paper 12-17-2009. 128KB Jun 04 2011 12:03:07 AM

ACTA UNIVERSITATIS APULENSIS

No 17/2009

ABOUT SOME BIVARIATE OPERATORS OF STANCU TYPE

Mircea D. Farcas¸
Abstract. In this paper, we will obtain a form of Bernstein-Stancu bivariate operators and finally we will give an approximation theorem for them.
2000 Mathematics Subject Classification: 41A10, 41A25, 41A36, 41A63.
1. Introduction
Let N be the set of positive integers, N0 = N ∪ {0} and ∆2 = {(x, y) ∈ R ×
R|x, y ≥ 0, x+y ≤ 1}. For m ∈ N, the operator Bm : C([0, 1]×[0, 1]) → C(∆2 )
defined for any function f ∈ C([0, 1] × [0, 1]) by


X
k j
(1)
,
(Bm f )(x, y) =
pm,k,j (x, y)f

m
m
k,j=0
k+j≤m

for any (x, y) ∈ ∆2 , where
pm,k,j (x, y) =

m!
xk y j (1 − x − y)m−k−j ,
k!j!(m − k − j)!

(2)

for any k, j ∈ N0 , k + j ≤ m and any (x, y) ∈ ∆2 is named the Bernstein
bivariate operator (see [11]).
Let eij : ∆2 → R be the functions test, defined by eij (x, y) = xi y j for any
(x, y) ∈ ∆2 , where i, j ∈ N0 . In the paper [10] the following representation for
the polynomials Bm epq is proved.
Lemma 1. The operators (Bm )m≥1 verify for any (x, y) ∈ ∆2 and any

m ∈ N, p, q ∈ N0 the following equality
p
q
1 X X [i+j]
(Bm epq )(x, y) = p+q
m
S(p, i)S(q, j)xi y j ,
m
i=0 j=0

107

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Mircea D. Farca¸s - About some bivariate operators of Stancu type
where S(p, i), S(q, j) are the Stirling’s numbers of second kind and m[k] =
= m(m − 1) . . . (m − k + 1), k ∈ N0 , m[0] = 1.
Let I1 , I2 ⊂ R be given intervals and f : I1 × I2 → R be a bounded
function. The function ωtotal (f ; · , ·) : [0, ∞) × [0, ∞) → R, defined for any
(δ1 , δ2 ) ∈ [0, ∞) × [0, ∞) by


ωtotal (f ; δ1 , δ2 ) = sup |f (x, y) − f (x′ , y ′ )| : (x, y), (x′ , y ′ ) ∈ I1 × I2 ,

(4)
|x − x′ | ≤ δ1 , |y − y ′ | ≤ δ2

is called the first order modulus of smoothness of function f or total modulus
of continuity of function f . For some further information on this measure of
smoothness see for example [6] or [15]. The following result is given in [14].
Theorem 1. Let L : C(I1 × I2 ) → B(I1 × I2 ) be a constant reproducing
linear positive operator. For any f ∈ C(I1 × I2 ), any (x, y) ∈ I1 × I2 and any
δ1 , δ2 > 0, the following inequality


p
|(Lf )(x, y) − f (x, y)| ≤ 1 + δ1−1 (L(· − x)2 )(x, y) ·


p
· 1 + δ2−1 (L(∗ − y)2 )(x, y) ωtotal (f ; δ1 , δ2 )

(5)

holds, where ” · ” and ” ∗ ” stand for the first and the second variable.
The purpose of this paper is to give a representation for the bivariate operators and GBS operators of Stancu type, to establish a convergence theorem
for these operators. We also give an approximation theorem for these operators in terms of the first modulus of smoothness and of the mixed modulus of
smoothness.
2.The construct of the bivariate operators of Stancu type.
Approximation and convergence theorems
Let α, β be given real parameters such that 0 ≤ α ≤ β. For m ∈ N, the
(α,β)
operator Pm
: C([0, 1]) → C([0, 1]) defined for any function f ∈ C([0, 1])
and any x ∈ [0, 1] by
(Pmα,β f )(x)

=

m
X

pm,k (x)f

k=0

108



k+α
m+β



,

(6)

Mircea D. Farca¸s - About some bivariate operators of Stancu type
is called the Bernstein-Stancu operator (see [1]).

Next, we will construct a type of Bernstein-Stancu bivariate operator, inspired by the Bernstein bivariate operator (1.1). Let α1 , β1 , α2 , β2 be given
real parameters such that 0 ≤ α1 ≤ β1 , 0 ≤ α2 ≤ β2 . For m ∈ N, the
(α ,α ,β ,β )
operator Sm 1 2 1 2 : C([0, 1] × [0, 1]) → C(∆2 ), defined for any function
f ∈ C([0, 1] × [0, 1]) by


X
k + α1 j + α2
(α1 ,α2 ,β1 ,β2 )
(Sm
f )(x, y) =
,
(7)
pm,k,j (x, y)f
,
m + β1 m + β2
k,j=0
k+j≤m

for any (x, y) ∈ ∆2 is a bivariate operator of Stancu type. Obviously, this
operator is linear and positive. For β1 = β2 = 0, we obtain the Bernstein
operator (1.1).
(α ,α ,β ,β )
Lemma 2. The operators (Sm 1 2 1 2 )m≥1 verify for any (x, y) ∈ ∆2 the
following equalities:
(α1 ,α2 ,β1 ,β2 )
(Sm
e00 )(x, y) = 1,
(8)
(α1 ,α2 ,β1 ,β2 )
(m + β1 )(Sm
e10 )(x, y) = mx + α1 ,

(9)

(α1 ,α2 ,β1 ,β2 )
(m + β2 )(Sm
e01 )(x, y) = my + α2 ,

(10)

(α1 ,α2 ,β1 ,β2 )
(m + β1 )2 (Sm
e20 )(x, y) = m(m − 1)x2 + (1 + 2α1 )mx + α12 ,
(α1 ,α2 ,β1 ,β2 )
(m + β2 )2 (Sm
e02 )(x, y) = m(m − 1)y 2 + (1 + 2α2 )my + α22 .

(11)
(12)

(α ,α ,β ,β )

Proof. We use the equalities (Sm 1 2 1 2 e00 )(x, y) = (Bm e00 )(x, y),
(α ,α ,β ,β )
(m + β1 )(Sm 1 2 1 2 e10 )(x, y) = m(Bm e01 )(x, y) + α1 (Bm e00 )(x, y) and
(α ,α ,β ,β )
(m + β1 )2 (Sm 1 2 1 2 e20 )(x, y) = m2 (Bm e20 )(x, y) + 2α1 m(Bm e01 )(x, y) +
+ α12 (Bm e00 )(x, y).

(α ,α2 ,β1 ,β2 )

Lemma 3. The operators (Sm 1
following equalities:

)m≥1 verify for any (x, y) ∈ ∆2 the

(α1 ,α2 ,β1 ,β2 )
(m + β1 )2 (Sm
(· − x)2 )(x, y) = mx(1 − x) + (β1 x − α1 )2 ,

(13)

(α1 ,α2 ,β1 ,β2 )
(∗ − y)2 )(x, y) = my(1 − y) + (β2 y − α2 )2 .
(m + β2 )2 (Sm

(14)

Proof. We use the equalities (8)-(12) and the relations

(α ,α ,β ,β )
(α ,α ,β ,β )
(α ,α ,β ,β )
(Sm 1 2 1 2 (·−x)2 )(x, y) = (Sm 1 2 1 2 e20 )(x, y)−2x(Sm 1 2 1 2 e10 )(x, y)+
(α ,α ,β ,β )
+ x2 (Sm 1 2 1 2 e00 )(x, y) and

109

Mircea D. Farca¸s - About some bivariate operators of Stancu type
(α ,α ,β ,β )

(α ,α ,β ,β )

(α ,α ,β ,β )

(Sm 1 2 1 2 (∗−y)2 )(x, y) = (Sm 1 2 1 2 e02 )(x, y)−2y(Sm 1 2 1 2 e01 )(x, y)+
(α ,α ,β ,β )
+ y 2 (Sm 1 2 1 2 e00 )(x, y).
(α ,α ,β ,β )

Lemma 4. The operators (Sm 1 2 1 2 )m≥1 verify for any (x, y) ∈ ∆2 the
following inequalities:
(α1 ,α2 ,β1 ,β2 )
4(m + β1 )2 (Sm
(· − x)2 )(x, y) ≤ m + 4β12

(15)

(α1 ,α2 ,β1 ,β2 )
4(m + β2 )2 (Sm
(∗ − y)2 )(x, y) ≤ m + 4β22 .

(16)

and
Proof. We use the relations (13), (14) and the inequalities x(1 − x) ≤ 1/4,
y(1 − y) ≤ 1/4, (β1 x − α1 )2 ≤ β12 , (β2 y − α2 )2 ≤ β22 , for any x, y ∈ [0, 1].
Theorem 2. If f ∈ C([0, 1] × [0, 1]), then for any (x, y) ∈ ∆2 and any
m ∈ N, we have the following inequalities:
s
!
2
m
+
4β
1
(α1 ,α2 ,β1 ,β2 )
·
|f (x, y) − (Sm
f )(x, y)| ≤ 1 + δ1−1
4(m + β1 )2
· 1 + δ2−1

s

m + 4β22
4(m + β2 )2

!

ωtotal (f ; δ1 , δ2 ),

(17)

for any δ1 , δ2 > 0 and
(α1 ,α2 ,β1 ,β2 )
|f (x, y) − (Sm
f )(x, y)| ≤
s
s
!
m + 4β12
m + 4β22
,
.
≤ 4ωtotal f ;
4(m + β1 )2
4(m + β2 )2

(18)

Proof. The
(2.12) results
q relation
q from2 Theorem 1.1 and Lemma 2.3; choosm+4β12
m+4β
ing by δ1 = 4(m+β1 )2 and δ2 = 4(m+β22)2 , we obtain the relation (2.13).
Corollary 1. If f ∈ C([0, 1] × [0, 1]), then

(α1 ,α2 ,β1 ,β2 )
lim Sm
f =f

m→∞

uniformly on ∆2 .

110

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Mircea D. Farca¸s - About some bivariate operators of Stancu type
3. Approximation and convergence theorems for gbs operators
of schurer type
In the following, let X and Y be compact real intervals. A function
f : X ×Y → R is called B-continuous (B¨ogel-continuous) function at (x0 , y0 ) ∈
X × Y if
lim
∆f [(x, y), (x0 , y0 )] = 0.
(x,y)→(x0 ,y0 )

Here ∆f [(x, y), (x0 , y0 )] = f (x, y) − f (x0 , y) − f (x, y0 ) + f (x0 , y0 ) denotes a
so-called mixed difference of f .
A function f : X × Y → R is called B-differentiable (B¨ogel-differentiable)
function at (x0 , y0 ) ∈ X × Y if it exists and if the limit is finite:
∆f [(x, y), (x0 , y0 )]
.
(x,y)→(x0 ,y0 ) (x − x0 )(y − y0 )
lim

The limit is named the B-differential of f at the point (x0 , y0 ) and is noted by
DB f (x0 , y0 ).
The definition of B-continuity and B-differentiability were introduced by
K. B¨ogel in the papers [7] and [8].
The function f : X × Y → R is B-bounded on X × Y if there exists K > 0
such that
|∆f [(x, y), (s, t)]| ≤ K
for any (x, y), (s, t) ∈ X × Y .

We shall use the function sets B(X × Y ) = f : X ×Y → R|f bounded on
X × Y with the usual
sup-norm k · k∞ , Bb (X × Y ) = f : X × Y → R|f B
bounded on X × Y and we set kf kB =
sup
|∆f [(x, y), (s, t)] | where
(x,y),(s,t)∈X×Y


f ∈ Bb (X × Y ), C
b (X × Y ) = f : X × Y → R|f B-continuous on X × Y
and Db (X × Y ) = f : X × Y → R|f B-differentiable on X × Y .
Let f ∈ Bb (X × Y ). The function ωmixed (f ; · , ·) : [0, ∞) × [0, ∞) → R,
defined by
ωmixed (f ; δ1 , δ2 ) = sup {|∆f [(x, y), (s, t)]| : |x − s| ≤ δ1 , |y − t| ≤ δ2 }

(20)

for any (δ1 , δ2 ) ∈ [0, ∞) × [0, ∞) is called the mixed modulus of smoothness.
For related topics, see [2], [3], [4] and [5].

111

Mircea D. Farca¸s - About some bivariate operators of Stancu type
Let L : Cb (X ×Y ) → B(X ×Y ) be a linear positive operator. The operator
U L : Cb (X × Y ) → B(X × Y ) defined for any function f ∈ Cb (X × Y ) and
any (x, y) ∈ X × Y by
(U Lf )(x, y) = (L(f (·, y) + f (x, ∗) − f (·, ∗))) (x, y)

(21)

is called GBS operator (”Generalized Boolean Sum” operator) associated to
the operator L, where ”·” and ”∗” stand for the first and respectively the second
variable. Let eij : X × Y → R be the functions test, defined by eij (x, y) = xi y j
for any (x, y) ∈ X × Y , where i, j ∈ N0 . The following theorem is proved in
[4].
Theorem 3. Let L : Cb (X × Y ) → B(X × Y ) be a linear positive operator
and U L : Cb (X × Y ) → B(X × Y ) the associated GBS operator. Then for any
f ∈ Cb (X × Y ), any (x, y) ∈ (X × Y ) and any δ1 , δ2 > 0, we have
|f (x, y) − (U Lf )(x, y)| ≤ |f (x, y)| |1 − (Le00 )(x, y)|+

p
p
+ (Le00 )(x, y) + δ1−1 (L(· − x)2 ) (x, y) + δ2−1 (L(∗ − y)2 ) (x, y)+
+δ1−1 δ2−1


p
2
2
(L(· − x) (∗ − y) ) (x, y) ωmixed (f ; δ1 , δ2 ).

(22)

For B-differentiable functions, we have (see [12]):

Theorem 4. Let L : Cb (X × Y ) → B(X × Y ) be a linear positive operator
and U L : Cb (X × Y ) → B(X × Y ) the associated GBS operator. Then for any
f ∈ Db (X × Y ) with DB f ∈ B(X × Y ), any (x, y) ∈ X × Y and any δ1 , δ2 > 0,
we have
|f (x, y) − (U Lf )(x, y)| ≤
p
≤ |f (x, y)||1−(Le00 )(x, y)|+3kDB f k∞ (L(· − x)2 (∗ − y)2 ) (x, y)+

p
p
+
(L(· − x)2 (∗ − y)2 )(x, y) + δ1−1 (L(· − x)4 (∗ − y)2 )(x, y)+
+δ2−1

+δ1−1 δ2−1

p

(L(· − x)2 (∗ − y)4 )(x, y)+



2
2
L(· − x) (∗ − y) (x, y) ωmixed (DB f ; δ1 , δ2 ).
112

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Mircea D. Farca¸s - About some bivariate operators of Stancu type
Lemma 5. There exists a natural number m1 ∈ N such that
(α1 ,α2 ,β1 ,β2 )
(Sm
(· − x)2 (∗ − y)2 )(x, y) ≤

1
,
4(m + β1 )(m + β2 )

1
,
4(m + β1 )2 (m + β2 )
1
(α1 ,α2 ,β1 ,β2 )
(· − x)2 (∗ − y)4 )(x, y) ≤
,
(Sm
4(m + β1 )(m + β2 )2
(α1 ,α2 ,β1 ,β2 )
(Sm
(· − x)4 (∗ − y)2 )(x, y) ≤

for any m ∈ N, m ≥ m1 and any (x, y) ∈ ∆2 .

(24)
(25)
(26)

(α ,α ,β ,β )

Proof. Using the relations (m + β1 )i (m + β2 )j (Sm 1 2 1 2 eij )(x, y) =
j   
i X
X
i
j
=
mν1 +ν2 α1i−ν1 α2j−ν2 (Bm eν1 ν2 )(x, y), for any i, j ∈ N0 we get
ν1
ν2
ν =0 ν =0
1

2

(α ,α ,β ,β )

(m + β1 )2 (m + β2 )2 (Sm 1 2 1 2 (· − x)2 (∗ − y)2 )(x, y) = Am2 + Bm + C, where
A, B, C are real numbers depending on x, y, α1 , β1 , α2 , β2 and
A = xy(1 − x)(1 − y) + 2x2 y 2 ≤ 3/16. Further on, we have
(α ,α ,β ,β )
(m + β1 )4 (m + β2 )2 (Sm 1 2 1 2 (· − x)2 (∗ − y)2 )(x, y) = Am3 + Bm2 + Cm + D,
where A, B, C, D are real numbers depending on x, y, α1 , β1 , α2 , β2 and
A = 3x(1 − x)[xy(1 − x)(1 − y) + 4x2 y 2 ] ≤ 15/64. We used the inequalities
x(1 − x) ≤ 1/4, for any x ∈ [0, 1] and xy(1 − x)(1 − y) ≤ 1/16, xy ≤ 1/4,
x2 y 2 ≤ 1/16, for any (x, y) ∈ ∆2 .
Theorem 5. If f ∈ Cb ([0, 1] × [0, 1]), then for any (x, y) ∈ ∆2 and any
m ∈ N, m ≥ m1 , the following inequalities
s
m + 4β12
(α1 ,α2 ,β1 ,β2 )
|(U Sm
f )(x, y) − f (x, y)| ≤ 1 + δ1−1
+
4(m + β1 )2
+δ2−1

s

1
m + 4β22
+ δ1−1 δ2−1 p
2
4(m + β2 )
2 (m + β1 )(m + β2 )

!

ωmixed (f ; δ1 , δ2 ),

(27)

for any δ1 , δ2 > 0 and

(α1 ,α2 ,β1 ,β2 )
|(U Sm
f )(x, y) − f (x, y)| ≤
!
r
r
5
m + 4β12
m + 4β22
,
≤ ωmixed f ;
2
m2
m2

113

(28)

Mircea D. Farca¸s - About some bivariate operators of Stancu type
hold, where
(α1 ,α2 ,β1 ,β2 )
(U Sm
f )(x, y)

=

X

 

k + α1
pm,k,j (x, y) f
,y +
m + β1

−f



k,j=0
k+j≤m

+f



j + α2
x,
m + β2



k + α1 j + α2
,
m + β1 m + β2



.

Proof. For the first inequality, we apply Theorem q
3 and Lemma 5. The
m+4β12
and
inequality (28) is obtained from (27) by choosing δ1 =
m2
q
2
m+4β2
.
δ2 =
m2
Corollary 2. If f ∈ Cb ([0, 1] × [0, 1]), then

(α1 ,α2 ,β1 ,β2 )
lim U Sm
f = f,

m→∞

(29)

uniformly on ∆2 .
Proof. It results from (3.6).
Theorem 6 Let the function f ∈ Db ([0, 1] × [0, 1]) with DB f ∈ B([0, 1] ×
[0, 1]). Then, for any (x, y) ∈ ∆2 and for any m ∈ N, m ≥ m1 , we have
3
(α1 ,α2 ,β1 ,β2 )
√
f )(x, y) − f (x, y)| ≤ √
|(U Sm
kDB f k∞ +
2 m + β1 m + β2

1
1
1
√
1 + δ1−1 √
+ δ2−1 √
+
+ √
2 m + β1 m + β2
m + β1
m + β2

1
−1 −1
√
+δ1 δ2 √
ωmixed (DB f ; δ1 , δ2 ),
(30)
2 m + β1 m + β2

for any δ1 , δ2 > 0 and

3
(α1 ,α2 ,β1 ,β2 )
√
f )(x, y) − f (x, y)| ≤ √
|(U Sm
kDB f k∞ +
2 m + β1 m + β2


7
1
1
√
+ √
.
(30)
ωmixed DB f ; √
,√
4 m + β1 m + β2
m + β1
m + β2
Proof. It results from Theorem 4 and Lemma 5.
114

Mircea D. Farca¸s - About some bivariate operators of Stancu type
Remark 1. Other construction for bivariate operators of Stancu type can
be found in the paper [6].
Remark 2. For β1 = β2 = 0, we find some results obtained in the paper
[13].
References
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ogel ¸si unele aplicat¸ii ˆın
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Math.-Mech., 18(2) (1973), 69-78 (Romanian)
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[5] Badea, C., Badea, I., Cottin, C., Gonska, H. H., Notes on the degree
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Mircea D. Farca¸s - About some bivariate operators of Stancu type
[13] Pop, O.T., Farca¸s, M.D., Approximation of B-continuous and B-differentiable functions by GBS operators of Bernstein bivariate polynomials, J.
Inequal. Pure Appl. Math., 7 (2006), 9pp (electronic)
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a ¸si mai multe variabile cu
ajutorul operatorilor liniari ¸si pozitivi, Ph. D. Thesis, Univ. ”Babe¸s-Bolyai”,
Cluj-Napoca, 1984 (Romanian)
[15] Timan, A.F., Theory of Approximation of Functions of Real Variable,
New York: Macmillan Co. 1963. MR22#8257
Author:
Mircea D. Farca¸s
National College ”Mihai Eminescu”
5 Mihai Eminescu Street
Satu Mare 440014, Romania
email:mirceafarcas2005@yahoo.com

116