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Revista de la Unión Matemática Argentina
Print version ISSN 0041-6932On-line version ISSN 1669-9637
Rev. Unión Mat. Argent. vol.48 no.1 Bahía Blanca Jan./June 2007
Voronovskaya Type Asymptotic Formula For Lupaş-Durrmeyer Operators
Naokant Deo
Abstract. In the present paper, we study some direct results in simultaneous approximation for linear combinations of Lupaş-Beta type operators.
2000 Mathematics Subject Classification. 41A35.
Key words and phrases. Lupaş operator, Linear combinations, Voronovskaya formula.
This research is supported by CAS-TWAS Postdoctoral Fellowship, (Chinese Academy of Sciences, Beijing, China and ICTP, Trieste, Italy).
Permanent address of author: Department of Applied Mathematics, Delhi College of Engineering, Bawana Road, Delhi-110042, India.
The Bernstein-Durrmeyer (the set of non-negative integers), were introduced by Durrmeyer [2] and independently by Lupaş [5]. For a function
they are defined by
![∫ ∑n 1 (Mnf )(x) = (n + 1) pn,k(x) pn,k(t)f(t)dt, x ∈ [0,1], k=0 0](/img/revistas/ruma/v48n1/1a063x.png)
where
![( ) n k n-k pn,k(x) = k x (1 - x) , 0 ≤ k ≤ n.](/img/revistas/ruma/v48n1/1a064x.png)
Later, starting with this integral modification of Bernstein polynomials, Heilmann [3] first defined modified Lupaş operators (see also Heilmann and Müller [4] as well as Sinha et al. [7]). More recently the present author [1] studied another modification of Lupaş operators. Now we consider Beta operator as a weight function on namely,
![]() | (1.1) |
where and
![( ) -----1----------tk----- n + k -----tk----- bn,k(t) = B(k + 1,n )(1 + t)n+k+1 = n k (1 + t)n+k+1 = nvn+1,k(x),](/img/revistas/ruma/v48n1/1a068x.png)
denoting the Beta function. Therefore
![]() |
Let us remark that many years ago W. Meyer-König and K. Zeller [6] have introduced, in order to approximate functions from
, the so-called Bernstein-power series
defined as
![]() |
with Because
, we see that
![( ) ∑∞ ∫ 1 ( ) (Lnf ) --y--- = n mn,k(y ) mn,k (T)f --T--- dT, y ∈ [0,1]. 1 - y k=0 0 1 - T](/img/revistas/ruma/v48n1/1a0617x.png)
The main object of this paper is to establish a Voronovskaya type asymptotic formula and an error estimate for the linear combination of the operators (1.1).
In this section, we shall give certain definition and lemmas which will be used in the sequel.
For every and
we have
![]() | (2.1) |
Lemma 2.1. Let (the set of non-negative integers), we define
![∑∞ ∫ ∞ μr,n,m(x ) = vn+r,k(x ) bn- r,k+r(t)(t - x)mdt k=0 0](/img/revistas/ruma/v48n1/1a0622x.png)
then
| (2.2) |
| (2.3) |
and there holds the recurrence relation:
![pict](/img/revistas/ruma/v48n1/1a0625x.png)
where Consequently, for each
![]() | (2.5) |
Proof. We can easily obtain (2.2) and (2.3) by using the definition of . For the proof of (2.4), we proceed as follows. First
![∑∞ ∫ ∞ φ2(x )μ ′r,n,m (x) = φ2(x)v′n+r,k(x) bn-r,k+r(t)(t - x )mdt - φ2(x)m μr,n,m- 1(x ). k=0 0](/img/revistas/ruma/v48n1/1a0630x.png)
Now, using relations
![2 ′ 2 ′ φ (x)vn,k(x) = (k - nx )vn,k(x) and φ (t)bn,k(t) = [k - (n + 1)t]bn,k(t),](/img/revistas/ruma/v48n1/1a0631x.png)
we obtain
![[ ] φ2 (x) μ′r,n,m(x ) + m μr,n,m -1(x) ∞ ∫ ∞ = ∑ [k - (n + r)x ]v (x ) b (t)(t - x)mdt n+r,k 0 n- r,k+r k=0 ∫ ∑∞ ∞ m = vn+r,k(x ) [(k + r) - (n + 1 - r)t]bn- r,k+r(t)(t - x) dt k=0 0 + [x - r(1 + 2x)]μr,n,m(x ) + (n + 1 - r)μr,n,m+1 (x ) ∞ ∫ ∞ = ∑ v (x ) t(1 + t)b′ (t)(t - x)mdt + [x - r(1 + 2x)]μ (x ) n+r,k 0 n- r,k+r r,n,m k=0 + (n + 1 - r)μr,n,m+1(x) ∑∞ ∫ ∞ [ ] = vn+r,k(x ) (2x + 1)(t - x) + (t - x)2 + (x + 1)x b′n- r,k+r(t)(t - x)mdt k=0 0 + [x - r(1 + 2x)]μr,n,m(x ) + (n + 1 - r)μr,n,m+1 (x ) = - (2x + 1)(m + 1)μr,n,m(x) - (m + 2)μr,n,m+1(x) - m φ2(x)μr,n,m- 1(x ) + [x - r(1 + 2x)]μr,n,m(x ) + (n + 1 - r)μr,n,m+1 (x ).](/img/revistas/ruma/v48n1/1a0632x.png)
This leads to (2.4). The proof of (2.5) easily follow from (2.2) and (2.4). □
Lemma 2.2. If is differentiable
times
on
then we get
![]() | (2.6) |
where
![r∏-1 n + l (n + r - 1)!(n - r - 1)! β (n, r) = ----------- = ---------------2-------. l=0 n - (l + 1) ((n - 1)!)](/img/revistas/ruma/v48n1/1a0638x.png)
Proof. By using the Leibniz theorem, we obtain
using again Leibniz theorem, we get
![(r) (n - 1)! ∑r (r ) bn-r,k+r(t) = ------------ (- 1)i bn,k+i(t). (n - r - 1)!i=0 i](/img/revistas/ruma/v48n1/1a0640x.png)
Thus
![∞ ∫ ∞ (L (r)f) (x ) = (n-+-r --1)!(n---r---1)!∑ v (x ) (- 1)rb(r) (t)f(t)dt. n ((n - 1)!)2 n+r,k 0 n-r,k+r k=0](/img/revistas/ruma/v48n1/1a0641x.png)
On integrating times by parts, we get the required result. □
3. Voronovskaya Asymptotic Formula
Theorem 3.1. Let integrable in
admits its
and
derivatives, which are bounded at a fixed point
and
as
for some
then
![[ 1 ( ) ] lim n ------- L (rn)f (x )- f (r)(x) = {1 + r + x(1+ 2r)}f(r+1 )(x) + 2φ2(x)f (r+2)(x), n→ ∞ β(n, r)](/img/revistas/ruma/v48n1/1a0651x.png)
where
![r∏-1 n + l (n + r - 1)!(n - r - 1)! ∘ --------- β (n,r) = -----------= ----------------2------ , φ(x) := x(1 + x) l=0 n - (l + 1) ((n - 1)!)](/img/revistas/ruma/v48n1/1a0652x.png)
Proof. Using Taylor's formula, we have
![]() | (3.1) |
where
Now, for arbitrary there exists a
such that
![]() | (3.2) |
Using the value of (2.6) in (3.1), we get
where
![∞ ∫ ∞ R (x) = 1-∑ v (x) b (t)(t - x )2ξ(t,x)dt n,r 2 n+r,k 0 n-r,k+r k=0](/img/revistas/ruma/v48n1/1a0659x.png)
In order to completely prove the theorem, it is sufficient to show that
![nRn,r(x) → 0 as n → ∞.](/img/revistas/ruma/v48n1/1a0660x.png)
Now
![nRn,r(x) = Qn,r,1(x ) + Qn,r,2(x )](/img/revistas/ruma/v48n1/1a0661x.png)
where
![n ∑∞ ∫ Qn,r,1(x) = -- vn+r,k(x) | | bn- r,k+r(t)(t - x)2ξ(t,x)dt 2 k=0 |t-x|≤δ](/img/revistas/ruma/v48n1/1a0662x.png)
and
Finally we estimate , using the assumption of theorem,
Thus, from (3.3) and (3.4), we have
![| | 2 nl→im∞ |nRn,r(x)| ≤ 2ɛφ (x).](/img/revistas/ruma/v48n1/1a0667x.png)
Since is arbitrary, therefore
![lim (nR (x)) = 0. n→∞ n,r](/img/revistas/ruma/v48n1/1a0669x.png)
This completes the proof. □
Theorem 3.2. Let and
and let
be the modulus of continuity of
then for
where the norm is sup-norm over ,
![η = 2(2r2 + 4r + n + 1)λ2 + 2 (2r2 + 5r + 2 + n)λ + (r2 + 3r + 2)](/img/revistas/ruma/v48n1/1a0678x.png)
and
![----------1----------- C (n, r) = (n - r - 1)(n - r - 2).](/img/revistas/ruma/v48n1/1a0679x.png)
Proof. Applying the Taylor formula
![∫ t{ } f (r)(t) - f(r)(x ) = (t - x)f(r+1)(x) + (f(r+1)(y ) - f (r+1)(x) dy. x](/img/revistas/ruma/v48n1/1a0680x.png)
Thus
Since,
![( | |) || (r+1) (r+1) || |y---x| ( (r+1) ) f (y) - f (x) < 1 + δ .ω f ;δ .](/img/revistas/ruma/v48n1/1a0682x.png)
Hence, by Schwartz's inequality
Further, choosing and using Lemma 2.1, we get the required result. □
Acknowledgement. The author is extremely grateful to the referee for making valuable suggestions leading to the better presentation of the paper.
[1] Deo N., Direct result on the Durrmeyer variant of Beta operators, Southeast Asian Bull. Math.,32 (2008) (in press). [ Links ]
[2] Durrmeyer J. L., Une formule d'inversion de la transformée de Laplace-applications à la théorie des moments, Thése de 3e cycle, Faculté des Sciences de l' Université de Paris, (1967). [ Links ]
[3] Heilmann M., Approximation auf [0,∞) durch das Verfahren der Operatoren vom Baskakov-Durrmeyer Typ, Dissertation, Universität Dortmund, (1987).
[4] Heilmann M. and Müller M. W., On simultaneous approximation by the method of Baskakov-Durrmeyer operators, Numer. Funct. Anal. and Optimiz., 10(1989), 127-138. [ Links ]
[5] Lupaş A., Die Folge der Betaoperatoren, Dissertation, Universität Stuttgart, (1972). [ Links ]
[6] Meyer-Köning W. and Zeller K. “Bernsteinsche Poetnzreihen”, Studia Math., 19(1960), 89-94, (see also E. W. Cheney and A. Sharma)
[7] Sinha R. P., Agrawal P. N. and Gupta V., On simultaneous approximation by modified Baskakov operators, Bull. Soc. Math. Belg. Ser. B, 42(2)(1991), 217-231. [ Links ]
Naokant Deo
School of Information Science and Engineering, Graduate University of Chinese Academy of Sciences, Zhongguancun Nan Yi Tiao No. 3, Haidian District, Beijing-100080, P. R. China.
dr_naokant_deo@yahoo.com
Recibido: 29 de septiembre de 2006
Aceptado: 28 de marzo de 2007