Journal of Inequalities in Pure and
Applied Mathematics
APPROXIMATION OF B-CONTINUOUS AND B-DIFFERENTIABLE
FUNCTIONS BY GBS OPERATORS OF BERNSTEIN
BIVARIATE POLYNOMIALS
volume 7, issue 3, article 92,
2006.
OVIDIU T. POP AND MIRCEA FARCAŞ
Vest University "Vasile Goldiş" of Arad Branch of Satu Mare
26 Mihai Viteazu Street
Satu Mare 440030, Romania.
EMail: ovidiutiberiu@yahoo.com
National College "Mihai Eminescu"
5 Mihai Eminescu Street
Satu Mare 440014, Romania.
EMail: mirceafarcas2005@yahoo.com
Received 13 September, 2005;
accepted 03 June, 2006.
Communicated by: S.S. Dragomir
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
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Abstract
In this paper we give an approximation of B-continuous and B-differentiable
functions by GBS operators of Bernstein bivariate polynomials.
2000 Mathematics Subject Classification: 41A10, 41A25, 41A35, 41A36, 41A63.
Key words: Linear positive operators, Bernstein bivariate polynomials, GBS operators, B-differentiable functions, approximation of B-differentiable functions by GBS operators, mixed modulus of smoothness.
Approximation of B-Continuous
and B-Differentiable Functions
by GBS Operators of Bernstein
Bivariate Polynomials
Contents
1
Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
3
6
Ovidiu T. Pop and Mircea Farcaş
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J. Ineq. Pure and Appl. Math. 7(3) Art. 92, 2006
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1.
Preliminaries
In this section, we recall some results which we will use in this article.
In the following, let X and Y be compact real intervals. A function f :
X × Y → R is called a B-continuous (Bögel-continuous) function in (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 a B-differentiable (Bögel-differentiable)
function in (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 in the point (x0 , y0 ) and is denoted
by DB f (x0 , y0 ).
The definitions of B-continuity and B-differentiability were introduced by
K. Bögel in the papers [5] and [6].
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
Approximation of B-Continuous
and B-Differentiable Functions
by GBS Operators of Bernstein
Bivariate Polynomials
Ovidiu T. Pop and Mircea Farcaş
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for any (x, y), (s, t) ∈ X × Y .
J. Ineq. Pure and Appl. Math. 7(3) Art. 92, 2006
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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)) |,
(x,y),(s,t)∈X×Y
where
f ∈ Bb (X × Y ),
Cb (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
Approximation of B-Continuous
and B-Differentiable Functions
by GBS Operators of Bernstein
Bivariate Polynomials
Ovidiu T. Pop and Mircea Farcaş
(1.1) 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.
For related topics, see [1], [2], [3] and [10].
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
(1.2)
(U Lf )(x, y) = (L(f (·, y) + f (x, ∗) − f (·, ∗))) (x, y)
is called the GBS operator ("Generalized Boolean Sum" operator) associated to
the operator L, where "·" and "∗" stand for the first and second variable.
Let the functions eij : X × Y → R, (eij )(x, y) = xi y j for any (x, y) ∈
X × Y , where i, j ∈ N. The following theorem is proved in [1].
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Theorem 1.1. 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 ∈ Cb (X × Y ), any (x, y) ∈ (X × Y ) and any δ1 , δ2 > 0, we have
(1.3)
|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)
p
−1 −1
+ δ1 δ2
(L(· − x)2 (∗ − y)2 ) (x, y) ωmixed (f ; δ1 , δ2 ).
In the following, we need the following theorem for estimating the rate of
the convergence of the B-differentiable functions (see [11]).
Theorem 1.2. 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
(1.4) |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)
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 ).
Approximation of B-Continuous
and B-Differentiable Functions
by GBS Operators of Bernstein
Bivariate Polynomials
Ovidiu T. Pop and Mircea Farcaş
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2.
Main Results
Let the sets ∆2 = {(x, y) ∈ R × R|x, y ≥ 0, x + y ≤ 1} and F(∆2 ) = {f |f :
∆2 → R}. For m a non zero natural number, let the operators Bm : F(∆2 ) →
F(∆2 ), defined for any function f ∈ F(∆2 ) by
X
k j
(2.1)
(Bm f )(x, y) =
pm,k,j (x, y)f
,
m m
k,j=0
k+j≤m
for any (x, y) ∈ ∆2 , where
(2.2)
pm,k,j (x, y) =
m!
xk y j (1 − x − y)m−k−j .
k!j!(m − k − j)!
Approximation of B-Continuous
and B-Differentiable Functions
by GBS Operators of Bernstein
Bivariate Polynomials
Ovidiu T. Pop and Mircea Farcaş
The operators are named Bernstein bivariate polynomials (see [8]).
Lemma 2.1. The operators (Bm )m≥1 are linear and positive on F(∆2 ).
Title Page
Proof. The proof follows immediately.
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For m a non zero natural number, let the GBS operator of Bernstein bivariate
polynomials U Bm (see [1]), U Bm : Cb (∆2 ) → B(∆2 ) defined for any function
f ∈ Cb (∆2 ) and any (x, y) ∈ ∆2 by
(2.3) (U Bm f )(x, y)
= (Bm (f (x, ∗) + f (· , y) − f (· , ∗))) (x, y)
X
j
k
k j
+f
,y − f
,
.
=
pm,k,j (x, y) f x,
m
m
m m
k,j=0
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Lemma 2.2. The operators (Bm )m≥1 verify for any (x, y) ∈ ∆2 the following:
(2.4)
(2.5)
(2.6)
(Bm e00 )(x, y) = 1;
x(1 − x)
Bm (· − x)2 (x, y) =
;
m
y(1 − y)
Bm (∗ − y)2 (x, y) =
;
m
(2.7)
Bm (· − x)2 (∗ − y)2 (x, y)
m−1
3(m − 2) 2 2 m − 2 2
2
=
x
y
−
(x
y
+
xy
)
+
xy;
m3
m3
m3
(2.8)
Bm (· − x)4 (∗ − y)2 (x, y)
5(3m2 − 26m + 24) 4 2 6(3m2 − 26m + 24) 3 2
=−
xy +
xy
m5
m5
3m2 − 41m + 42 2 2
6(m2 − 7m + 6) 3
−
x
y
−
xy
m5
m5
3m2 − 26m + 24 4
3m2 − 17m + 14 2
+
x
y
+
xy
m5
m5
m−2 2 m−1
−
xy +
xy
m5
m5
and
(2.9)
Bm (· − x)2 (∗ − y)4 (x, y)
5(m2 − 26m + 24) 2 4 6(3m2 − 26m + 24) 2 3
=−
xy +
xy
m5
m5
Approximation of B-Continuous
and B-Differentiable Functions
by GBS Operators of Bernstein
Bivariate Polynomials
Ovidiu T. Pop and Mircea Farcaş
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6(m2 − 7m + 6) 3 3m2 − 41m + 42 2 2
xy −
xy
m5
m5
3m2 − 26m + 24 4 3m2 − 17m + 14 2
+
xy +
xy
m5
m5
m−2 2
m−1
−
x
y
+
xy
m5
m5
−
for any non zero natural number m.
Proof. Let (x, y) ∈ ∆2 and m be a non zero natural number. We have
(Bm e00 )(x, y) =
X
k,j=0
k+j≤m
m!
xk y j (1 − x − y)m−k−j
k!j!(m − k − j)!
Approximation of B-Continuous
and B-Differentiable Functions
by GBS Operators of Bernstein
Bivariate Polynomials
Ovidiu T. Pop and Mircea Farcaş
m
= (x + y + 1 − x − y) = 1,
Title Page
so (2.4) holds,
(Bm e10 )(x, y) =
X
k,j=0
k+j≤m
m!
k
xk y j (1 − x − y)m−k−j
k!j!(m − k − j)!
m
X
=x
k=1,j=0
k+j≤m
(m − 1)!
xk−1 y j (1 − x − y)m−k−j
(k − 1)!j!(m − k − j)!
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it results that
(2.10)
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(Bm e10 )(x, y) = x
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and similarly
(Bm e01 )(x, y) = y.
(2.11)
In the same way, using the formulas
k 2 = k(k − 1) + k,
k 3 = k(k − 1)(k − 2) + 3k(k − 1) + k,
k 4 = k(k − 1)(k − 2)(k − 3) + 6k(k − 1)(k − 2) + 7k(k − 1) + k,
we obtain
(2.12)
(2.13)
1
m−1 2
(Bm e20 )(x, y) =
x + x,
m
m
(m − 1)(m − 2) 3 3(m − 1) 2
1
x +
x + 2 x,
(Bm e30 )(x, y) =
2
2
m
m
m
(m − 1)(m − 2)(m − 3) 4
x
m3
6(m − 1)(m − 2) 3 7(m − 1) 2
1
+
x +
x + 3x
3
3
m
m
m
and similarly the relations (Bm e02 )(x, y), (Bm e03 )(x, y), (Bm e04 )(x, y).
We have
(2.14) (Bm e40 )(x, y) =
(Bm e11 )(x, y)
X
m−1
(m − 1)!
k
y
xk y j−1 (1 − x − y)m−k−j
=
m
k!(j − 1)!(m − k − j)!
m−1
k=0,j=1
Approximation of B-Continuous
and B-Differentiable Functions
by GBS Operators of Bernstein
Bivariate Polynomials
Ovidiu T. Pop and Mircea Farcaş
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k+j≤m
=
m−1
y(Bm−1 e10 )(x, y),
m
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(Bm e21 )(x, y)
2 X
2
m−1
(m−1)!
k
k j−1
m−k−j
=
y
x y (1−x−y)
m
k!(j −1)!(m−k−j)!
m−1
k=0,j=1
k+j≤m
=
m−1
m
2
y(Bm−1 e20 )(x, y),
and in the same way, we write (Bm e31 )(x, y), (Bm e41 )(x, y), (Bm e32 )(x, y),
(Bm e42 )(x, y). Taking (2.12) - (2.14) into account, we obtain
(2.15)
(2.16)
m−1
xy,
m
m−1
(m − 1)(m − 2) 2
(Bm e21 )(x, y) =
x y+
xy,
2
m
m2
(Bm e11 )(x, y) =
Approximation of B-Continuous
and B-Differentiable Functions
by GBS Operators of Bernstein
Bivariate Polynomials
Ovidiu T. Pop and Mircea Farcaş
Title Page
(2.17)
(m − 1)(m − 2)(m − 3) 3
(Bm e31 )(x, y) =
xy
m3
3(m − 1)(m − 2) 2
m−1
+
x y+
xy,
3
m
m3
(m − 1)(m − 2)(m − 3)(m − 4) 4
xy
m4
6(m − 1)(m − 2)(m − 3) 3
+
xy
m4
7(m − 1)(m − 2) 2
m−1
+
x y+
xy,
4
m
m4
(2.18) (Bm e41 )(x, y) =
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(2.19) (Bm e22 )(x, y) =
(m − 1)(m − 2)(m − 3) 2 2
xy
m3
m−1
(m − 1)(m − 2) 2
+
(x y + xy 2 ) +
xy,
3
m
m3
(m − 1)(m − 2)(m − 3)(m − 4) 3 2
xy
m4
3(m − 1)(m − 2)(m − 3) 2 2
(m − 1)(m − 2)(m − 3) 3
+
x y+
xy
4
m
m4
3(m − 1)(m − 2) 2
(m − 1)(m − 2) 2 m − 1
x y+
xy +
xy,
+
4
m
m4
m4
(2.20) (Bm e32 )(x, y) =
Approximation of B-Continuous
and B-Differentiable Functions
by GBS Operators of Bernstein
Bivariate Polynomials
Ovidiu T. Pop and Mircea Farcaş
(2.21)
(Bm e42 )(x, y)
(m − 1)(m − 2)(m − 3)(m − 4)(m − 5) 4 2
xy
=
m5
(m − 1)(m − 2)(m − 3)(m − 4) 4
xy
+
m5
6(m − 1)(m − 2)(m − 3)(m − 4) 3 2
+
xy
m5
6(m − 1)(m − 2)(m − 3) 3
+
xy
m5
7(m − 1)(m − 2)(m − 3) 2 2
+
xy
m5
(m−1)(m−2) 2 m−1
7(m−1)(m−2) 2
+
x y+
xy +
xy
5
m
m5
m5
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and similarly the relations (Bm e12 )(x, y), (Bm e13 )(x, y), (Bm e14 )(x, y),
(Bm e23 )(x, y), (Bm e24 )(x, y).
Now, we have
(Bm (· − x)2 )(x, y) = (Bm e20 )(x, y) − 2x(Bm e10 )(x, y) + x2 (Bm e02 )(x, y),
(Bm (·−x)2 (∗−y)2 )(x, y) = (Bm e22 )(x, y)−2y(Bm e21 )(x, y)+y 2 (Bm e20 )(x, y)
− 2x(Bm e12 )(x, y) + 4xy(Bm e11 )(x, y) − 2xy 2 (Bm e10 )(x, y)
+ x2 (Bm e02 )(x, y) − 2x2 y(Bm e01 )(x, y) + x2 y 2 (Bm e00 )(x, y),
(Bm (· − x)4 (∗ − y)2 )(x, y)
= (Bm e40 )(x, y) − 2y(Bm e41 )(x, y) + y 2 (Bm e40 )(x, y)
− 4x(Bm e32 )(x, y) + 8xy(Bm e31 )(x, y) − 4xy 2 (Bm e30 )(x, y)
+ 6x2 (Bm e22 )(x, y) − 12x2 y(Bm e21 )(x, y) + 6x2 y 2 (Bm e20 )(x, y)
− 4x3 (Bm e12 )(x, y) + 8x3 y(Bm e11 )(x, y) − 4x3 y 2 (Bm e10 )(x, y)
+ x4 (Bm e02 )(x, y) − 2x4 y(Bm e01 )(x, y) + x4 y 2 (Bm e00 )(x, y)
and taking (2.9) – (2.21) into account, we obtain (2.5), (2.7) and (2.8). Similarly
we obtain (2.9).
Lemma 2.3. The operators (Bm )m≥1 verify for any (x, y) ∈ ∆2 the following
inequalities:
1
(2.22)
(Bm (· − x)2 )(x, y) ≤
,
4m
1
(2.23)
(Bm (∗ − y)2 )(x, y) ≤
,
4m
Approximation of B-Continuous
and B-Differentiable Functions
by GBS Operators of Bernstein
Bivariate Polynomials
Ovidiu T. Pop and Mircea Farcaş
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for any non zero natural number m,
(2.24)
Bm (· − x)2 (∗ − y)2 (x, y) ≤
9
,
4m2
for any natural number m, m ≥ 2,
(2.25)
(2.26)
9
Bm (· − x)4 (∗ − y)2 (x, y) ≤ 3 ,
m
9
Bm (· − x)2 (∗ − y)4 (x, y) ≤ 3 ,
m
for any natural number m, m ≥ 8.
Proof. Because x(1 − x) ≤
From (2.7), we have
1
4
for any x ∈ [0, 1], (2.22) and (2.23) results.
Approximation of B-Continuous
and B-Differentiable Functions
by GBS Operators of Bernstein
Bivariate Polynomials
Ovidiu T. Pop and Mircea Farcaş
Title Page
2
2
Bm (· − x) (∗ − y) (x, y)
2(m − 2) 2 2 m − 2
1
=
xy +
x(1 − x)y(1 − y) + 3 xy
3
3
m
m
m
1
2(m − 2) m − 2
≤
+
+ 3
m3
16m3
m
33m − 50
=
,
16m3
from where (2.24) results.
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From (2.8), we have
Bm (· − x)4 (∗ − y)2 (x, y)
6(3m2 − 26m + 24) 3 2
3m2 − 26m + 24 4
=
x
y
(1
−
x)
+
x y(y + 1)
m5
m5
3m2 − 17m + 14 2
6(m2 − 7m + 6) 3
−
x
y
+
x y(1 − y)
m5
m5
24m − 28 2 2 m − 2
+
xy +
xy(1 − y) + xy.
m5
m5
But
3m2 − 26m + 24 2
3m2 − 26m + 24 4
x
y(y
+
1)
≤
2
xy
m5
m5
10m − 12 2
6m2 − 42m + 36 2
=
x y−
xy
5
m
m5
10m − 12 3 2
6m2 − 42m + 36 2
≤
x
y
−
xy
m5
m5
and then, from the inequalities above, we obtain
(2.27) Bm (· − x)4 (∗ − y)2 (x, y)
6m2 − 42m + 36 2
6(3m2 − 26m + 24) 3 2
≤
x
y
(1
−
x)
+
x y(1 − y)
m5
m5
3m2 − 17m + 14 2
10m − 12 2 2
+
x y(1 − y) +
x y (1 − y)
5
m
m5
14m − 16 2 2 m − 2
+
xy +
xy(1 − y) + xy.
m5
m5
Approximation of B-Continuous
and B-Differentiable Functions
by GBS Operators of Bernstein
Bivariate Polynomials
Ovidiu T. Pop and Mircea Farcaş
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Because x(1 − x) ≤ 14 , y(1 − y) ≤ 14 , xy ≤ 1 for any x, y ∈ [0, 1], from (2.27)
we have
Bm (· − x)4 (∗ − y)2 (x, y)
6(3m2 − 26m + 24) 6m2 − 42m + 36
≤
+
4m5
4m5
2
3m − 17m + 14 10m − 12 14m − 16 m − 2
+
+
+
+
+1
4m5
4m5
m5
4m5
27m2 − 148m + 170
=
,
m5
Approximation of B-Continuous
and B-Differentiable Functions
by GBS Operators of Bernstein
Bivariate Polynomials
from where (2.25) results.
Ovidiu T. Pop and Mircea Farcaş
Theorem 2.4. Let the function f ∈ Cb (∆2 ). Then, for any (x, y) ∈ ∆2 , any
natural number m, m ≥ 2, we have
(2.28) |f (x, y) − (U Bm f )(x, y)|
−1 1
−1 1
−1 −1 3
≤ 1 + δ1 √ + δ2 √ + δ1 δ2
ωmixed (f ; δ1 , δ2 )
2m
2 m
2 m
for any δ1 , δ2 > 0 and
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(2.29)
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7
1
1
|f (x, y) − (U Bm f )(x, y)| ≤ ωmixed f ; √ , √
2
m
m
.
Proof. For the first inequality we apply Theorem 1.1 and Lemma 2.3. The inequality (2.29) is obtained from (2.28) by choosing δ1 = δ2 = √1m .
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J. Ineq. Pure and Appl. Math. 7(3) Art. 92, 2006
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Corollary 2.5. If f ∈ Cb (∆2 ), then
(2.30)
lim (U Bm f )(x, y) = f (x, y)
m→∞
uniformly on ∆2 .
Proof. Because f ∈ Cb (∆
that f is uniform B-continuous on
2 ), there results
1
1
∆2 and then lim ωmixed f ; √m , √m = 0 (see [2] or [3]). From (2.29), there
m→∞
results the conclusion.
Theorem 2.6. Let the function f ∈ Db (∆2 ) with DB f ∈ B(∆2 ). Then for any
(x, y) ∈ ∆2 , any natural number m, m ≥ 8, we have
Approximation of B-Continuous
and B-Differentiable Functions
by GBS Operators of Bernstein
Bivariate Polynomials
Ovidiu T. Pop and Mircea Farcaş
9
(2.31) |f (x, y) − (U Bm f )(x, y)| ≤
kDb f k∞
2m
3
3
3
−1
−1
−1 −1 9
√ +δ
√ +δ δ
+
+δ
ωmixed (DB f ; δ1 , δ2 )
2m 1 m m 2 m m 1 2 4m2
for any δ1 , δ2 > 0 and
(2.32) |f (x, y) − (U Bm f )(x, y)|
3
1
1
≤
6kDB f k∞ + 13ωmixed DB f ; √ √
.
4m
m m
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Proof. It results from Theorem 1.2 and Lemma 2.3.
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References
[1] C. BADEA AND C. COTTIN, Korovkin-type theorems for generalised
boolean sum operators, Colloquia Mathematica Societatis János Bolyai,
58, Approximation Theory, Kecskemét (Hunagary), 1990, 51–67.
[2] C. BADEA, Modul de continuitate în sens Bögel şi unele aplicaţii în
aproximarea printr-un operator Bernstein, Studia Univ. "Babeş-Bolyai",
Ser. Math.-Mech., 18(2) (1973), 69–78 (Romanian).
[3] C. BADEA, I. BADEA, C. COTTIN AND H.H. GONSKA, Notes on the
degree of approximation of B-continuous and B-differentiable functions,
J. Approx. Theory Appl., 4 (1988), 95–108.
Approximation of B-Continuous
and B-Differentiable Functions
by GBS Operators of Bernstein
Bivariate Polynomials
Ovidiu T. Pop and Mircea Farcaş
[4] D. BĂRBOSU, Aproximarea funcţiilor de mai multe variabile prin sume
booleene de operatori liniari de tip interpolator, Ed. Risoprint, ClujNapoca, 2002 (Romanian).
[5] K. BÖGEL, Mehrdimensionale Differentiation von Funtionen mehrerer
Veränderlicher, J. Reine Angew. Math., 170 (1934), 197–217.
[6] K. BÖGEL, Über die mehrdimensionale differentiation, integration und
beschränkte variation, J. Reine Angew. Math., 173 (1935), 5–29.
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[7] K. BÖGEL, Über die mehrdimensionale differentiation, Jber. DMV, 65
(1962), 45–71.
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[8] G.G. LORENTZ, Bernstein Polynomials, University of Toronto Press,
Toronto, 1953.
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J. Ineq. Pure and Appl. Math. 7(3) Art. 92, 2006
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[9] M. NICOLESCU, Contribuţii la o analiză de tip hiperbolic a planului, St.
Cerc. Mat., III, 1-2, 1952, 7–51 (Romanian).
[10] M. NICOLESCU, Analiză Matematică, II, E. D. P. Bucureşti, 1980 (Romanian).
[11] O.T. POP, Approximation of B-differentiable functions by GBS operators
(to appear in Anal. Univ. Oradea).
[12] D.D. STANCU, Gh. COMAN, O. AGRATINI AND R. TRÎMBIŢAŞ, Analiză Numerică şi Teoria Aproximării, I, Presa Universitară Clujeană, ClujNapoca, 2001 (Romanian).
Approximation of B-Continuous
and B-Differentiable Functions
by GBS Operators of Bernstein
Bivariate Polynomials
Ovidiu T. Pop and Mircea Farcaş
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