An. Şt. Univ. Ovidius Constanţa
Vol. 11(2), 2003, 163–170
ON THE ORDER OF APPROXIMATION OF
FUNCTIONS BY THE BIDIMENSIONAL
OPERATORS FAVARD-SZÁSZ-MIRAKYAN
Ioana Taşcu and Anca Buie
Abstract
We will present an approximation result concerning the order of approximation of bivariate function by means of the bidimensional operator of Favard-Szász-Mirakyan.
1
Introduction
Let X be the interval [0, +∞) and RX the space of real functions defined on
X. The Favard-Szász-Mirakyan operator Mm defined on RX is given by
∞
k
(mx)k
f
(Mn f )(x) = e−mx
.
(1.1)
k!
m
k=0
Consider a bivariate function defined an I = [0, +∞) × [0, +∞).
The parametric extensions for the operators (1.1) are given by
(x Mm f ) (x, y) = e−mx
∞
k k
(mx)
f
,y ,
k!
m
(1.2)
∞
j j
(ny)
f x,
.
j!
n
j=0
(1.3)
k=0
(y Mn f ) (x, y) = e−ny
Since Mm is a positive linear operator, it follows that its parametric extensions x Mm , y Mn are also positive and linear operators.
Key Words: Taylor series, Mirakyan operator, Voronovskaja theorem, parametric extension, modulus of smoothness
Mathematical Reviews subject classification: 41A58, 41A45, 41A63
163
164
Ioana Taşcu and Anca Buie
Proposition 1.1 The parametric extension x Mm , y Mn satisfy the relation
x Mm
· y Mn =y Mn · x Mm .
Their product is the bidimensional operator Mm,n which, for any function
f ∈ RI , gives the approximant
∞
∞ k
j
k j
(mx) (ny)
f
,
Mm,n (f ) (x, y) = e−mx e−ny
.
(1.4)
k!
j!
m n
j=0
k=0
Properties of the bidimensional operator of Mirakyan were studied in [1],
[2], [3], [4]. Next, we will present some of them.
Theorem 1.1 The bidimensional operator of Favard-Szász-Mirakyan has the
following properties:
(i) It is linear and positive.
(ii) Mm,n (e0,0 ) (x, y) = 1;
Mm,n (e1,0 ) (x, y) = x;
Mm,n (e0,1 ) (x, y) = y;
x
;
m
y
Mm,n (e0,2 ) (x, y) = y 2 + .
m
Mm,n (e2,0 ) (x, y) = x2 +
(iii) If a > 0, b > 0 and f ∈ C ([0, a] × [0, b]), then the sequence
{Mm,n (f )}(m,n)∈N ∗ ×N ∗ is uniformly convergent to f on [0, a] × [0, b].
(iv) If a > 0, b > 0 and f ∈ C ([0, a] × [0, b]), then we have the estimation
a
b
,
,
|f (x, y) − Mm,n (f ) (x, y)| ≤ 4ω
m
n
where by ω we denote the first modulus of smoothness.
2
Main results
Lemma 2.1 If δ > 0 and a > 0, then
lim me
m→∞
−mx
k
−x|≥δ
|m
(mx)
k!
k
2
k
− x = 0.
m
ON THE ORDER OF APPROXIMATION OF FUNCTIONS BY THE
BIDIMENSIONAL OPERATORS FAVARD-SZÁSZ-MIRAKYAN
Proof. By (1.4) we have the next inequality
2
(mx)k k
m
4
me−mx
− x ≤ 2 Mm,n (x − ·) ; x, y .
k!
m
δ
k
−x|≥δ
|m
165
(2.1)
A simple computation yields
x
x2
x3
+ x4 ,
+
7
+
6
m3
m2
m
x2
x
Mm,n φ4 (x, y) =
+
3
+ x3 .
m2
m
These equalities and (i), (ii) from Theorem 1.1. imply:
1 4
Mm,n (x − ·) ; x, y = 3 3mx2 + x .
m
By (2.1) and (2.2), we have
2
(mx)k k
3mx2 + x
−mx
−x ≤
me
.
k!
m
m2 δ 2
k
−x|≥δ
|m
Mm,n φ3 (x, y) =
(2.2)
The proof of the lemma is complete.
Theorem 2.1 Consider a > 0, b > 0, (x, y) ∈ [0, a] × [0, b] and
f ∈ C ([0, a] × [0, b]). Assume that
(i) the function f has the second partial derivatives;
(ii) the function f has the mixt partial derivatives in (x, y).
Then
lim min {m, n} [Mm,n (f ) (x, y) − f (x, y)] ≤
m,n→∞
x ∂2f
y ∂2f
(x,
y)
+
(x, y) .
2 ∂x2
2 ∂y 2
The equality holds when m = n.
Proof. Using the corresponding Taylor series, we obtain
∂f
1
∂f
(x, y) + (t − y)
(x, y)
f (s, t) = f (x, y) +
(s − x)
1!
∂x
∂y
2
1
∂2f
2 ∂ f
+
(x, y)
(x, y) + (s − x)(t − y)
(s − x)
2
2!
∂x
∂x ∂y
2
∂2f
2 ∂ f
+ (s − x) (t − y)
(x, y) + (t − y)
(x, y)
∂y∂x
∂y 2
2
+ (s − x) µ1 (s − x) + (s − x) (t − y) µ2 (s − x, t − y)
2
+ (t − y) µ3 (t − y) ,
(2.3)
166
Ioana Taşcu and Anca Buie
where the mappings µ1 , µ2 , µ3 are bounded and
lim µ1 (h) = 0,
h→0
lim
h1 ,h2 →0
µ2 (h1 , h2 ) = 0,
lim µ3 (h) = 0 .
h→0
k
j
k
j
(mx) (ny)
In (2.3), we choose s = , t = and multiply by
, we get
m
n
k!
j!
∂f
(x, y) Mm,n ((· − x) ; x, y)
∂x
∂f
1 ∂2f
2
(x, y) Mm,n ((∗ − y) ; x, y) +
+
(x,
y)
M
;
x,
y
(·
−
x)
m,n
∂y
2 ∂x2
2
1 ∂ f
+
(x, y) Mm,n ((· − x) (∗ − y) ; x, y)
2 ∂x∂y
1 ∂2f
(x, y) Mm,n ((· − x) (∗ − y) ; x, y)
+
2 ∂y∂x
1 ∂2f
2
+
(∗
−
y)
(x,
y)
M
;
x,
y
+ Rm,n f (x, y) , (2.4)
m,n
2 ∂y 2
(Mm,n f ) (x, y) − f (x, y) =
where
2 k
k
− x µ1
−x
m
m
k=0
∞
∞
(mx)k (ny)j k
j
k
j
+ e−mx e−ny
−x
− y µ2
− x, − y
k!
j!
m
n
m
n
k=0 j=0
2 ∞
j j
j
(nx)
+ e−nx
− x µ3
− x . (2.5)
j!
n
n
j=0
∞
(mx)k
Rm,n f (x, y) = e−mx
k!
Since
Mm,n ((· − x) ; x, y)
=
0
Mm,n ((∗ − y) ; x, y)
=
0
Mm,n ((· − x) (∗ − y) ; x, y) =
2
=
Mm,n (· − x) ; x, y
2
Mm,n (∗ − y) ; x, y
=
0
x
m
y
,
n
the relation (2.4) is equivalent to the following
Mm,n (f ) (x, y) − f (x, y) =
x ∂2f
y ∂ 2f
(x,
y)
+
(x, y) + Rm,n f (x, y).
2
2
2m ∂x
2n ∂y
(2.6)
ON THE ORDER OF APPROXIMATION OF FUNCTIONS BY THE
BIDIMENSIONAL OPERATORS FAVARD-SZÁSZ-MIRAKYAN
167
Now, by multiplying (2.6) by min {m, n} , and crossing to limit with respect
to m, n we obtain
lim min {m, n} Mm,n (f ) (x, y) − f (x, y) ≤
m,n→∞
≤
y ∂2f
x ∂2f
(x, y) +
(x, y) + lim min {m, n} Rm,n f (x, y).
2
2
m,n→∞
2 ∂x
2 ∂y
We claim that
(2.7)
lim min {m, n} Rm,n f (x, y) = 0.
m,n→∞
Indeed, let ε > 0. Since lim µ1 (h) = 0, there exists δ > 0 such that, for any
h→0
h, with |h| < δ , we have |µ1 (h)| < ε. From lim µ3 (k) = 0, we obtain that
k→0
there exist an δ > 0 such that, for any k with |k| < δ , we have µ3 (k) < ε.
Considering δ = max {δ , δ }, for every h, k with |h| < δ and |k| < δ, we
have
|µ1 (h)| < ε and |µ3 (k)| < ε .
Let us use the notations
I1
I2
J1
J2
k
k ∈ N; m
− x < δ ,
k
=
k ∈ N; m
− x ≥ δ ,
=
j ∈ N ; nj − y < δ ,
=
j ∈ N ; nj − y ≥ δ .
=
Since the maps µ1 , µ2 and µ3 are bounded, we can write
2 (mx)k k
k
Rm,n f (x, y) ≤ e−mx
− x µ1
− x k!
m
m
k∈I1
2
(mx)k k
+ (sup |µ1 |) e−mx
− x + (sup |µ2 |) Mm,n (|· − x| |∗ − y| ; x, y)
k!
m
k∈I2
2
2
(ny)j j
(ny)j j
j
−nx
−nx
− x µ3
− y +(sup |µ3 |) e
−x
+e
j!
n
n
j!
n
j∈J1
j∈J2
(2.8)
k
j
For k ∈ I1 and j ∈ J1 , we have µ1
− x < ε and µ3
− y < ε.
m
n
168
Ioana Taşcu and Anca Buie
Moreover, Mm,n (|· − x| |∗ − y| ; x, y) = 0. Therefore (2.8) becomes
2
k
−x
m
k∈I1
2
(mx)k k
−mx
−x
+ (sup |µ1 |) e
k!
m
k∈I2
2
2
(ny)j j
(ny)j j
− x + (sup |µ3 |) e−nx
−x
+ εe−nx
j!
n
j!
n
j∈J1
j∈J2
2
(mx)k
(mx)k k
2 −mx
−mx
+ (sup |µ1 |) e
−x
≤ εδ e
k!
k!
m
k∈I1
k∈I2
2
(ny)j
(ny)j j
+ (sup |µ3 |) e−ny
−y
+ εδ 2 e−ny
j!
j!
n
j∈I1
j∈I2
2
(mx)k k
−x
≤ 2εδ 2 + (sup |µ1 |) e−mx
k!
m
k∈I2
2
(ny)j j
−y
+ (sup |µ3 |) e−ny
.
j!
n
(mx)k
Rm,n f (x, y) ≤ εe−mx
k!
j∈I2
Multiplying the previous inequality with min {m, n}, we obtain
min {m, n} Rm,n f (x, y) ≤ 2εδ 2 min {m, n}
2
(mx)k k
+ (sup |µ1 |) min {m, n} e−mx
−x
k!
m
k∈I2
2
(ny)j j
−ny
−y
+ (sup |µ3 |) min {m, n} e
.
j!
n
(2.9)
j∈I2
But
min {m, n} e−mx
2
(mx)k k
− x ≤ me−mx
k!
m
k∈I2
| mk −x|≥δ
(mx)
k!
k
2
k
−x
m
and
min {m, n} e
−ny
2
(ny)j j
−y
≤ ne−ny
j!
n
j∈I2
|
j
n −y
|≥δ
j
(ny)
j!
j
−y
n
2
.
ON THE ORDER OF APPROXIMATION OF FUNCTIONS BY THE
BIDIMENSIONAL OPERATORS FAVARD-SZÁSZ-MIRAKYAN
169
Now, by using the Lemma 2.1, we obtain
lim min {m, n} e
−mx
m,n→∞
| mk −x|≥δ
and
lim min {m, n} e−ny
m,n→∞
|
j
n −y
|≥δ
(mx)k
k!
(ny)j
j!
2
k
−x =0
m
j
−y
n
2
= 0.
So, there exist m0 , n0 ∈ N such that for every m, n ∈ N, with m ≥ m0 and
n ≥ n0 , we have
(mx)k k
1
−mx
−x <ε
min {m, n} e
k!
m
(sup |µ1 |)
k
−x|≥δ
|m
and
min {m, n} e
−ny
| nj −y|≥δ
(ny)j
j!
j
−y
n
<ε
1
.
(sup |µ3 |)
Then, according to (2.13), we can conclude that there exist m0 , n0 ∈ N
such that for every m, n ∈ N, with m ≥ m0 and n ≥ n0 , the next inequality
holds
min {m, n} Rm,n f (x, y) < 2ε δ 2 min {m, n} + 1 .
(2.10)
This is equivalent with
lim min {m, n} Rm,n f (x, y) = 0.
m,n→∞
References
[1] Agratini, O., Aproximarea prin operatori liniari, Presa Universitară Clujeană, Cluj-Napoca, 2000.
[2] Bărbosu, D., Bivariate operators of Mirakyan type, Bulletin for Applied
Computer Mathematics, BAM-1794/2000 XCIV, 169-176.
[3] Bărbosu, D., Aproximarea funcţiilor de mai multe variabile prin sume
booleene de operatori liniari de tip interpolator, Ed. Risoprint, ClujNapoca, 2002.
[4] Ciupa, A., On the remainder in an approximation formula by Favard-Szasz
operators for functions of two variables, Bul. Şt. Univ. Baia Mare, seria B,
Mat-Inf., vol.X (1994), 63-66.
170
Ioana Taşcu and Anca Buie
[5] Stancu, D.D., Coman, Gh., Agratini, O., Trâmbiţaş, R., Analiză numerică
şi teoria aproximării, Presa Univ. Clujeană, Cluj-Napoca, 2001.
North University of Baia Mare
Department of Mathematics and Computer Science
Victoriei 78, 430122, Baia Mare, Romania
E-mail: itascu@ubm.ro