e. M(∅) =0 , M(Ω) =1a n d Mis countably additive), by
the formula μ(A) =γ(M(A)), where γ:[ 0, 1] →[0, 1] is an increasing and concave function, with γ(0) =0 ,
γ(1) = 1 (see, e.g., [5, pp. 16–17, Example 2.1]). Note that in fact if Mis only finitely additive, then
μ(A) =γ(M(A)) still is submodular.
Also, any possibility measure μis monotone and submodular. Indeed, while the monotonicity is immediate
from the axiom μ(A ∪B) =m a x {μ(A ), μ(B)}, the submodularity is immediate from the property μ(A ∩B) ≤
min{μ(A ), μ(B)}.
Recall here that a set function μ :P(Ω) →[0, 1] (P(Ω) denotes the family of all subsets of Ω) is
called a possibility measure on the non-empty set Ω, if it satisfies the axioms μ(∅) =0 , μ(Ω) =1a n d
μ(/uniontext
i∈IAi) =s u p {μ(A i); i ∈I}for all Ai∈Ω, and any I, family of indices.
It is known that any given possibility distribution (on Ω), that is a function λ :Ω→[0, 1], such
that sup{λ(s ); s ∈Ω} =1 , induces a possibility measure μλ:P(Ω) →[0, 1], given by the formula
μλ(A) =s u p {λ(s); s ∈A}, for all A ⊂Ω, A /negationslash=∅, μλ(∅) = 0 (see, e.g., [6, Chapter 1]).
3. Main results
Let BSdbe the sigma algebra of all Borel measurable subsets in P(Sd)a n d μ :BSd→[0, +∞)b e a
normalized, monotone and submodular set function on BSd.
We say that μis strictly positive if μ(A ∩Sd) >0, for every open set A ⊂Rnwith A ∩Sd/negationslash=∅.
Also, by definition, the support of μ, denoted by supp(μ), is the set of all x ∈Sdwith the property that
for every open neighborhood Nx∈B Sdof x, we have μ(N x) >0.
Denote by C+(Sd)t h e space of all positive-valued continuous functions on Sdand by L∞
μ(Sd)t h e space
of all real-valued BSd-measurable functions f, such that there exists a set E⊂Sd(depending on f) with
μ(E) =0a n d fis bounded on Sd\E.

S.G. Gal, B.D. Opris / J. Math. Anal. Appl. 424 (2015) 1374–1379 1377
Denote
Mn,μ(f)(x)=/summationdisplay
|α|=nc(α, μ)·Bα(x),x ∈Sd,n∈N,
where applying Remark 2.2 (i), we easily get
c(α, μ)=(C)/integraltext
Sdf(t)Bα(t)dμ(t)
(C)/integraltext
SdBα(t)dμ(t)=(C)/integraltext
Sdf(t)Pα(t)dμ(t)
(C)/integraltext
SdPα(t)dμ(t).
It is worth noting here that we did not loose any generality by the normalization condition on the set
valued function μand that the condition supp(μ) \∂Sd/negationslash=∅guarantees that (C) /integraltext
SdBα(t)dμ(t) >0, for
allBα.
For the proof of the main results, we need the following auxiliary result.
Lemma 3.1. Let us suppose that μis a normalized, monotone and submodular set function. If we define
Tn:C+(Sd) →R+by
Tn(f)=( C)/integraldisplay
Sdf(t)Pα(t)dμ(t),f ∈C+/parenleftbig
Sd/parenrightbig
,n∈N,|α|=n,
then for all f, g∈C+(Sd), we have
/vextendsingle/vextendsingleTn(f)−Tn(g)/vextendsingle/vextendsingle≤T
n/parenleftbig
|f−g|/parenrightbig
=(C)/integraldisplay
Sd/vextendsingle/vextendsinglef(t)−g(t)/vextendsingle/vextendsingle·P
α(t)dμ(t).
Proof. Since Pα(t) ≥0f o r all t ∈Sd, the functional Tnhas the following properties: is positively ho-
mogeneous (from Remark 2.2 (i)), monotonically increasing (from Remark 2.2 (iii)) and sublinear (from
Remark 2.2 (ii)).
Let f, g∈C+(Sd). We have f=f−g+g≤|f−g| +g, which successively implies Tn(f) ≤Tn(|f−g|) +
Tn(g), that is Tn(f) −Tn(g) ≤Tn(|f−g|).
Writing now g=g−f+f≤|f−g| +fand applying the above reasonings, it follows Tn(g) −Tn(f) ≤
Tn(|f−g|), which combined with the above inequality gives |Tn(f) −Tn(g)| ≤Tn(|f−g|).2
The first main result is an analogous result to Theorem 1 in [1]and refers to uniform approximation.
Theorem 3.2. Let μbe a normalized, monotone, submodular and strictly positive set function on BSd, such
that supp(μ) \∂Sd/negationslash=∅. For every f∈C+(Sd)we have
lim
n→∞/vextenddouble/vextenddoubleM
n,μ(f)−f/vextenddouble/vextenddouble
C(Sd)=0,
where /bardblF/bardblC(Sd)=m a x {|F(x)|; x ∈Sd}.
Proof. We follow the reasonings in the proof of Theorem 1 in [1], by keeping the notations there.
In this sense, let Πdbe the set of all permutations of the set {d, d −1, …, 1, 0}and denote a =α
n=
(α0
n, …, αd
n) ∈Sd, in barycentric coordinates. For π∈Πd, let us consider (in barycentric coordinates)
Sπ=/braceleftbig
a=(a0,a1, …, a d)∈Sd;aπ(d)≤aπ(d−1)≤…≤aπ(1)≤aπ(0)/bracerightbig
.
Also, for η>0, a ∈Sdand π∈Πd, let us define the d-dimensional open cube

1378 S.G. Gal, B.D. Opris / J. Math. Anal. Appl. 424 (2015) 1374–1379
Uπ(a;η)=/braceleftbig
x∈Rd;aπ(d)−η<x π(d)<a π(d)+η, …, a π(1)−η<x π(1)<a π(1)+η/bracerightbig
and the d-dimensional closed simplex
Vπ(a;η)=/braceleftbig
x∈Rd;aπ(d)≤xπ(d)≤aπ(d)+η, …, a π(1)≤xπ(1)≤aπ(1)+η,
aπ(d)+…+aπ(1)≤xπ(d)+…+xπ(1)≤aπ(d)+…+aπ(1)+η/bracerightbig
.
Note that if a ∈Sdand 0 <η≤1
d+1, then Vπ(a; η) ⊆Sd(see [1, p. 327]).
For ε >0, from the uniform continuity of fon Sd, there exists δ>0, such that for all x, y∈Sdwith
/bardblx −y/bardbl∞<δ, it follows |f(x) −f(y)| <ε, where /bardblx −y/bardbl∞:= max {|xi−yi|; i =1, …, d}.
Take δ=m i n {δ/d, 1/(d +1 ), 1/6}, M=/bardblf/bardblC(Sd), |α| =n, π∈Πd, a =α
n∈Sd.
Following the ideas of the proof in [1, p. 328], we can write
/vextendsingle/vextendsinglec(α, μ)−f(a)/vextendsingle/vextendsingle
=/vextendsingle/vextendsingle/vextendsingle/vextendsingle(C)/integraltext
Sdf(t)Pα(t)dμ(t)
(C)/integraltext
SdPα(t)dμ(t)−(C)/integraltext
Sdf(a)·Pα(t)dμ(t)
(C)/integraltext
SdPα(t)dμ(t)/vextendsingle/vextendsingle/vextendsingle/vextendsingle(2)
=|(C)/integraltext
Sdf(t)Pα(t)dμ(t)−(C)/integraltext
Sdf(a)·Pα(t)dμ(t)|
(C)/integraltext
SdPα(t)dμ(t)
≤(C)/integraltext
Sd|f(t)−f(a)|· Pα(t)dμ(t)
(C)/integraltext
SdPα(t)dμ(t)(3)
≤(C)/integraltext
Sd∩Uπ(a;δ)|f(t)−f(a)|· Pα(t)dμ(t)
(C)/integraltext
SdPα(t)dμ(t)+(C)/integraltext
Sd\Uπ(a;δ)|f(t)−f(a)|Pα(t)dμ(t)
(C)/integraltext
SdPα(t)dμ(t)(4)
≤ε+2M·(C)/integraltext
Sd\Uπ(a;δ)Pα(t)dμ(t)
(C)/integraltext
Sd∩Uπ(a;δ)Pα(t)dμ(t)(5)
≤ε+2M·(C)/integraltext
Sd\Uπ(a;δ)Pα(t)dμ(t)
(C)/integraltext
Vπ(a;δ2)Pα(t)dμ(t)(6)
≤ε+2M·max{Pα(x); x∈Sd\Uπ(a;δ)}·μ(Sd)
min{Pα(x); x∈Vπ(a;δ2)}·μ(V π(a;δ2))(7)
≤ε+2M·max{Pα(x); x∈Sd\Uπ(a;δ)}
min{Pα(x); x∈Vπ(a;δ2)}·infa∈Sπμ(V π(a;δ2)).
Note that above, (2)is obtained from Remark 2.2 (i), (3)is obtained from Lemma 3.1 , (4)is obtained
from the second inequality in Remark 2.2 (iv) (since μis subadditive too), (5), (6)are obtained from Re-
mark 2.2 (iii), (i) and from the first inequality in Remark 2.2 (iv), while (7)is obtained from Remark 2.2 (iii),
(i) and (v).
Since in the proof of Lemma 2 in [1], only the monotonicity and the strict positivity of the measure is
involved, it analogously follows that
inf
a∈Sπμ/parenleftbig
Vπ/parenleftbig
a;δ2/parenrightbig/parenrightbig
>0.
Moreover, because in the rest of the proof of Theorem 1 in [1], the measure is not longer involved (see
Lemmas 3, 4 and 5 in [1]), we immediately get the proof of Theorem 3.2 too.2
The second main result is an analogue result to Theorem 1 in [2]and refers to pointwise convergence. In
this sense, analyzing the reasonings in the proof of Theorem 1 in [2]and using the same properties of the
Choquet integral as in the proof of the above Theorem 3.2 , we easily get the following.

S.G. Gal, B.D. Opris / J. Math. Anal. Appl. 424 (2015) 1374–1379 1379
Theorem 3.3. Let μbe a normalized, monotone, submodular set function on BSd, such that supp(μ) \∂Sd/negationslash=∅.
If f∈L∞
μ(Sd)and f(x) ≥0, for all x ∈Sd, then at any point x ∈supp(μ) where fis continuous, we have
lim
n→∞/vextendsingle/vextendsingleMn,μ(f)(x)−f(x)/vextendsingle/vextendsingle=0.
Remark 3.4. According to Remark 2.2 (vi), an example of submodular set function μsatisfying all the
requirements in the statements of Theorems 3.2 and 3.3can simply be defined by μ(A) =/radicalbig
ν(A), where ν
is a Borel probability measure as in [1]and [2]. Also, it is worth noting that due to the nonlinearity of the
Choquet integral (see Remark 2.2 (ii)), unlike the case in [1,2], the Bernstein–Durrmeyer operator in the
present paper is nonlinear.
Remark 3.5. The positivity of function fin Theorems 3.2 and 3.3is necessary because of the positive
homogeneity of the Choquet integral applied in the proof of relation (2). However, if fis of arbitrary sign
on Sd, then it is immediate that the statements of Theorems 3.2 and 3.3can be restated for the slightly
modified Bernstein–Durrmeyer operator defined by
M∗
n,μ(f)(x)=M n,μ(f−m)(x)+ m,
where mis a lower bound for f, that is f(x) ≥m, for all x ∈Sd.
Acknowledgment
The authors thank the referee for the valuable remarks. The work of the first author was supported by
a grant of the Romanian National Authority for Scientific Research, CNCS–UEFISCDI, project number
PN-II-ID-PCE-2011-3-0861.
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