Download APPROXIMATION OF B-DIFFERENTIABLE FUNCTIONS BY GBS

Analele Universităţii Oradea
Fasc. Matematica, Tom XIV (2007), 15–31
APPROXIMATION OF B-DIFFERENTIABLE
FUNCTIONS BY GBS OPERATORS
OVIDIU T. POP
Abstract. In this paper we give an approximation of B-differentiable functions by GBS operators theorem, and then, through particular cases, we shall obtain statements verified by the GBS operators of Bernstein-Stancu type, GBS operators of Durrmeyer-Stancu
type and GBS operators of Kantorovich type.
1. Introduction
In this section, we recall some notions and results which we will use in
this paper (see [14]).
Define the natural number m0 by
(
max{1, −[β]}, iff β ∈ R\Z
m0 =
(1.1)
max{1, 1 − β}, iff β ∈ Z.
For the real number β, we have
m + β ≥ γβ
for any natural number m, m ≥ m0 , where
(
max{1 + β, {β}}, iff β ∈ R\Z
γβ = m0 + β =
max{1 + β, 1},
iff β ∈ Z.
(1.2)
(1.3)
2000 Mathematics Subject Classification. 41A10, 41A25, 41A35, 41A36, 41A63.
Key words and phrases. Linear positive operators, Bernstein-Stancu operators,
Durrmeyer-Stancu operators, GBS operators, B-differentiable functions, approximation of B-differentiable functions by GBS operators, mixed modulus of smoothness.
15
16
OVIDIU T. POP
For the real numbers α, β, α ≥ 0, we note


iff α ≤ β
 1,
(α,β)
µ
=
α−β

, iff α > β.
 1+
γβ
(1.4)
Remark 1.1. For the real numbers α and β, α ≥ 0, we have that
1 ≤ µ(α,β) .
Lemma 1.1. For the real numbers α and β, α ≥ 0, we have
0≤
k+α
≤ µ(α,β)
m+β
(1.5)
for any natural number m, m ≥ m0 and for any k ∈ {0, 1, . . . , m}.
In the following we consider the real numbers α and β, α ≥ 0, m0 and
µ(α,β) defined by (1.1) - (1.4).
(α,β)
Let the operators Pm
: C([0, µ(α,β) ]) → C([0, 1]), defined for any
function f ∈ C([0, µ(α,β) ]) by
m
X
k+α
(α,β)
,
(1.6)
(Pm f )(x) =
pm,k (x)f
m+β
k=0
for any natural number m, m ≥ m0 and any x ∈ [0, 1].
These operators are named Bernstein-Stancu operators, introduced
and studied in 1969 by D. D. Stancu in the paper [19]. In [19], the
domain of definition of the Bernstein-Stancu operators is C([0, 1]) and
the numbers α and β verify the condition 0 ≤ α ≤ β.
(0,0)
Remark 1.2. For α = β = 0, the operators Pm
operators.
are the Bernstein
Lemma 1.2. a) There exists m(2) ∈ N such that
1
,
Pm(α,β) ϕ2x (x) ≤
m
for any x ∈ [0, 1], any natural number m, m ≥ m(2).
(1.7)
APPROXIMATION OF B-DIFFERENTIABLE FUNCTIONS
17
b) There exists m(4) ∈ N such that
1
Pm(α,β) ϕ4x (x) ≤ 2 ,
m
(1.8)
for any x ∈ [0, 1], any natural number m, m ≥ m(4), where, for x ∈ [0, 1],
ϕx : [0, 1] → R, ϕx (t) = |t − x|, for any t ∈ [0, 1].
Proof. For the proof see [15].
(α,β)
Let the operators Dm : L1 [0, µ(α,β) ] → C([0, 1]), defined for any
function f ∈ L1 [0, µ(α,β) ] by
(α,β)
Dm
f
(x) = (m + 1)
m
X
k=0
Z1
pm,k (t)f
pm,k (x)
mt + α
m+β
dt,
(1.9)
0
for any natural number m, m ≥ m0 and any x ∈ [0, 1].
These operators are called Durrmeyer-Stancu type operators. In the
paper [17], the domain of definition of these operators is C([0, 1]) and
the numbers α and β verify the condition 0 ≤ α ≤ β.
(0,0)
Remark 1.3. For α = β = 0, the operators Dm
operators.
are the Durrmeyer
Lemma 1.3. a) There exists m0 (2) ∈ N such that
1
(α,β) 2
,
Dm
ϕx (x) ≤
m
(1.10)
for any x ∈ [0, 1], any natural number m, m ≥ m0 (2).
b) There exists m0 (4) ∈ N such that
1
(α,β) 4
Dm
ϕx (x) ≤ 2 ,
m
for any x ∈ [0, 1], any natural number m, m ≥ m0 (4).
(1.11)
18
OVIDIU T. POP
Proof. For x ∈ [0, 1], let ψx : [0, 1] → R be a function defined for any
t ∈ [0, 1] by ψx (t) = t − x. Then
2
(α,β)
Dm
ϕx
(x) = (m + 1)
m
X
Z1
pm,k (x)
k=0
=
=
m
m+β
m
m+β
2
(m + 1)
2 "
m
X
pm,k (t)
mt + α
−x
m+β
2
dt =
0
Z1
pm,k (x)
k=0
(Dm ψx2 )(x)+2
α − βx
pm,k (t) t − x +
m
2
dt =
0
α−βx
α−βx
(Dm ψx )(x)+
m
m
2
#
(Dm ψx0 )(x)
and similarly
4
(α,β)
Dm
ϕx
4 "
m
α − βx
Dm ψx3 (x)+
Dm ψx4 (x) + 4
(x) =
m+β
m
2
3
α − βx
α − βx
2
+6
Dm ψx (x) + 4
Dm ψx1 (x)+
m
m
#
4
α − βx
Dm ψx0 (x) .
+
m
Applying Proposition II.3 from [10], it results that
α − βx
lim m
(Dm ψx ) (x) = lim m
m→∞
m→∞
m
α − βx
m
2
Dm ψx0 (x) = 0
and
lim m
m→∞
2
α − βx
m
4−i
Dm ψxi (x) = 0,
i ∈ {0, 1, 2, 3}.
Next, taking into account Application 3.2 from [16], we get (1.10) and
(1.11).
APPROXIMATION OF B-DIFFERENTIABLE FUNCTIONS
19
For m a non zero natural number, let the operators Km : L1 ([0, 1]) →
C([0, 1]) defined for any function f ∈ L1 ([0, 1]) by
k+1
(Km f )(x) = (m + 1)
m
X
Zm+1
pm,k (x)
k=0
f (t)dt,
(1.12)
k
m+1
for any x ∈ [0, 1].
The operators Km , where m is a non zero natural number are named
Kantorovich operators, introduced and studied in 1930 by L. V. Kantorovich (see [20]).
Lemma 1.4. We have
1
Km ϕ2x (x) <
,
m
(1.13)
3
,
(1.14)
2m2
for any natural number m, m ≥ 3 and any x ∈ [0, 1], where, for x ∈ [0, 1],
ϕx : [0, 1] → R, ϕx (t) = |t − x| for any t ∈ [0, 1].
Km ϕ4x (x) <
Proof. For the proof see [16].
2. Preliminaries
In the following, let X and Y be compact real intervals. A function
f : X × Y → R is called a B-continuous function in (x0 , y0 ) ∈ X × Y iff
for any ε > 0, there exists δ > 0 such that
|∆f [(x, y), (x0 , y0 )]| < ε,
for any (x, y) ∈ X × Y , with |x − x0 | < δ and |y − y0 | < δ. 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 .
20
OVIDIU T. POP
A function f : X × Y → R is called a B-differentiable function in
(x0 , y0 ) ∈ X × Y iff it exists and if the limit is finite
∆f [(x, y), (x0 , y0 )]
.
(x,y)→(x0 ,y0 ) (x − x0 )(y − y0 )
lim
This limit is named the B-differential of f in the point (x0 , y0 ) and is
noted by DB f (x0 , y0 ).
The definition of B-continuity and B-differentiability were introduced
by K. Bögel in the papers [7] and [8].
The function f : X × Y → R is B-bounded on X × Y iff 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 |f : X × Y → R, f
bounded on X × Y } with the usual sup-norm k · k∞ , Bb (X × Y ) =
{f |f : X × Y → R, f B-bounded on X × Y } and we set kf kB =
sup
|∆f [(x, y), (s, t)]| where f ∈ Bb (X × Y ), Cb (X × Y ) = {f |f :
(x,y),(s,t)∈X×Y
X × Y → R, f B-continuous on X × Y } and Db (X × Y ) = {f |f :
X × Y → R, f B-differentiable on X × Y }.
We recall the following results (see [13]).
Theorem 2.1. Let f : X × Y → R be a function. If f admits the
00
in a neighborhood of the point (x0 , y0 ) ∈ X × Y and
derivatives fx0 , fxy
00
the derivative fxy
is continuous in (x0 , y0 ), then f is B-differentiable in
(x0 , y0 ) and
00
DB f (x0 , y0 ) = fxy
(x0 , y0 ).
(2.1)
Theorem 2.2. (K. Bögel). Let f : [a, b]×[a0 , b0 ] → R be a function. If
f is B-differentiable on [a, b] × [a0 , b0 ], there exists (ξ, η) ∈ (a, b) × (a0 , b0 )
such that
∆f (a, b), (a0 , b0 ) = (a0 − a)(b0 − b)DB f (ξ, η).
(2.2)
APPROXIMATION OF B-DIFFERENTIABLE FUNCTIONS
21
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 } ,
(2.3)
for any (δ1 , δ2 ) ∈ [0, ∞) × [0, ∞) is called the mixed modulus of smoothness.
Important properties of ωmixed were established by C. Badea, C. Cottin
and I. Badea in the papers [1] and [2].
The mixed modulus of smoothness for bivariate functions has properties similarly to the properties of the first modulus of smoothness for the
univariate functions. We will use the following lemma.
Lemma 2.3. Let f ∈ Bb (X × Y ) be a function. Then
ωmixed (f ; δ1 , δ2 ) ≤ ωmixed (f ; δ10 , δ20 )
(2.4)
for any (δ1 , δ2 ), (δ10 , δ20 ) ∈ [0, ∞) × [0, ∞) such that δ1 ≤ δ10 and δ2 ≤ δ20 ,
∆f [(x, y), (s, t)] ≤ ωmixed (f ; |x − s|, |y − t|)
(2.5)
and
∆f [(x, y), (s, t)] ≤ 1 + δ1−1 |x − s| 1 + δ2−1 |y − t| ωmixed (f ; δ1 , δ2 )
(2.6)
for any (x, y), (s, t) ∈ X × Y , any δ1 , δ2 > 0.
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 ), any (x, y) ∈ X × Y by
(U Lf )(x, y) = (L(f (·, y) + f (x, ∗) − f (·, ∗))) (x, y)
(2.7)
is called GBS operator (”Generalized Boolean Sum” operator) associated
to the operator L, where ”·” and ”∗” stand for the first and second
variable.
22
OVIDIU T. POP
3. Main results
In the following we consider the real numbers α1 , α2 , β1 , β2 , α1 ≥ 0,
α2 ≥ 0 and the numbers m1 , m2 , µ(α1 ,β1 ) , µ(α2 ,β2 ) defined by
(
max{1, −[βi ]}, iff βi ∈ R\Z
mi =
,
(3.1)
max{1, 1 − βi }, iff βi ∈ Z
(
max{1 + βi , {βi }}, iff βi ∈ R\Z
γi = mi + βi =
,
(3.2)
max{1 + βi , 1},
iff βi ∈ Z


iff αi ≤ βi
 1,
(αi ,βi )
=
,
(3.3)
µ
αi − βi

, iff αi > βi
 1+
γβi
where i ∈ {1, 2}.
In the following we consider the natural numbers m, n, m ≥ m1 and
n ≥ m2 .
(α ,β )(α ,β )
The bivariate operator of Bernstein-Stancu type Pm,n1 1 2 2 and the
(α ,β )(α ,β )
(α ,β )(α ,β )
GBS operator of Bernstein-Stancu type U Pm,n1 1 2 2 , Pm,n1 1 2 2 ,
(α ,β )(α ,β )
U Pm,n1 1 2 2 : C [0, µ(α1 ,β1 ) ] × [0, µ(α2 ,β2 ) ] → C([0, 1] × [0, 1]), are
defined for any function f ∈ C [0, µ(α1 ,β1 ) ] × [0, µ(α2 ,β2 ) ] , any (x, y) ∈
[0, 1] × [0, 1] and any natural numbers m, n, m ≥ m1 , n ≥ m2 by
(α1 ,β1 )(α2 ,β2 )
Pm,n
f (x, y) =
(3.4)
n
m X
X
k + α1 j + α2
,
,
pm,k (x)pn,j (y)f
=
m + β1 n + β2
k=0 j=0
(α1 ,β1 )(α2 ,β2 )
U Pm,n
f (x, y) =
(3.5)
m X
n
X
k + α1
j + α2
=
pm,k (x)pn,j (y) f
, y + f x,
−
m
+
β
n
+
β
1
2
k=0 j=0
k + α1 j + α2
−f
,
m + β1 n + β2
(see [6] or [15]).
APPROXIMATION OF B-DIFFERENTIABLE FUNCTIONS
23
(α ,β )(α ,β )
The bivariate operator of Durrmeyer-Stancu type Dm,n1 1 2 2 and
(α ,β )(α ,β )
(α ,β )(α ,β )
GBS operator of Durrmeyer-Stancu type U Dm,n1 1 2 2 , Dm,n1 1 2 2 ,
(α ,β )(α ,β )
U Dm,n1 1 2 2 : L1 [0, µ(α1 ,β1 ) ] × [0, µ(α2 ,β2 ) ] → C([0, 1] × [0, 1]) are de
fined for any function f ∈ C [0, µ(α1 ,β1 ) ] × [0, µ(α2 ,β2 ) ] , any (x, y) ∈
[0, 1] × [0, 1] and any natural numbers m, n, m ≥ m1 , n ≥ m2 by
m X
n
X
(α1 ,β1 )(α2 ,β2 )
f (x, y) = (m + 1)(n + 1)
Dm,n
pm,k (x)pn,j (y)·
(3.6)
k=0 j=0
Z1
Z1
·
pm,k (s)pn,j (t)f
0
ms + α1 nt + α2
,
m + β1 n + β2
dsdt,
0
(α1 ,β1 )(α2 ,β2 )
U Dm,n
f (x, y) =
= (m+1)(n+1)
m X
n
X
(3.7)
Z1 Z1
pm,k (x)pn,j (y)
k=0 j=0
pm,k (s)pn,j (t)·
0 0
nt + α2
ms + α1 nt + α2
ms+α1
, y +f x,
,
−f
dsdt
· f
m + β1
n + β2
m + β1 n + β2
(see [18]).
For m, n non zero natural numbers, let the bivariate operator of Kantorovich type Km,n and the GBS operator of Kantorovich type U Km,n ,
Km,n , U Km,n : L1 ([0, 1] × [0, 1]) → C([0, 1] × [0, 1]) are defined for any
function f ∈ L1 ([0, 1] × [0, 1]) and any (x, y) ∈ [0, 1] × [0, 1] by
(Km,n f )(x, y) =
(3.8)
k+1
= (m + 1)(n + 1)
m X
n
X
k=0 j=0
j+1
Zm+1 Zn+1
f (s, t)dsdt
pm,k (x)pn,j (y)
j
k
m+1 n+1
24
OVIDIU T. POP
and
(U Km,n f )(x, y) =
= (m + 1)(n + 1)
(3.9)
m X
n
X
pm,k (x)pn,j (y)·
k=0 j=0
k+1
j+1
Zm+1 Zn+1
·
[f (s, y) + f (x, t) − f (s, t)] dsdt.
j
k
m+1 n+1
We prove the following theorem for estimating the rate of the convergence
of the B-differentiable functions.
Theorem 3.1. 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)| ≤
(3.10)
≤ |f (x, y)||1−(Le00 )(x, y)|+3kDB f k∞ (L |· − x| |∗ − y|)(x, y)+
+ (L |· − x| |∗ − y|)(x, y) + δ1−1 (L(· − x)2 |∗ − y|)(x, y)+
+ δ2−1 (L |· − x| (∗ − y)2 )(x, y)+
+
δ1−1 δ2−1 (L(·
− x) (∗ − y) )(x, y) ωmixed (DB f ; δ1 , δ2 )
2
|f (x, y) − (U Lf )(x, y)| ≤
2
(3.11)
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)+
p
+ δ2−1 (L(· − x)2 (∗ − y)4 )(x, y)+
2
2
−1 −1
+ δ1 δ2 (L(· − x) (∗ − y) )(x, y) ωmixed (DB f ; δ1 , δ2 ),
APPROXIMATION OF B-DIFFERENTIABLE FUNCTIONS
25
where e00 : X × Y → R, e00 (x, y) = 1, for any (x, y) ∈ X × Y .
Proof. Let (s, t) ∈ X × Y . We start from identity
f (s, y) + f (x, t) − f (s, t) = f (x, y) − ∆f [(x, y), (s, t)]
and applying the L operator, we have
(U Lf )(x, y) = f (x, y)(Le00 )(x, y) − (L∆f [(x, y), (·, ∗)]) (x, y),
from where
|f (x, y) − (U Lf )(x, y)| ≤
(3.12)
≤ |f (x, y)||1 − (Le00 )(x, y)| + |(L∆f [(x, y), (·, ∗)])(x, y)| .
By Theorem 2.2, there exists (ξ, η) ∈ (x, y) × (s, t) such that
∆f [(x, y), (s, t)] = (s − x)(t − y)DB f (ξ, η)
and on the other hand, we have
DB f (ξ, η) = ∆DB f [(x, y), (ξ, η)] + DB f (x, η) + DB f (ξ, y) − DB f (x, y).
Since DB f ∈ Bb (X × Y ), from the relations above, it results that
|(L∆f [(x, y), (·, ∗)])(x, y)| = |(L(s − x)(t − y)DB f (ξ, η))(x, y)| =
= (L(s − x)(t − y)(∆DB f [(x, y), (ξ, η)] + DB f (x, η) + DB f (ξ, y)−
− DB f (x, y)))(x, y) ≤ (L(s − x)(t − y)∆DB f [(x, y), (ξ, η)])(x, y)+
+ (L(s − x)(t − y)(DB f (x, η) + DB f (ξ, y) − DB f (x, y)))(x, y) ≤
≤ (L|s − x||t − y||∆DB f [(x, y), (ξ, η)]|)(x, y)+
+ L|s − x||t − y|(|DB f (x, η)| + |DB f (ξ, y)| + |DB f (x, y)| (x, y) ≤
≤ L|s − x||t − y|ωmixed (DB f ; |ξ − x|, |η − y|) (x, y)+
+ 3kDB f k∞ L|s − x||t − y| (x, y).
26
OVIDIU T. POP
Taking into account that ωmixed verifies the relations (2.4) - (2.6) and
that the inequalities |ξ − x| ≤ |s − x|, |η − y| ≤ |t − y| hold, we have that
(L∆f [(x, y), (·, ∗)])(x, y) ≤
≤ (L|s−x||t−y|ωmixed (DB f ; |s−x|, |t−y|)) (x, y)+
3kDB f k∞ (L|s − x||t − y|)(x, y)
≤ (L|s − x||t − y|(1 + δ1−1 |s − x|)(1 + δ2−1 |t − y|)·
· ωmixed (DB f ; δ1 , δ2 ))(x, y) + 3kDB f k∞ (L|s−x||t−y|)(x, y) =
= 3kDB f k∞ (L|s − x||t − y|)(x, y) + (L(|s − x||t − y|+
δ1−1 (s − x)2 |t − y| + δ2−1 |s − x|(t − y)2 + δ1−1 δ2−1 (s − x)2 (t − y)2 ))(x, y)·
· ωmixed (DB f ; δ1 , δ2 ),
from where
+ δ2−1
|(L∆f [(x, y) , (·, ∗)]) (x, y)| ≤ 3 kDB f k∞ (L |· − x| |∗ − y|) +
(3.13)
+ (L |s − x| |t − y|) (x, y) + δ1−1 L (s − x)2 |t − y| (x, y) +
L |s − x| (t − y)2 (x, y) + δ1−1 δ2−1 L (s − x)2 (t − y)2 (x, y)
ωmixed (DB f ; δ1 , δ2 )
By Cauchy’s inequality, it follows that
p
(L∆f [(x, y), (·, ∗))(x, y) ≤ 3kDB f k∞ (L(· − x)2 (∗ − y)2 )(x, y)+
(3.14)
p
p
+
(L(· − x)2 (∗ − y)2 )(x, y) + δ1−1 (L(· − x)4 (∗ − y)2 )(x, y)+
p
+ δ2−1 (L(· − x)2 (∗ − y)4 )(x, y)+
−1 −1
2
2
+ δ1 δ2 (L(· − x) (∗ − y) )(x, y) ωmixed (DB f ; δ1 , δ2 ).
From (3.12) - (3.14), the inequalities (3.10) and (3.11) result.
APPROXIMATION OF B-DIFFERENTIABLE FUNCTIONS
27
Corollary 3.2. Let L : Cb (X × Y ) → B (X × Y ) be a linear positive
operator satisfying
L(· − x)2i (∗ − y)2j (x, y) = L(· − x)2i (x, y) L(∗ − y)2j (x, y)
(3.15)
for any (x, y) ∈ X × Y , any i, j ∈ {1, 2} and U L : Cb (X×Y ) → B(X×Y )
the associated GBS operator. Then, for any f ∈ Db (X × Y ) with DB f ∈
Bb (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)|+
(3.16)
p
+ 3kDB k∞ (L(· − x)2 ) (x, y) (L(∗ − y)2 ) (x, y)+
hp
+
(L(· − x)2 ) (x, y) (L(∗ − y)2 ) (x, y)+
p
+ δ1−1 (L(· − x)4 ) (x, y) (L(∗ − y)2 ) (x, y)+
p
+ δ2−1 (L(· − x)2 ) (x, y) (L(∗ − y)4 ) (x, y)+
i
+ δ1−1 δ2−1 L(· − x)2 (x, y) L(∗ − y)2 (x, y) ωmixed (DB f ; δ1 , δ2 ).
Proof. It results from relations (3.11) and (3.15).
In the following, we give three application of the Corollary 3.2, where
m(2), m(4), m0 (2), m0 (4) are the ones from Lemma 1.2 and Lemma 1.3.
Remark 3.1. On verify immediately that the operators
(α1 ,β1 )(α2 ,β2 )
(α1 ,β1 )(α2 ,β2 )
Pm,n
Dm,n
m≥m1 ,
m≥m1
n≥m2
n≥m2
and (Km,n )m,n≥1 fulfil the condition (3.15).
Theorem 3.3. For any natural numbers m, n, m ≥ max{m(2), m(4)},
n ≥ max{n(2), n(4)}, any function f ∈ Db ([0, 1] × [0, 1]) with
DB f ∈ B([0, 1] × [0, 1]), any (x, y) ∈ [0, 1] × [0, 1], we have
1
(α1 ,β1 )(α2 ,β2 )
f (x, y) − U Pm,n
+
(3.17)
f (x, y) ≤ 3kDB f k∞ √
mn
1
1
1
−1
−1
−1 −1 1
√ +δ
√ +δ δ
+ √
+δ
ωmixed (DB f ; δ1 , δ2 )
mn 1 m n 2 n m 1 2 mn
28
OVIDIU T. POP
for any δ1 , δ2 > 0 and
(α1 ,β1 )(α2 ,β2 )
f (x, y) − U Pm,n
f (x, y) ≤
1
1
1
√
.
≤ 3kDB f k∞ + 4ωmixed DB f ; √ , √
m
n
mn
(3.18)
Proof. For the first inequality (3.17), we apply Corollary 3.2 and Lemma
1
1.2. The inequality (3.18) is obtained from (3.17) by choosing δ1 = √
m
1
and δ2 = .
n
Remark 3.2. If α1 = α2 = β1 = β2 = 0, then µ(0,0) = 1, m1 = m2 = 1
and we obtain the bivariate (Bm,n )m,n≥1 operators of Bernstein and the
GBS (U Bm,n )m,n≥1 operators of Bernstein (see [2]).
Remark 3.3. If p, q are natural numbers, α1 = α2 = 0, β1 = −p, β2 =
−q and changing m by m + p, n by n + q, then µ(0,−p) = 1 + p, µ(0,−q)
=
em,n,p,q
1 + q, m1 = 1 + p, m2 = 1 + q, we obtain the bivariate B
m,n≥1
e
operators
operators of Bernstein-Schurer and the GBS U Bm,n,p,q
m,n≥1
of Bernstein-Schurer (see [5]).
Remark 3.4. If α1 ≥ 0, α2 ≥ 0, p, q are natural numbers and changing β1 by β1 − p, β2 by
β2 − q,m by m + p, n by n + q, we obtain the
(α
,β
,α
,β
)
1
1
2
2
operators of Schurer-Stancu and the GBS
bivariate Sem,n,p,q
m≥m1
n≥m2
(α1 ,β1 ,α2 ,β2 )
operators of Schurer-Stancu (see [4]).
U Sem,n,p,q
m≥m
1
n≥m2
Theorem 3.4. For any natural numbers m, n, m ≥ max{m0 (2), m0 (4)},
n ≥ max{n0 (2), n0 (4)}, any function f ∈ Db ([0, 1] × [0, 1]) with DB f ∈
B([0, 1] × [0, 1]), any (x, y) ∈ [0, 1] × [0, 1], we have
1
(α1 ,β1 )(α2 ,β2 )
f (x, y) − U Dm,n
+
(3.19)
f (x, y) ≤ 3kDB f k∞ √
mn
1
1
1
−1
−1
−1 −1 1
√ +δ
√ +δ δ
+ √
+δ
ωmixed (DB f ; δ1 , δ2 )
mn 1 m n 2 n m 1 2 mn
APPROXIMATION OF B-DIFFERENTIABLE FUNCTIONS
for any δ1 , δ2 > 0 and
(α1 ,β1 )(α2 ,β2 )
f (x, y) − U Dm,n
f (x, y) ≤
1
1
1
√
.
≤ 3kDB f k∞ + 4ωmixed DB f ; √ , √
m
n
mn
29
(3.20)
Proof. For the first inequality (3.19), we apply Corollary 3.2 and Lemma
1
1.3. The inequality (3.20) is obtained from (3.19) by choosing δ1 = √
m
1
and δ2 = √ .
n
Remark 3.5. If α1 = α2 = β1 = β2 = 0, we obtain the bivariate
(Dm,n )m,n≥1 operators of Durrmeyer and GBS (U Dm,n )m,n≥1 operators
of Durrmeyer.
Remark 3.6. If p, q are natural numbers, α1 = α2 = 0, β1 = −p,
β2 = −q
m by m + p, n by n + q, we obtain the bi and changing
e m,n,p,q
variate D
operators of Durrmeyer-Schurer and the GBS
m,n≥1
e m,n,p,q
operators of Durrmeyer-Schurer.
UD
m,n≥1
Remark 3.7. If α1 ≥ 0, α2 ≥ 0, p, q are natural numbers and changing β1 by β1 − p, β
2 by β2 − q, m by m + p,n by n + q,we obtain the bivari(α
,β
)(α
,β
)
1
1
2
2
e m,n,p,q
operators of Durrmeyer-Schurer-Stancu and the
ate D
m≥m1
n≥m2
(α1 ,β1 )(α2 ,β2 )
e m,n,p,q
GBS U D
operators of Durrmeyer-Schurer-Stancu.
m≥m
1
n≥m2
Theorem 3.5. For any natural numbers m, n ≥ 3, any function
f ∈ Db ([0, 1] × [0, 1]) with DB f ∈ B([0, 1] × [0, 1]), any (x, y) ∈ [0, 1] ×
[0, 1], we have
1
1
|f (x, y) − (U Km,n f )(x, y)| ≤ 3kDB f k∞ √
+ √
+
(3.21)
mn
mn
r
r
3 −1 1
3 −1 1
−1 −1 1
√ +
√ + + δ1 δ2
+
δ
δ
ωmixed (DB f ; δ1 , δ2 )
2 1 m n
2 2 m n
mn
30
OVIDIU T. POP
for any δ1 , δ2 > 0 and
|f (x, y) − (U Km,n f )(x, y)| <
(3.22)
√
1
1
1
√
< 3kDB f k∞ + (2 + 6) ωmixed DB f ; √ , √
.
m
n
mn
Proof. For the inequality (3.21), we apply Corollary 3.2 and Lemma 1.4.
1
The inequality (3.22) is obtained from (3.21) by choosing δ1 = √ and
m
1
δ2 = √ .
n
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31
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Received 6 July 2006
National College ”Mihai Eminescu”, 5 Mihai Eminescu Street, Satu
Mare 440014, Romania
Vest University ”Vasile Goldiş” of Arad, Branch of Satu Mare, 26
Mihai Viteazul Street, Satu Mare 440030, Romania
E-mail address: [email protected]