Volume 10 (2009), Issue 1, Article 7, 8 pp.
APPROXIMATION OF B-CONTINUOUS AND B-DIFFERENTIABLE FUNCTIONS
BY GBS OPERATORS DEFINED BY INFINITE SUM
OVIDIU T. POP
NATIONAL C OLLEGE "M IHAI E MINESCU "
5 M IHAI E MINESCU S TREET
S ATU M ARE 440014, ROMANIA
ovidiutiberiu@yahoo.com
Received 27 June, 2008; accepted 18 March, 2009
Communicated by S.S Dragomir
A BSTRACT. In this paper we start from a class of linear and positive operators defined by infinite
sum. We consider the associated GBS operators and we give an approximation of B-continuous
and B-differentiable functions with these operators. Through particular cases, we obtain statements verified by the GBS operators of Mirakjan-Favard-Szász, Baskakov and Meyer-König and
Zeller.
Key words and phrases: Linear positive operators, GBS operators, B-continuous and B-differentiable functions, approximation of B-continuous and B-differentiable functions by GBS operators.
2000 Mathematics Subject Classification. 41A10, 41A25, 41A35, 41A36, 41A63.
1. I NTRODUCTION
In this section, we recall some notions and results which we will use in this article. Let N be
the set of positive integers and N0 = N ∪ {0}.
In the following, let X and Y be real intervals.
A function f : X × Y → R is called a B-continuous function in (x0 , y0 ) ∈ X × Y if and
only if
lim
∆f [(x, y), (x0 , y0 )] = 0,
(x,y)→(x0 ,y0 )
where
∆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 a B-continuous function on X × Y if and only if it is
B-continuous in any point of X × Y .
A function f : X × Y → R is called a B-differentiable function in (x0 , y0 ) ∈ X × Y if and
only if it exists and if the limit is finite
∆f [(x, y), (x, y0 )]
.
(x,y)→(x0 ,y0 ) (x − x0 )(y − y0 )
lim
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2
OVIDIU T. P OP
This limit is called the B-differential of f in the point (x0 , y0 ) and is noted by DB f (x0 , y0 ).
A function f : X × Y → R is called a B-differentiable function on X × Y if and only if it
is B-differentiable in any point of X × Y .
The definition of B-continuity and B-differentiability was introduced by K. Bögel in the
papers [8] and [9].
The function f : X × Y → R is B-bounded on X × Y if and only if there exists k > 0 so
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 is B-bounded on X × Y },
Cb (X × Y ) = {f |f : X × Y → R, f is B-continuous on X × Y } and Db (X × Y ) = {f |f :
X × Y → R, f is 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 }
for any (δ1 , δ2 ) ∈ [0, ∞) × [0, ∞) is called the mixed modulus of smoothness.
Theorem 1.1. Let X and Y be compact real intervals and f ∈ Bb (X × Y ).
Then lim ωmixed (f ; δ1 , δ2 ) = 0 if and only if f ∈ Cb (X × Y ).
δ1 ,δ2 →0
For any x ∈ X consider the function ϕx : X → R, defined by ϕx (t) = |t − x|, for any t ∈ X.
For additional information, see the following papers: [1], [3], [15] and [19].
Let m ∈ N and the operator Sm : C2 ([0, ∞)) → C([0, ∞)) defined for any function f ∈
C2 ([0, ∞)) by
∞
X
k
(mx)k
−mx
f
,
(1.1)
(Sm f )(x) = e
k!
m
k=0
f (x)
2
x→∞ 1+x
for any x ∈ [0, ∞), where C2 ([0, ∞)) = f ∈ C([0, ∞)) : lim
exists and is finite . The
operators (Sm )m≥1 are called the Mirakjan-Favard-Szász operators, introduced in 1941 by G.
M. Mirakjan in the paper [13].
These operators were intensively studied by J. Favard in 1944 in the paper [11] and O. Szász
in the paper [20].
From [18], the following three lemmas result.
Lemma 1.2. For any m ∈ N, we have that
(1.2)
x
Sm ϕ2x (x) =
,
m
(1.3)
3mx2 + x
Sm ϕ4x (x) =
m3
for any x ∈ [0, ∞) and
(1.4)
a
Sm ϕ2x (x) ≤
,
m
(1.5)
a(3a + 1)
Sm ϕ4x (x) ≤
m2
for any x ∈ [0, a], where a > 0.
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Let m ∈ N and the operator Vm : C2 ([0, ∞)) → C([0, ∞)), defined for any function f ∈
C2 ([0, ∞)) by
k ∞ X
x
m+k−1
k
−m
(1.6)
(Vm f )(x) = (1 + x)
f
k
1+x
m
k=0
for any x ∈ [0, ∞).
The operators (Vm )m≥1 are called Baskakov operators, introduced in 1957 by V. A. Baskakov
in the paper [5].
Lemma 1.3. For any m ∈ N, we have that
x(1 + x)
(1.7)
Vm ϕ2x (x) =
,
m
3(m + 2)x4 + 6(m + 2)x3 + (3m + 7)x2 + x
Vm ϕ4x (x) =
m3
for any x ∈ [0, ∞) and
a(1 + a)
(1.9)
Vm ϕ2x (x) ≤
,
m
(1.8)
a(9a3 + 18a2 + 10a + 1)
Vm ϕ4x (x) ≤
m2
for any x ∈ [0, a], where a > 0.
(1.10)
W. Meyer-König and K. Zeller have introduced a sequence of linear positive operators in
paper [12]. After a slight adjustment, given by E. W. Cheney and A. Sharma in [10], these
operators take the form Zm : B([0, 1)) → C([0, 1)), defined for any function f ∈ B([0, 1)) by
∞ X
m+k
k
m+1 k
,
(1.11)
(Zm f )(x) =
(1 − x)
x f
m
+
k
k
k=0
for any m ∈ N and for any x ∈ [0, 1).
These operators are called the Meyer-König and Zeller operators.
In the following we consider Zm : C([0, 1]) → C([0, 1]), for any m ∈ N.
Lemma 1.4. For any m ∈ N and any x ∈ [0, 1], we have that
x(1 − x)2
2x
2
(1.12)
Zm ϕx (x) ≤
1+
m+1
m+1
and
2
.
(1.13)
Zm ϕ2x (x) ≤
m
The inequality of Corollary 5 from [4], in the condition (1.14) becomes inequality (1.15).
Inequality (1.16) is demonstrated in [16].
Theorem 1.5. 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. Supposing that the operator L has the
property
(1.14)
L(· − x)2i (∗ − y)2j (x, y) = L(· − x)2i (x, y) L(∗ − y)2j (x, y)
for any (x, y) ∈ X × Y and any i, j ∈ {1, 2}, where "·" and "∗" stand for the first and second
variable. Then:
J. Inequal. Pure and Appl. Math., 10(1) (2009), Art. 7, 8 pp.
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OVIDIU T. P OP
(i) For any function f ∈ Cb (X × Y ), any (x, y) ∈ X × Y and any δ1 , δ2 > 0, we have that
(1.15) |f (x, y) − (U Lf )(x, y)| ≤ |f (x, y)||1 − (Le00 )(x, y)|
h
p
p
+ (Le00 )(x, y)+δ1−1 (L(·−x)2 )(x, y)+δ2−1 (L(∗−y)2 )(x, y)
i
p
+ δ1−1 δ2−1 (L(· − x)2 )(x, y)(L(∗ − y)2 )(x, y) ωmixed (f ; δ1 , δ2 ).
(ii) For any f ∈ Db (X ×Y ) with DB f ∈ B(X ×Y ), any (x, y) ∈ X ×Y and any δ1 , δ2 > 0,
we have that
(1.16)
|f (x, y) − (U Lf )(x, y)|
p
≤ |f (x, y)||1 − (Le00 )(x, y)| + 3kDB f 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
−1 −1
2
2
+ δ1 δ2 (L(· − x) )(x, y)(L(∗ − y) )(x, y) ωmixed (DB f ; δ1 , δ2 ).
2. P RELIMINARIES
Let I, J, K ⊂ R be intervals, J ⊂ K and I ∩ J 6= ∅. We consider the sequence of nodes
((xm,k )k∈N0 )m≥1 so that xm,k ∈ I ∩ J, k ∈ N0 , m ∈ N and the functions ϕm,k : K → R with
the property that ϕm,k (x) ≥ 0, for any k ∈ N0 , m ∈ N and x ∈ J.
Definition 2.1. If m ∈ N, we define the operator L∗m : E(I) → F (K) by
∞
X
∗
(2.1)
(Lm f ) (x) =
ϕm,k (x)f (xm,k )
k=0
for any function f ∈ E(I) and any x ∈ K, where E(I) and F (K) are subsets of the set of real
functions defined on I, respectively on K.
Proposition 2.1. The operators (L∗m )m≥1 are linear and positive on E(I ∩ J).
Proof. The proof follows immediately.
L∗m,n
Definition 2.2. If m, n ∈ N, the operator
: E(I × I) → F (K × K) defined for any
function f ∈ E(I × I) and any (x, y) ∈ K × K by
∞ X
∞
X
∗
(2.2)
Lm,n f (x, y) =
ϕm,k (x)ϕn,j (y)f (xm,k , xn,j )
k=0 j=0
∗
is called the bivariate operator of L - type.
Proposition 2.2. The operators L∗m,n m,n≥1 are linear and positive on E[(I × I) ∩ (J × J)].
Proof. The proof follows immediately.
Definition 2.3. If m, n ∈ N, the operator U L∗m,n : E(I × I) → F (K × K) defined for any
function f ∈ E(I × I) and any (x, y) ∈ K × K by
(2.3)
∞ X
∞
X
U L∗m,n f (x, y) =
ϕm,k (x)ϕn,j (y) f (xm,k , y) + f (x, xn,j ) − f (xm,k , xn,j )
k=0 j=0
is called a GBS operator of L∗ - type.
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3. M AIN R ESULTS
Lemma 3.1. For any m, n ∈ N, i, j ∈ N0 and (x, y) ∈ K × K, the identity
(3.1)
L∗m,n (· − x)2i (∗ − y)2j (x, y) = L∗m (· − x)2i (x) L∗n (∗ − y)2j (y)
holds.
Proof. We have that
L∗m,n (·
2i
2j
− x) (∗ − y)
(x, y) =
=
∞ X
∞
X
ϕm,k (x)ϕn,j (y)(xm,k − x)2i (xn,j − y)2j
k=0 j=0
∞
X
2i
ϕm,k (x)(xm,k − x)
k=0
∞
X
ϕn,j (y)(xn,j − y)2j
j=0
= L∗m (· − x)2i (x) L∗n (∗ − y)2j (y),
so (3.1) holds.
pFor the operators constructed in this section, we note that δm (x) =
(L∗m ϕ4x ) (x), where x ∈ I ∩ J, m ∈ N, m 6= 0.
Then, by taking Lemma 3.1 into account, Theorem 1.5 becomes:
p
(L∗m ϕ2x ) (x), δm,x =
Theorem 3.2.
(i) For any function f ∈ Cb (I × I), any (x, y) ∈ (I × I) ∩ (J × J), any m, n ∈ N, any
δ1 , δ2 > 0, we have that
(3.2) |f (x, y) − (U L∗m,n f )(x, y)|
≤ |f (x, y)||1 − (Le00 )(x, y)| + (Le00 )(x, y) + δ1−1 δm (x) + δ2−1 δn (y)
+ δ1−1 δ2−1 δm (x)δn (y) ωmixed (f ; δ1 , δ2 ).
(ii) For any function f ∈ Db (I × I) with DB f ∈ B(I × I), any (x, y) ∈ (I × I) ∩ (J × J),
any m, n ∈ N, any δ1 , δ2 > 0, we have that
(3.3) |f (x, y) − (U L∗ f )(x, y)| ≤ |f (x, y)||1 − (Le00 )(x, y)|
+ 3kDB f k∞ δm (x)δn (y) + δm (x)δn (y) + δ1−1 δm,x δn (y)
2
(x)δn2 (y) ωmixed (DB f ; δ1 , δ2 ).
+ δ2−1 δm (x)δn,y + δ1−1 δ2−1 δm
In the following, we give examples of operators and of the associated GBS operators.
Application 1. If I = J = K = [0, ∞), E(I) = C2 ([0, ∞)), F (K) = C([0, ∞)), ϕm,k (x) =
k
k
, xm,k = m
, x ∈ [0, ∞), m, k ∈ N0 , m 6= 0, then we obtain the Mirakjan-Favarde−mx (mx)
k!
Szász operators.
Theorem 3.3. Let a, b ∈ R, a > 0 and b > 0. Then:
(i) For any function f ∈ C([0, ∞) × [0, ∞)), any (x, y) ∈ [0, a] × [0, b] and m, n ∈ N, we
have that
√ √ 1
1
(3.4)
|f (x, y)−(U Sm,n f )(x, y)| ≤ 1+ a 1+ b ωmixed f ; √ , √
.
m
n
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OVIDIU T. P OP
(ii) For any function f ∈ Db ([0, ∞) × [0, ∞)) ∩ C([0, ∞) × [0, ∞)) with DB f ∈ B([0, a] ×
[0, b]), any (x, y) ∈ [0, a] × [0, b], any m, n ∈ N, we have that
"
√
√
(3.5) |f (x, y) − (U Sm,n f )(x, y)| ≤ ab 3kDB f k∞ + 1 + 3a + 1
+
√
#
√ 1
1
1
√
.
3b + 1 + ab ωmixed DB f ; √ , √
mn
n
m
Proof. It results from Theorem 3.2, by choosing δ1 =
√1
m
, δ2 =
√1
n
and Lemma 1.2.
Theorem 3.4. If f ∈ C([0, ∞) × [0, ∞)), then the convergence
(3.6)
lim (U Sm,n f )(x, y) = f (x, y)
m,n→∞
is uniform on any compact [0, a] × [0, b], where a, b > 0.
Proof. It results from Theorem 1.1 and Theorem 3.3.
Application 2. If I = J = K = [0, ∞), E(I) = C2 ([0, ∞)), F (K) = C([0, ∞)), ϕm,k (x) =
x k
k
, xm,k = m
, x ∈ [0, ∞), m, k ∈ N0 , m 6= 0, then we obtain the
(1 + x)−m m+k−1
1+x
k
Baskakov operators.
Theorem 3.5. Let a, b ∈ R, a > 0 and b > 0. Then:
(i) For any function f ∈ C([0, ∞) × [0, ∞)), any (x, y) ∈ [0, a] × [0, b] and any m, n ∈ N,
we have that
(3.7)
|f (x, y) − (U Vm,n f )(x, y)|
p
p
1
1
≤ 1 + a(1 + a) 1 + b(1 + b) ωmixed f ; √ , √
.
m
n
(ii) For any function f ∈ Db ([0, ∞) × [0, ∞)) ∩ C([0, ∞) × [0, ∞)) with DB f ∈ B([0, a] ×
[0, b]), any (x, y) ∈ [0, a] × [0, b], any m, n ∈ N, we have that
(
p
(3.8) |f (x, y) − (U Vm,n f )(x, y)| ≤ ab(1 + a)(1 + b) 3kDB k∞
h
√
√
+ 1 + 9a3 + 18a2 + 10a + 1 + 9b3 + 18b2 + 10b + 1
)
i
p
1
1
1
√
.
+ ab(1 + a)(1 + b) ωmixed DB f ; √ , √
m
n
mn
Proof. It results from Theorem 3.2, by choosing δ1 =
√1
m
, δ2 =
√1
n
and Lemma 1.3.
Theorem 3.6. If f ∈ C([0, ∞) × [0, ∞)), then the convergence
(3.9)
lim (U Vm,n f )(x, y) = f (x, y)
m,n→∞
is uniform on any compact [0, a] × [0, b], where a, b > 0.
Proof. It results from Theorem 1.1 and Theorem 3.5.
m+k
k
Application 3. If I = J = K = [0, 1], E(I) = F (K) = C([0, 1]), ϕm,k (x) =
(1 −
k
m+1 k
x)
x , xm,k = m , x ∈ [0, 1], m, k ∈ N0 , m 6= 0, then we obtain the Meyer-König and Zeller
operators.
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A PPROXIMATION
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B-C ONTINUOUS AND B- DIFFERENTIABLE F UNCTIONS
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Theorem 3.7. For any function f ∈ C([0, 1] × [0, 1]), any (x, y) ∈ [0, 1] × [0, 1] and any
m, n ∈ N, we have that
√
1
1
.
(3.10)
|f (x, y) − (U Zm,n f )(x, y)| ≤ (3 + 2 2)ωmixed f ; √ , √
n
m
Proof. It results from Theorem 3.2, by choosing δ1 =
√1
m
, δ2 =
√1
n
and Lemma 1.4.
Theorem 3.8. If f ∈ C([0, 1] × [0, 1]), then the convergence
lim (U Zm,n f )(x, y) = f (x, y)
(3.11)
m,n→∞
is uniform on [0, 1] × [0, 1].
Proof. It results from Theorem 1.1 and Theorem 3.7.
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