Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 4, Issue 2, Article 34, 2003
RATE OF CONVERGENCE OF SUMMATION-INTEGRAL TYPE OPERATORS
WITH DERIVATIVES OF BOUNDED VARIATION
VIJAY GUPTA, VIPIN VASISHTHA, AND M.K. GUPTA
S CHOOL OF A PPLIED S CIENCES ,
N ETAJI S UBHAS I NSTITUTE OF T ECHNOLOGY,
S ECTOR 3 DWARKA , N EW D ELHI -110045,
I NDIA .
vijay@nsit.ac.in
D EPARTMENT OF M ATHEMATICS ,
H INDU C OLLEGE ,
M ORADABAD -244001, I NDIA
D EPARTMENT OF M ATHEMATICS ,
C H . C HARAN S INGH U NIVERSITY,
M EERUT-255004,
I NDIA
mkgupta2002@hotmail.com
Received 28 January, 2003; accepted 29 March, 2003
Communicated by A. Fiorenza
A BSTRACT. In the present paper, we estimate the rate of convergence of the recently introduced
generalized sequence of linear positive operators Gn,c (f, x) with derivatives of bounded variation.
Key words and phrases: Linear positive operators, Bounded variation, Total variation, Rate of convergence.
2000 Mathematics Subject Classification. 41A25, 41A30.
1. I NTRODUCTION
Let DBγ (0, ∞), (γ ≥ 0) be the class of all locally integrable functions defined on (0, ∞),
satisfying the growth condition |f (t)| ≤ M tγ , M > 0 and f 0 ∈ BV on every finite subinterval
of [0, ∞). Then for a function f ∈ DBγ (0, ∞) we consider the generalized family of linear
positive operators which includes some well known operators as special cases. The generalized
ISSN (electronic): 1443-5756
c 2003 Victoria University. All rights reserved.
013-03
2
V IJAY G UPTA , V IPIN VASISHTHA ,
AND
M.K. G UPTA
sequence of operators is defined by
(1.1)
Gn,c (f, x) = n
∞
X
∞
Z
pn,k (x; c)
pn+c,k−1 (t; c) f (t) dt
0
k=1
+ pn,0 (x; c) f (0) ,
where pn,k (x; c) = (−1)k
xk (k)
φ
k! n,c
x ∈ [0, ∞)
(x),
(i) φn,c (x) = e−nx for c = 0,
(ii) φn,c (x) = (1 + cx)−n/c for c ∈ N,
and {φn,c }n∈N be a sequence of functions defined on an interval [0, b] , b > 0 having the following properties for every n ∈ N, k ∈ N0 :
(i)
(ii)
(iii)
(iv)
φn,c ∈ C ∞ ([a, b]) ;
φn,c (0) = 1;
(k)
φn,c is completely monotone (−1)k φn,c (x) ≥ 0;
(k+1)
(k)
There exists an integer c such that φn,c = −nφn+c,c , n > max {0, −c} .
Remark 1.1. We may remark here that the functions φn,c have various applications in different
fields, like potential theory, probability theory, physics and numerical analysis. A collection of
most interesting properties of such functions can be found in [10, Ch. 4].
It is easily verified that the operators (1.1) are linear positive operators. Also Gn,c (1, x) = 1.
The generalized new sequence Gn,c was recently introduced by Srivastava and Gupta [9].
For c = 0 and φn,c (x) = e−nx the operators Gn,c reduce to the Phillips operators (see e.g.
[7], [8]), which are defined by
Z ∞
∞
X
pn,k−1 (t; 0) f (t) dt + e−nx f (0) , x ∈ [0, ∞),
(1.2) Gn,0 (f, x) = n
pn,k (x; 0)
0
k=1
−nx
where pn,k (x; 0) = e k! (nx)k .
For c = 1 and φn,c (x) = (1 + cx)−n/c the operators Gn,c reduce to the new sequence of
summation integral type operators [6], which are defined by
(1.3)
Gn,1 (f, x) = n
∞
X
∞
Z
pn,k (x; 1)
pn+1,k−1 (t; 1) f (t) dt
0
k=1
+ (1 + x)−n f (0) ,
x ∈ [0, ∞),
where
pn,k (x; 1) =
n+k−1
k
xk (1 + x)n−k .
Remark 1.2. It may be noted that for c = 1, we get the Baskakov basis functions pn,k (x; 1)
which are
to the well known Meyer-Konig and Zeller basis functions mn,k (t) =
kclosely related
n
n+k−1
x
t (1 − t) , t ∈ [0, 1] because by replacing the variable t with 1+x
in the above MKZ
k
basis functions we get the Baskakov basis functions. Zeng [11] obtained the exact bound for
the Meyer Konig Zeller basis functions. Very recently Gupta et al. [6] used the bound of Zeng
[11] and estimated the rate of convergence for the operators Gn,1 (f, x) on functions of bounded
variation.
J. Inequal. Pure and Appl. Math., 4(2) Art. 34, 2003
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R ATE OF C ONVERGENCE
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The operators (1.3) are slightly modified form of the operators introduced by Agarwal and
Thamer [1], which are defined by
(1.4)
G∗n,1
(f, t) = (n − 1)
∞
X
Z
pn,k (x; 1)
∞
pn,k−1 (t; 1) f (t) dt
0
k=1
+ (1 + x)−n f (0) ,
x ∈ [0, ∞),
where pn,k (x; 1) is as defined by (1.3) above.
Recently Gupta [5] estimated the rate of approximation for the sequence (1.4) for bounded
variation functions. Although the operators defined by (1.3) and (1.4) above are almost the
same, but the main advantage to consider the operators in the form (1.3) rather than the form
(1.4) is that some approximation properties become simpler in the analysis for the form (1.3) in
comparison to the form (1.4). The rate of approximation with derivatives of bounded variation
has been studied by several researchers. Bojanic and Cheng ([2], [3]) estimated the rate of
convergence with derivatives of bounded variation for Bernstein and Hermite-Fejer polynomials
by using different methods.
Alternatively we may rewrite the operators (1.1) as
Z ∞
(1.5)
Gn,c (f, x) =
Kn (x, t; c) f (t) dt,
0
where
Kn (x, t; c) = n
∞
X
pn,k (x; c) pn+c,k−1 (t; c) + pn,0 (x; c) pn,0 (t; c) δ (t) ,
k=1
δ (t) being the Dirac delta function. Also let
Z
(1.6)
t
Kn (x, s; c) ds
βn (x, t; c) =
0
then
Z
βn (x, ∞; c) =
∞
Kn (x, s; c) ds = 1.
0
In the present paper we extend the results of [4] and [6] and study the rate of convergence
by means of the decomposition technique of functions with derivatives of bounded variation.
More precisely the functions having derivatives of bounded variation on every finite subinterval
on the interval [0, ∞) be defined as
Z x
f (x) = f (0) +
ψ (t) dt,
0 < a ≤ x ≤ b,
0
where ψ is a function of bounded variation on [a, b] and c is a constant.
We denote the auxiliary function fx , by


f (t) − f (x− ) , 0 ≤ t < x;




0,
t = x;
fx (t) =




 f (t) − f (x+ ) , x < t < ∞.
J. Inequal. Pure and Appl. Math., 4(2) Art. 34, 2003
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V IJAY G UPTA , V IPIN VASISHTHA ,
AND
M.K. G UPTA
2. AUXILIARY R ESULTS
In this section we give certain results, which are necessary to prove the main result.
Lemma 2.1. [9]. Let the function µn,m (x) , m ∈ N 0 , be defined as
Z ∞
∞
X
µn,m (x; c) = n
pn,k (x; c)
pn+c,k−1 (t; c) (t − x)m dt + (−x)m pn,0 (x; c) .
k=1
0
Then
cx
,
(n − c)
x (1 + cx) (2n − c) + (1 + 3cx) cx
,
µn,2 (x; c) =
(n − c) (n − 2c)
µn,0 (x; c) = 1, µn,1 (x; c) =
and there holds the recurrence relation
[n − c (m + 1)] µn,m+1 (x; c)
= x (1 + cx) µ(1)
n,m (x; c) + 2mµn,m−1 (x; c) + [m (1 + 2cx) + cx] µn,m (x; c) .
Consequently for each x ∈ [0, ∞), we have from this recurrence relation that
µn,m (x; c) = O n−[(m+1)/2] .
Remark 2.2. In particular, given any number λ > 2 and x > 0 from Lemma 2.1, we have for
c ∈ N 0 and n sufficiently large
(2.1)
λx (1 + cx)
Gn,c (t − x)2 , x ≡ µn,2 (x; c) ≤
.
n
Remark 2.3. It is also noted from (2.1), that
(2.2)
2
Gn,c (|t − x| , x) ≤ Gn,c (t − x) , x
12
p
≤
λx (1 + cx)
√
.
n
Lemma 2.4. Let x ∈ (0, ∞) and Kn (x, t) be defined by (1.5). Then for λ > 2 and for n
sufficiently large, we have
Ry
(i) βn (x, y; c) = 0 Kn (x, t; c) dt ≤ λx(1+cx)
, 0 ≤ y < x,
n(x−y)2
R∞
λx(1+cx)
(ii) 1 − βn (x, z; c) = z Kn (x, t; c) dt ≤ n(z−x)2 , x < z < ∞.
Proof. First, we prove (i). In view of (2.1), we have
Z y
Z y
(x − t)2
−2
Kn (x, t; c) dt ≤
µn,2 (x; c)
2 Kn (x, t; c) dt ≤ (x − y)
0
0 (x − y)
λx (1 + cx)
≤
.
n (x − y)2
The proof of (ii) is similar.
3. M AIN R ESULT
In this section we prove the following main theorem.
J. Inequal. Pure and Appl. Math., 4(2) Art. 34, 2003
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R ATE OF C ONVERGENCE
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Theorem 3.1. Let f ∈ DBγ (0, ∞) , γ > 0, and x ∈ (0, ∞). Then for λ > 2 and for n
sufficiently large, we have

√
x+ √xn
n] x+ x
[X
_k
x _
λ (1 + cx) 

((f 0 )x ) + √
((f 0 )x )
|Gn,c (f, x) − f (x)| ≤

n
n √x
x
k=1
x− k
+
+
+
+
where
Wb
a
x−
n
λ (1 + cx) f (2x) − f (x) − xf 0 x+ + |f (x)|
p n
λx (1 + cx)
√
M 2γ O n−γ/2 + f 0 x+ n
p
1 λx (1 + cx) 0 + √
f x − f 0 x− 2
n
0 +
cx
f x + f 0 x− ,
2 (n − c)
(fx ) denotes the total variation of fx on [a, b].
Proof. We have
Z
∞
Gn,c (f, x) − f (x) =
Kn (x, t; c) (f (t) − f (x)) dt
Z ∞ Z t
0
=
Kn (x, t; c) f (u) du dt.
0
0
x
Using the identity
f 0 (u) =
1 0 +
1
f x + f 0 x− + (f 0 )x (u) + f 0 x+ − f 0 x− sgn (u − x)
2
2
1 0 +
0
0
−
+ f (x) − f x + f x
χx (u) ,
2
it is easily verified that
Z ∞ Z t
1 0 +
0
0
−
f (x) − f x + f x
χx (u) du K (x, t; c) dt = 0.
2
0
x
Also
Z
∞
0
Z
t
1 0 +
0
−
f x −f x
sgn (u − x) du Kn (x, t; c) dt
2
1 0 +
=
f x − f 0 x− Gn,c (|t − x| , x)
2
t
1 0 +
0
−
f x +f x
du K (x, t; c) dt
2
1 0 +
=
f x + f 0 x− Gn,c ((t − x) , x) .
2
x
and
Z
0
∞
Z
x
J. Inequal. Pure and Appl. Math., 4(2) Art. 34, 2003
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V IJAY G UPTA , V IPIN VASISHTHA ,
AND
M.K. G UPTA
Thus we have
|Gn,c (f, x) − f (x)|
Z ∞ Z t
Z x Z t
0
0
≤
(f )x (u) du Kn (x, t; c) dt −
(f )x (u) du Kn (x, t; c) dt
x
x
x
0
1 0 + + f x − f 0 x− Gn,c (|t − x| , x)
2
1 0 + + f x + f 0 x− Gn,c ((t − x) , x)
2
1
= |An (f, x; c) + Bn (f, x; c)| + f 0 x+ − f 0 x− Gn,c (|t − x| , x)
2
1 0 + + f x + f 0 x− Gn,c ((t − x) , x) .
2
(3.1)
To complete the proof of the theorem it is sufficient to estimate the terms An (f, x;√c) and
Bn (f, x; c). Applying integration by parts, using Lemma 2.4 and taking y = x − x/ n, we
have
Z x Z t
0
|Bn (f, x; c)| = (f )x (u) du dt (βn (x, t; c)) ,
Z
x
Z
0
0
x
y
Z x
βn (x, t; c) (f )x (t) dt ≤
+
0
0
|(f 0 )x (t)| |βn (x, t; c)| dt
y
λx (1 + cx)
≤
n
Z
λx (1 + cx)
n
Z
≤
0
0
x
y_
t
x
y_
1
((f )x )
dt +
(x − t)2
0
((f 0 )x )
t
Z
y
x
x_
((f 0 )x ) dt
t
x
1
x _
√
((f 0 )x ) .
dt
+
2
n
(x − t)
x− √x
n
Let u = x/ (x − t). Then we have
λx (1 + cx)
n
Z
0
x
y_
t
√
λx (1 + cx)
1
((f )x )
2 dt =
n
(x − t)
0
Z
1
x
_
n
((f 0 )x ) du
x
x− u
√
≤
[ n] x
λ (1 + cx) X _
n
((f 0 )x ) .
x
k=1 x− u
Thus
√
(3.2)
|Bn (f, x; c)| ≤
[ n] x
λ (1 + cx) X _
J. Inequal. Pure and Appl. Math., 4(2) Art. 34, 2003
n
x
x _
((f 0 )x ) .
((f )x ) + √
n √x
x
k=1 x− u
0
x−
n
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R ATE OF C ONVERGENCE
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On the other hand, we have
(3.3) |An (f, x; c)|
Z ∞ Z t
0
=
(f )x (u) du Kn (x, t; c) dt
Zx ∞ Zx t
0
= (f )x (u) du Kn (x, t; c) dt
2x
x
Z 2x Z t
0
+
(f )x (u) du dt (1 − βn (x, t; c))
x
Z ∞ x
Z
0 + ∞
≤ (f (t) − f (x)) Kn (x, t; c) dt + f x (t − x) Kn (x, t; c) dt
2x
2xZ 2x
Z 2x
+ (f 0 )x (u) du |1 − βn (x, 2x; c)| +
|(f 0 )x (t)| |1 − βn (x, t; c)| dt
Zx
Zx ∞
M ∞
|f
(x)|
Kn (x, t; c) tγ |t − x| dt +
Kn (x, t; c) (t − x)2 dt
≤
x 2x
x2
2x
Z
0 + ∞
λ
(1
+ cx) + f x Kn (x, t; c) |t − x| dt +
f (2x) − f (x) − xf 0 x+ nx
2x
√
x
x+ √n
[ n] x+ x
_
λ (1 + cx) X _k
x
+
((f 0 )x ) .
((f 0 )x ) + √
n
n
x
k=1 x
Next applying Hölder’s inequality, and Lemma 2.1, we proceed as follows for the estimation of
the first two terms in the right hand side of (3.3):
Z
Z
M ∞
|f (x)| ∞
γ
(3.4)
Kn (x, t; c) t |t − x| dt +
Kn (x, t; c) (t − x)2 dt
x 2x
x2
2x
21
Z ∞
12 Z ∞
M
2
2γ
≤
Kn (x, t; c) t dt
Kn (x, t; c) (t − x) dt
x
2x
0
Z
|f (x)| ∞
+
Kn (x, t; c) (t − x)2 dt
2
x
2x
p
λx (1 + cx)
λ (1 + cx)
γ
−γ/2
√
+ |f (x)|
.
≤ M2 O n
nx
n
Also the third term of the right side of (3.3) is estimated as
Z
0 + ∞
f x Kn (x, t; c) |t − x| dt
2x
Z
0 + ∞
≤ f x
Kn (x, t; c) |t − x| dt
0
≤ f 0 x+ Z
∞
21 Z
Kn (x, t; c) (t − x) dt
2
0
= f 0 x+ ∞
12
Kn (x, t; c) dt
0
p
λx (1 + cx)
√
.
n
Combining the estimates (3.1) – (3.4), we get the desired result.
This completes the proof of Theorem 3.1.
J. Inequal. Pure and Appl. Math., 4(2) Art. 34, 2003
http://jipam.vu.edu.au/
8
V IJAY G UPTA , V IPIN VASISHTHA ,
AND
M.K. G UPTA
Remark 3.2. For negative values of c, the operators Gn,c may be defined in different ways.
Here we consider one such example, when c = −1 then φn,c (x) = (1 − x)n , the operator
reduces to
Z 1
n
X
pn,k (x; −1)
Gn,−1 (f, x) = n
pn−1,k−1 (t; −1) f (t) dt
0
k=1
+ (1 − x)n f (0) ,
where
x ∈ [0, 1] ,
n
xk (1 − x)n−k .
k
The rate of convergence for the operators Gn−1 (f, x) is analogous so we omit the details.
pn,k (x; −1) =
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J. Inequal. Pure and Appl. Math., 4(2) Art. 34, 2003
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