APPROXIMATION PROPERTIES OF A STANCU
S-VARIATE OPERATOR OF BETA TYPE
by
Lucia A. Căbulea and Mihaela Todea
Abstract: In this paper the main result consist in introducing and investigating the
approximation properties of a new beta operator of a second kind , ,..., 1 ,...,1,..., 1 1 1 2
1
+ +
= s
s s s
n n
x n x n n n
n T
S
which is an integral linear positive operator reproducing the linear functions. Then we have estimations of the orders of approximation, by using the modulus of continuity of first and second orders.
1. INTRODUCTION [1]
In his recent paper [18], Professor D. D. Stancu has introduced a new beta operator of second kind Sn defined by the following formula:
(1)
( )
∫
∞
+ + −
∞ ∈ +
+ =
0
1 1
)), , 0 ( ( , )
1 ( ) ( ) 1 , (
1 )
( dt x
t t t f n
nx B x f
S nx n
nx n
where f ∈ M[0,∞) and n is a natural number ≥ 2.
Here M[0,∞) denotes the space of bounded and locally integral functions on [0,∞). Actually, Sn is applicable to functions of polynomial growth if we restrict ourselves to n ≥ n0 with n0∈ N sufficiently great.
The operator (1) is constructed, roughly speaking, by applying a beta second kind transform
∫
∞ +−
∞
∈
+
=
0 1,
,
(
[
0
,
)),
)
1
(
)
(
)
,
(
1
M
g
dt
t
t
t
g
q
p
B
g
T
p qp q
p
to the Baskakov operator
(2)
(
(
0
,
))
)
1
(
1
)
,
(
0
∞
∈
+
+
−
=
∞ +=
∑
x
n
k
f
x
x
k
k
n
x
f
B
n kk
k n
Sn is a positive linear operator reproducing the linear functions. It is mentioned in [18] that Sn is distinct from the other beta-type operators considered earlier in the papers [10], [19], [8] and [2].
Furthermore, Professor D. D. Stancu has given estimations for the order of approximation by using the moduli of continuity of first and second orders. He has also established an asymptotic formula of Voronovskaja-type
(3)
(
)
2
)
1
(
))
(
)
)(
((
lim
n
S
nf
x
f
x
x
x
f
x
n
′′
+
=
−
∞ →
for all f ∈ M[0,∞) admitting a bounded derivative of second order at x (x >0).
2. THE BETA SECOND-KIND S-VARIATE
TRANSFORM s
s
q q q
p p p
T 1,, 2,...,,..., 2 1
Let us denote by M([0,∞)×[0,∞)×…×[0,∞)) the linear space
of functions g(t1, t2, …, ts), defined for ti≥ 0, i ≥ 0, bounded and Lebesque measurable in I1×I2×…×Is, where Is = [0,∞).
We shall define a linear transform by using the beta s-variate distribution of second kind, with the positive parameters p1, p2,…, ps and q1, q2,…, qs, which has the probability density
(4)
,
)
1
)(
,
(
)
,...,
,
(
1
1 2
1 ,..., ,
,..., ,
2 1
2
1
∏
= +
−
+
=
si
q p i i i
p i s
q q q
p p
p i i
i s
s
t
q
p
B
t
t
t
t
b
where ti> 0, i = 1,2,…, s and s
s p p p
q q q
b 1 2,..., ,..., 2 , 1
,
(t1,t2,…,ts) =0 otherwise, by B(p,q), i=1,2,…,s is denoted the beta function. By using distribution (4) we can define the beta s-variate second-kind transform s
s
q q q
p p p
T 1,,2,...,,..., 2
1 , of a function g ∈ M([0,∞)×[0,∞)×…×[0,∞)), by
(5) s
s
q q q
p p p
T 1,,2,...,,..., 2
1 g =
=
∫ ∫ ∫
∞ ∞ ∞ =0 0 0 1 1 2
,..., ,
,..., ,
1, ,..., ) (( , ,..., ) ...
(
... 1 2
2
1 t s s
q q q
p p p s
t t b t t t dt dt dt
t t
g s
s
=
∏
∫ ∫ ∫
∏
= +
−
=
∞ ∞ ∞
+
s
i
s q
p i p i s
i
s
dt
dt
dt
t
t
t
t
t
g
q
p
B
i ii
1
2 1 1
1 0 0 0
2
1
...
)
1
(
)
,...,
,
(
...
)
,
(
1
such that s s
q q q
p p p
T 1,, 2,...,,..., 2
One observes that s s
q q q
p p p
T 1,, 2,...,,..., 2
1 is a linear positive functional. We need to state and prove:
THEOREM 1. The moment of order (r1, r2,…,rs) (r1, r2,…,rs ≥ 1) of the functional s
s
q q q
p p p
T 1,,2,...,,..., 2
1 has the following value (6)
s s
s
s r r r
q q q
p p p s s r
r
r p q p q p q T e
v , ,..., 1 1 2 2 ,, ,...,,..., , ,..., 2 1 2 1
2 1 2
1 ( , ; , ;...; , )= =
=
∏
= − − −
− + +
s
i i i i i
i i i
i
r q q
q
r p p
p
1 ( 1)( 2)...( )
) 1 )...(
1 (
Proof: We have
s s
i
s
i
q p i
r p i i
i r
r r q q q
p p
p
dt
dt
t
t
q
p
B
e
T
i i i i
s s
s
...
)
1
(
...
)
,
(
1
1
1 0 0 0 1
1 ,...,
, ,..., ,
,...,
, 1 2
2 1
2
1
∏
∫ ∫ ∫ ∏
=
∞ ∞ ∞
= +
− +
+
=
==
∏
∫
=
∞
+ − +
+
s
i
i q p i
r p i i
i
dt
t
t
q
p
B
i ii i
1 0
1
)
1
(
)
,
(
1
.
If we make the change of integration variable yi = ti / (1+ti), we have ti = yi / (1-yi), dti = (1-yi)-2dyi (i = 1, 2, …., s) and we obtain
∏
= + −= s
i i i
i i i i r
r r q q q
p p p
q p B
r q r p B e
T
s s s
1 ,..., , ,..., ,
,..., ,
) , (
) ,
(
2 1 2 1
2
1 .
By using the relation
B(p+r,q-r) =
(
,
)
)
)...(
2
)(
1
(
)
1
)...(
1
(
q
p
B
r
q
q
q
r
p
p
p
−
−
−
+
+
−
we obtain the desired result (6). 3. THE FUNCTIONAL
) (
) , ;...; , ; ,
( , ,...,
,..., ,
,..., , 2
2 1 1 ,...,
, 1 2
2 1
2 1 2
1 p q p q p q T B f
F
s s
s
s n n n
q q q
p p p s s f
n n
n =
Now let us apply the transform (5) to the s-variant Baskakov operator
s
n n n
(7) + + − = =
∑ ∏
∞ = = s s k i k i k k k s i i i i s n n n n k n k f t t k k n t t t f B i i s s ,..., ) 1 ( 1 ) ,..., , )( ( 1 1 0 ,..., , 1 2 1 ,..., , 2 1 2 1We may state and prove
THEOREM 2. The s
s q q q p p p
T
1,, 2,...,,...,2
1 transform of Bn1,n2,...,ns f can be
expressed under the following form
(8)
(
,
;
,
;...;
,
)
=
(
, ,...,)
=
,..., , ,..., , 2 2 1 1 ,...,
, 1 2
2 1
2 1 2
1
p
q
p
q
p
q
T
B
f
F
s s
s
s n n n
q q q p p p s s f n n n
∑ ∏
,...,∞ =0 = + − ++ ++ −− ++ + ++ −− , 1 1
1 2 1 ,..., ) 1 )...( 1 )( ( ) 1 )...( 1 ( ) 1 )...( 1 ( 1 s k k k s i s s i i i i i i i i i i i i i i i i i i i n k n k f k n q p q p q p n q q q q p p p k k n Proof:
We can write successively
∑ ∏
∞ = =
+
−
=
0 ,..., , 1 ,..., , ,..., , ,..., , 2 1 2 1 2 1 2 1)
,
(
1
1
)
(
s s s s k k k si i i i
i i n n n q q q p p p
q
p
B
k
k
n
f
B
T
⋅
+
⋅
∫ ∫ ∫ ∏
∞ ∞ ∞ = + + + − + s s s i s k q p n i k p in
k
n
k
n
k
f
dt
dt
dt
t
t
i i i i i i,...,
,
...
)
1
(
...
2 2 1 1 0 0 01
2 1 1
.
If we make the change of variable yi = ti / (1+ti), i = 1,2,…,s in the integral
∫
∞ + + +− +
+
=
0 1,
)
1
(
)
,
(
i i i i i i ii p q n k
i i k p i i i k n
t
dt
t
q
p
I
, we get=
+
+
=
−
=
∫
+ −(
1
)
+ −(
,
)
)
,
(
1 0 1 1, i i i i i
q n i p k i i i k
n
p
q
y
y
dy
B
k
p
n
q
I
i i i ii i
)
,
(
)
1
)...(
1
)(
(
)
1
)...(
1
(
)
1
)...(
1
(
i i i i i i i i i i i i i i i i i iq
p
B
k
n
q
p
q
p
q
p
n
q
q
q
q
p
p
p
−
+
+
+
−
+
+
+
+
−
+
+
−
=
and we obtain the formula (8).
parameter-dependent operator s s
n n n
Lα,,α,...,,...,α 2 1
2
1 , as a generalization of the Baskakov operator. By using the factorial powers, with the step hi = -αi, it can be expressed under the following compact form
(
)
(
)
[ ] [ ]
[ ] + ⋅
⋅
⋅
+ −
=
− +
− −
∞
= =
∑ ∏
s s k
n i
n k
i
k k k
s
i i
i i s
n n n
n k n k n k f x
x
k k n x
x x f L
i i i
i i i i
s s
s
,..., , )
1 (
1
1 ,...,
,
2 2 1 1 ,
, ,
0 ,...,
, 1
2 1 ,...,
, ,..., ,
2 1 2
1 2 1
α α α
α α α
4. THE BETA SECOND-KIND LINEAR POSITIVE OPERATOR
1 ,..., 1 ,...,
1 1 1
+ + s
s s
n n
x n x n
T
Now we introduce a new beta second-kind s-variate
approximating operator. If in (5) we choose pi = nixi and qi = ni + 1, then to any function f ∈M([0,∞)×[0,∞)×…×[0,∞)) we associate the linear positive operator
s
n n n
S 1,2,..., , defined by (10)
(
)
∏
=∫ ∫ ∫ ∏
∞ ∞ ∞= + +
− + +
+ +
=
= =
s
i
s
i
s s
n x n i
x n i i
i u
n n
x n x n s n
n n
dt dt dt t t t f t
t n
x n B
f T
x x x f S
i i i i i s
s s s
1 0 0 0 1
2 1 2
1 1 1 1 ,..., 1 ,..., 2
1 ,..., ,
... )
,..., , ( )
1 ( ... )
1 , (
1
) ,..., ,
( 1
1 1 2
1
This represents an extension to s-variables of the operator (1) of D. D. Stancu. Because
(
)
∫ ∫ ∫
∞ ∞ ∞ + + +=
=
0 0 0 1 2 1 2
1 ,..., 1 , 1
,..., , 0
,..., 0 , 0 ,...,
,
(
,
)
...
(
,
,...,
)
...
1
2 1
2 2 1 1 2
1 s s
n n n
x n x n x n n
n
n
e
x
y
b
t
t
t
dt
dt
dt
S
ss s s
and according to (6) we have
(
Sn1,n2,...,nse1,0,...,0)
(x1,x2,...,xs)= x1,(
Sn1,n2,...,nse0,1,...,0)
(x1,x2,...,xs)= x2 ,…,(
Sn1,n2,...,nse0,0,...,1)
(x1,x2,...,xs)= xsand(
Sn1,n2,...,nse1,1,...,1)
(x1,x2,...,xs)=x1x2...xs.{
}
i i i
s s
s
x x
x x x f x x x f f
δ
δ
δ
δ
ϖ
≤ −
− =
'
' ' 2 ' 1 2
1 2
1, ,..., ) sup ( , ,..., ) ( , ,..., ) ,
; (
i=1,2,…,s where (x1,x2,…,xs) and ( , ,..., ) ' ' 2 '
1 x xs
x are the points of
(0,∞)×(0,∞)×…×(0,∞) and δi∈R+, i = 1,2,…,s.
By using a basic property of the modulus of continuity we can write
)
,...,
;
(
)
(
1
1
)
,...,
,
)(
(
)
,...,
,
(
1 1
2 2
2 1 ,..., , 2
1 1 2
s s
i
i i n i
s n
n n s
f
t
x
S
x
x
x
f
S
x
x
x
f
i
s
δ
δ
ϖ
δ
+
−
≤
≤
−
∑
=Because
1
)
1
(
)
(
)
(
22
−
+
=
−
=
i i i i
i n n
n
x
x
x
t
S
x
i i
δ
, i = 1,2,…,sif we take
1
1
−
=
i i
n
δ
we can stateTHEOREM 3. If f ∈ C([0,∞)×[0,∞)×…×[0,∞)), such that
〈∞
)
,...,
,
;
(
1 2,..., ,2
1n n s
n
f
x
x
x
S
s , then for any ni > , i = 1,2,…,s we have the following
inequality
−
−
−
+
+
≤
≤
−
∑
=1
1
,...,
1
1
,
1
1
;
)
1
(
1
)
,...,
,
)(
(
)
,...,
,
(
2 1
1
2 1 ,...,
, 2
1 1 2
s s
i
i i
s n
n n s
n
n
n
f
x
x
x
x
x
f
S
x
x
x
f
s
ϖ
A similar inequality can be obtained if we assume that the function f has a bounded continuous derivative for xi≥ 0, i = 1,2,…,s.
According to the Bohman-Korovkin convergence criterion, we can deduce COROLLARY 1. If f ∈ M([0,∞)×[0,∞)×…×[0,∞)) and is continuous at all points of I1×I2×…×Is (Ii⊂ [0,∞), i = 1,2,…,s), then Sn1,n2,...,ns f converges uniformly on
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