IJMMS 2004:9, 459–468
PII. S0161171204205038
http://ijmms.hindawi.com
© Hindawi Publishing Corp.
RATE OF CONVERGENCE OF BOUNDED VARIATION
FUNCTIONS BY A BÉZIER-DURRMEYER VARIANT
OF THE BASKAKOV OPERATORS
VIJAY GUPTA and ULRICH ABEL
Received 6 May 2002
We consider a Bézier-Durrmeyer integral variant of the Baskakov operators and study the
rate of convergence for functions of bounded variation.
2000 Mathematics Subject Classification: 41A36, 41A25, 26A45.
1. Introduction. Let W (0, ∞) be the class of functions f which are locally integrable
on (0, ∞) and are of polynomial growth as t → ∞, that is, for some positive r , there
n of the Baskakov operators
holds f (t) = O(t r ) as t → ∞. The Durrmeyer variant V
associates to each function f ∈ W (0, ∞) the series
n (f ; x) = (n − 1)
V
∞
∞
pn,k (x)
k=0
0
pn,k (t)f (t)dt,
x ∈ [0, ∞),
(1.1)
where
pn,k (x) =
n+k−1 k
x (1 + x)−n−k
k
(1.2)
is the Baskakov basis function. Note that (1.1) is well defined, for n ≥ r + 2, provided
that f (t) = O(t r ) as t → ∞. The operators (1.1) were first introduced by Sahai and
Prasad [9]. They termed these operators as modified Lupaş operators. In 1991, Sinha et
n as modified Baskakov
al. [10] improved and corrected the results of [9] and denoted V
operators. The rate of convergence of the operators (1.1) on functions of bounded
variation was studied in [8, 11].
We mention that Agrawal and Thamer [2] considered the variant
Mn (f ; x) = (n − 1)
∞
k=1
∞
pn,k (x)
0
pn,k−1 (t)f (t)dt + (1 + x)−n f (0)
(1.3)
of the operators (1.1) and studied its properties in subsequent papers [3, 4, 5]. See also
[1]. The rate of convergence of the operators discussed by Agrawal and Thamer was
studied by the first author in [7].
460
V. GUPTA AND U. ABEL
For each function f ∈ W (0, ∞) and α ≥ 1, we consider the Bézier–type Baskakovn,α as
Durrmeyer operators V
n,α (f ; x) = (n − 1)
V
∞
∞
(α)
Qn,k (x)
k=0
0
pn,k (t)f (t)dt,
(1.4)
where
(α)
α
α
Qn,k (x) = Jn,k
(x) − Jn,k+1
(x),
Jn,k (x) =
∞
pn,j (x).
(1.5)
j=k
n,α are positive linear operators and V
n,α (1; x) = 1. In the special
It is obvious that V
n,1 . Some basic properties
case α = 1, the operators Vn,α reduce to the operators Vn ≡ V
of Jn,k are as follows:
(i) Jn,k (x) − Jn,k+1 (x) = pn,k (x) (k = 0, 1, 2, . . .);
(x) = npn+1,k−1 (x) (k = 1, 2, 3, . . .);
(ii) Jn,k
x
(iii) Jn,k (x) = n 0 pn+1,k−1 (t)dt (k = 1, 2, 3, . . .);
(iv) 0 < · · · < Jn,k+1 (x) < Jn,k (x) < · · · < Jn,1 (x) < Jn,0 (x) ≡ 1 (x > 0);
(v) Jn,k is strictly increasing on [0, ∞).
In this paper, we study the rate of convergence for the new sequence of operators
(1.4), for functions f of bounded variation. Our result essentially generalizes and imn,α (f ; x)
proves the results of [8, 11]. Furthermore, we find the limit of the sequence V
for bounded locally integrable functions f having a discontinuity of the first kind at
x ∈ (0, ∞).
2. The main results. As a main result, we derive the following estimate on the rate
of convergence.
Theorem 2.1. Assume that f ∈ W (0, ∞) is a function of bounded variation on every
finite subinterval of (0, ∞). Furthermore, let α ≥ 1, λ > 2, and x ∈ (0, ∞) be given. Then,
for each r ∈ N, there exists a constant M(f , α, r , x) such that for sufficiently large n, the
n,α satisfy the estimate
Bézier–type Baskakov-Durrmeyer operators V
α
1
V
f
(x+)
+
f
(x−)
(f
;
x)
−
n,α
α+1
α+1
α(10 + 11x) f (x+) − f (x−)
≤ (2.1)
2 nx(1 + x)
√
n x+x/ k
2αλ(1 + x) + x M(f , α, r , x)
,
gx +
+
√
nx
nr
k=1
x−x/ k
where



f (t) − f (x−)
gx (t) = 0


f (t) − f (x+)
and
b
a (gx )
is the total variation of gx on [a, b].
(0 ≤ t < x),
(t = x),
(x < t < ∞),
(2.2)
A BÉZIER-DURRMEYER VARIANT OF THE BASKAKOV OPERATORS
461
Remark 2.2. The exponent r in the O-term of (2.1) can be chosen arbitrary large.
As an immediate consequence of Theorem 2.1, we obtain in the special case α = 1
the following estimate which improves the results of [8, 11].
Corollary 2.3. Under the assumptions of Theorem 2.1, there holds, for sufficiently
large n,
1
V
n (f ; x) − 2 f (x+) + f (x−) (10 + 11x) f (x+) − f (x−)
≤ 2 nx(1 + x)
+
√
n x+x/
k
2λ(1 + x) + x
nx
k=1
√
x−x/ k
(2.3)
M(f , 1, r , x)
gx +
,
nr
where gx is defined as in Theorem 2.1.
Theorem 2.4. Let x ∈ (0, ∞). If f ∈ L(0, ∞) has a discontinuity of the first kind at x,
then
n,α (f ; x) =
lim V
n→∞
α
1
f (x+) +
f (x−).
α+1
α+1
(2.4)
3. Auxiliary results. In order to prove our main result, we will need the following
lemmas. Throughout the paper, for each real x, let ψx (t) = t − x.
Lemma 3.1 (see [6]). Let {ξi }∞
i=1 be a sequence of independent and identically distributed random variables with finite variance such that the expectation E(ξi ) = a1 ∈
R ≡ (−∞, ∞), and the variance V (ξi ) = b12 > 0. Assume that E|ξi − a1 |3 < ∞. Then there
√
exists a constant c with 1/ 2π < c < 0.82 such that, for all n = 1, 2, 3, . . . and all t ∈ R,
3
t
n
1 1
E ξi − a1 −u2 /2
√
P
√
e
du ≤ c
.
ξi − a1 ≤ t −
√ 3
b1 n
2π −∞
nb
(3.1)
1
k=1
Lemma 3.2 (see [10]). For each fixed x ∈ [0, ∞) and m ∈ N0 , the central moments
n (ψxm ; x) of the Baskakov-Durrmeyer operators (1.1) satisfy
V
n ψxm ; x = O n−(m+1)/2
V
(n → ∞).
(3.2)
In particular,
n (1; x) = 1,
V
2(n − 1)x(1 + x)
2(1 + 2x)2
n ψx2 ; x =
+
.
V
(n − 2)(n − 3)
(n − 2)(n − 3)
(3.3)
Remark 3.3. Note that, given any λ > 2 and any x > 0, for all n sufficiently large,
we have the estimate
λx(1 + x)
n ψx2 ; x <
V
.
n
(3.4)
462
V. GUPTA AND U. ABEL
Lemma 3.4 (see [13]). For all x > 0 and n, k ∈ N, there holds
√
α 1+x
(α)
α
.
(x)pn,k (x) ≤ Qn,k (x) ≤ αpn,k (x) < √
Jn,k
2enx
(3.5)
Throughout, let
Kn,α (x, t) = (n − 1)
(α)
Qn,k (x)pn,k (t),
Kn,α (x, t)dt.
0
(3.6)
k=0
y
λn,α (x, y) =
∞
(3.7)
With this definition, for each function f ∈ W (0, ∞), there holds, for all sufficiently
large n,
n,α (f ; x) =
V
∞
0
Kn,α (x, t)f (t)dt.
(3.8)
Kn,α (x, u)du = 1.
(3.9)
Note that, in particular,
λn,α (x, ∞) =
∞
0
Lemma 3.5. For each λ > 2 and, for all sufficiently large n, there exist, for all x ∈
(0, ∞),
y
λαx(1 + x)
n(x − y)2
0
∞
λαx(1 + x)
Kn,α (x, t)dt ≤
1 − λn,α (x, z) =
n(z − x)2
z
λn,α (x, y) =
Kn,α (x, t)dt ≤
(0 ≤ y < x),
(3.10)
(x < z < ∞).
(3.11)
Proof. First we prove (3.10). There holds
y
y
0
(x − t)2
dt
(x − y)2
0
n,α ψx2 ; x
≤ (x − y)−2 V
n,1 ψx2 ; x ,
≤ α(x − y)−2 V
Kn,α (x, t)dt ≤
Kn,α (x, t)
(3.12)
where we applied Lemma 3.4. Now (3.10) is a consequence of Remark 3.3. The proof of
(3.11) is similar.
Lemma 3.6 (see [13]). Let {ξi }∞
i=1 be a sequence of independent random variables with
the same geometric distribution
P ξ1 = k =
x
1+x
k
1
1+x
(k ∈ N),
(3.13)
where x > 0 is a parameter. Then,
E ξ1 = x,
2
E ξ1 − Eξ1 = x(1 + x),
3
E ξ1 − Eξ1 ≤ 3x(1 + x)2 .
(3.14)
A BÉZIER-DURRMEYER VARIANT OF THE BASKAKOV OPERATORS
463
Lemma 3.7. For all x ∈ (0, ∞) and k = 0, 1, 2, . . . , there hold
α
α(10 + 11x)
J (x) − J α
,
n,k
n−1,k+1 (x) ≤
2 nx(1 + x)
α
α(10 + 11x)
J (x) − J α
.
n,k
n−1,k (x) ≤
2 nx(1 + x)
(3.15)
(3.16)
Proof. First we prove (3.15). Proceeding along the lines of [8, Lemma 2.8] and [12],
it is easily checked that
ξ1 − Eξ1 3
x
Jn,k (x) − Jn−1,k+1 (x) ≤ 2(0.82)E
3/2 + √ 2π nx(1 + x)
n x(1 + x)
x
2(0.82) · 3x(1 + x)2
3/2 + √ 2 nx(1 + x)
n x(1 + x)
10 + 11x
,
≤ 2 nx(1 + x)
≤
(3.17)
where we used Lemmas 3.1 and 3.6. Application of the inequality |aα − bα | ≤ α|a − b|,
for 0 ≤ a, b ≤ 1, and α ≥ 1 yields (3.15). The proof of (3.16) is similar.
4. Proofs of the main results
Proof of Theorem 2.1. Our starting point is the identity
f (t) =
1
α
α − 1 f (x+) − f (x−)
f (x+) +
f (x−) + sign(t − x) +
α+1
α+1
α+1
2
f (x+) + f (x−)
+ gx (t) + δx (t) f (x) −
,
2
(4.1)
n,α (δx ; x) =
where δx (t) = 1 (t = x) and δx (t) = 0 (t ≠ x) (see [12, Equation (28)]). Since V
0, we conclude that
α
1
V
n,α (f ; x) − α + 1 f (x+) + α + 1 f (x−) α − 1 1
f (x+) − f (x−) + V
n,α sign(t − x); x +
n,α gx ; x .
V
≤ 2
α+1
(4.2)
First, we obtain
x
∞
∞
(α)
n,α sign(t − x); x = (n − 1)
V
Qn,j (x)
pn,j (t)dt −
pn,j (t)dt
j=0
= (n − 1)
∞
x
0
x
∞
(α)
Qn,j (x)
pn,j (t)dt − 2
pn,j (t)dt
0
j=0
= 1 − 2(n − 1)
∞
j=0
(α)
Qn,j (x)
0
x
0
pn,j (t)dt.
(4.3)
464
V. GUPTA AND U. ABEL
Using
k
pn−1,j (x) = (n − 1)
j=0
∞
x
pn,k (t)dt,
(4.4)
we conclude that
j
∞
(α)
n,α sign(t − x); x = 1 − 2
V
Qn,j (x) 1 −
pn−1,k (x)
j=0
= −1 + 2
∞
k=0
pn−1,k (x)
k=0
= −1 + 2
∞
∞
(α)
Qn,j (x)
(4.5)
j=k
α
pn−1,k (x)Jn,k
(x)
k=0
since
∞
(α)
j=0 Qn,j (x)
= 1. Therefore, we obtain
∞
∞
α−1
2 (α+1)
α
n,α sign(t − x); x +
=2
pn−1,k (x)Jn,k
(x) −
Q
(x)
V
α+1
α + 1 k=0 n−1,k
k=0
since
∞
(α+1)
k=0 Qn−1,k (x)
(4.6)
= 1. By the mean value theorem, it follows that
(α+1)
α+1
α+1
α
Qn−1,k (x) = Jn−1,k
(x) − Jn−1,k+1
(x) = (α + 1)pn−1,k (x)γn,k
(x),
(4.7)
where Jn−1,k+1 (x) < γn,k (x) < Jn−1,k (x). Hence,
∞
α
α−1
α
n,α sign(t − x); x +
V
pn−1,k (x) Jn,k
(x) − γn,k
(x) ,
=2
α+1
k=0
(4.8)
α
α
α
α
α
α
Jn,k
(x) − Jn−1,k
(x) < Jn,k
(x) − γn,k
(x) < Jn,k
(x) − Jn−1,k+1
(x).
(4.9)
where
Lemma 3.7 implies that
α − 1 α(10 + 11x)
≤ V
sign(t
−
x);
x
+
n,α
α+1
nx(1 + x)
for x ∈ (0, ∞).
(4.10)
n,α (gx ; x). We
In order to complete the proof of the theorem, we need an estimate of V
use the integral representation (3.8) and decompose [0, ∞) into three parts as follows:
√
x+x/√n ∞
x−x/ n
n,α gx ; x =
V
+
+
Kn,α (x, t)gx (t)dt
√
√
0
x−x/ n
x+x/ n
(4.11)
= I 1 + I2 + I3 ,
say.
√
√
We start with I2 . For t ∈ [x − x/ n, x + x/ n], we have
√
n
x+x/
gx (t) ≤
√
x−x/ n
gx ,
(4.12)
A BÉZIER-DURRMEYER VARIANT OF THE BASKAKOV OPERATORS
465
and therefore
√
√
n
x+x/
I 2 ≤
n x+x/ k
1 gx .
(gx ) ≤
√
√
n k=1
x−x/ n
(4.13)
x−x/ k
√
Next we estimate I1 . Let y = x −x/ n. Using integration by parts with (3.7), we have
y
I1 =
0
y
gx (t)dt λn,α (x, t) = gx (y)λn,α (x, y) −
Since |gx (y)| = |gx (y) − gx (x)| ≤
x
y (gx ),
x
I 1 ≤
gx λn,α (x, y) +
y
0
λn,α (x, t)dt gx (t).
(4.14)
we conclude that
y
0
λn,α (x, t)dt −
x
gx
.
(4.15)
t
√
Since y = x − x/ n ≤ x, (3.10) implies that
x
λαx(1 + x) λαx(1 + x)
I1 ≤
gx +
n(x − y)2 y
n
y
0
x
1
dt −
gx .
(x − t)2
t
(4.16)
Integrating the last term by parts, we get
y x x
λαx(1 + x)
t gx
−2
I 1 ≤
x
dt .
gx + 2
n
(x
− t)3
0
0
(4.17)
√
Replacing the variable y in the last integral by x − x/ n, we obtain
x−x/√n x
0
n−1
gx (x − t)−3 dt =
t
k=1
≤
x/√k
x
−3
gx t dt
√
x/ k+1 x−t
n
1 2
2x k=1
x
√
x−x/ k
gx .
(4.18)
Hence,
n
2λα(1 + x) I 1 ≤
nx
k=1
x
√
x−x/ k
gx .
(4.19)
Finally, we estimate I3 . We let
x (t) =
g

gx (t)
(0 ≤ t ≤ 2x),
g (2x) (2x < t < ∞)
x
(4.20)
466
V. GUPTA AND U. ABEL
and divide I3 = I31 + I32 , where
I31 =
I32 =
∞
x (t)dt,
√ Kn,α (x, t)g
x+x/ n
∞
2x
(4.21)
Kn,α (x, t) gx (t) − gx (2x) dt.
√
With y = x + x/ n, the first integral can be written in the form
I31 = lim
R→+∞
x (R) λn,α (x, R) − 1
gx (y) 1 − λn,α (x, y) + g
R
+
y
x (t) .
1 − λn,α (x, t) dt g
(4.22)
By (3.11), we conclude that
y t
R
g
x (R)
1
λαx(1 + x)
x gx
+
+
d
g
lim
t
x
2
R→+∞ (y − x)2
n
(R − x)2
y (t − x)
x
t
y 2x
λαx(1 + x)
1
x gx
gx
.
+
d
=
2 t
n
(y − x)2
y (t − x)
x
|I31 | ≤
(4.23)
In a similar way as above we obtain
2x
y
√
t
y 2x
n−1
x+x/
k 1
x gx
−2
−2
dt
+x
gx
≤x
gx −
gx
(t − x)2
(y − x)2
x
x
x
k=1
(4.24)
which implies the estimate
√
k
n x+x/
2λα(1 + x) I31 ≤
nx
x
k=1
gx .
(4.25)
We proceed with I32 . By assumption, there exists an integer r such that f (t) = O(t 2r )
as t → ∞. Thus, for a certain constant M > 0, depending only on f , x, and r , we have
∞
(α)
I32 ≤ M(n − 1)
Q (x)
∞
n,k
k=0
≤ αM(n − 1)
∞
k=0
2x
pn,k (t)t 2r dt
(4.26)
∞
pn,k (x)
2x
pn,k (t)t
2r
dt,
where we used Lemma 3.4. Obviously, t ≥ 2x implies t ≤ 2(t − x) and it follows that
∞
I32 ≤ 22r αM(n − 1)
pn,k (x)
k=0
∞
0
n ψx2r ; x .
pn,k (t)(t − x)2r dt = 22r αM V
(4.27)
A BÉZIER-DURRMEYER VARIANT OF THE BASKAKOV OPERATORS
467
By Lemma 3.2, the central moments of the Baskakov-Durrmeyer operators (1.1) satisfy
n (ψx2r ; x) = O(n−r )(n → ∞), and we obtain
V
(n → ∞).
I32 = O n−r
(4.28)
Collecting the estimates (4.13), (4.19), (4.25), and (4.28) yields with regard to (4.11)
√
n x+x/
k 2λα(1 + x) + x V
n,α gx ; x ≤
gx + O n−r
√
nx
k=1
(n → ∞).
(4.29)
x−x/ k
Finally, combining (4.2), (4.10), and (4.29), we obtain (2.1). This completes the proof
of Theorem 2.1.
Proof of Theorem 2.4. Since the function ψx2 given by ψx2 (t) = (t − x)2 is of
bounded variation on every finite subinterval of [0, ∞), we deduce from Theorem 2.1
that, for all x ∈ (0, ∞),
n,α ψx2 ; x = 0.
lim V
n→∞
(4.30)
If f ∈ L∞ (0, ∞), then gx defined as in (2.2) is also bounded and is continuous at the
point x. By the Korovkin theorem, we conclude that
n,α gx ; x = gx (x) = 0.
lim V
n→∞
(4.31)
Therefore, the right-hand side of inequality (4.2) tends to zero as n → ∞. This completes
the proof of Theorem 2.4.
Acknowledgment. The authors are thankful to the referee for the valuable comments that helped to improve this paper. Moreover, the referee provided further important references.
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Vijay Gupta: School of Applied Sciences, Netaji Subhas Institute of Technology, Azad Hind Fauj
Marg, Sector-3, Dwarka, New Delhi-110045, India
E-mail address: vijay@nsit.ac.in
Ulrich Abel: Fachbereich MND, Fachhochschule Giessen-Friedberg, University of Applied Sciences, Wilhelm-Leuschner-Straße 13, 61169 Friedberg, Germany
E-mail address: Ulrich.Abel@mnd.fh-friedberg.de