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Revista de la Unión Matemática Argentina

versión On-line ISSN 1669-9637

Rev. Unión Mat. Argent. v.48 n.1 Bahía Blanca ene./jun. 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.

1. Introduction

The Bernstein-Durrmeyer Mn, n ∈ N0 (the set of non-negative integers), were introduced by Durrmeyer [2] and independently by Lupaş [5]. For a function f ∈ L1[0,1] 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

where

 ( ) n k n-k pn,k(x) = k x (1 - x) , 0 ≤ k ≤ n.

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 C [0, ∞ ) namely,

 ∞ ∫ ∞ ∑ (Lnf ) (x ) = vn,k(x) 0 bn,k(t)f (t)dt, x ∈ [0,∞ ), k=0
(1.1)

where  ( ) v (x) = n+k- 1 ---xk--- n,k k (1+x)n+k 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),

B (.,.) denoting the Beta function. Therefore

 ∫ ∞∑ ∞ (Lnf )(x) := n vn,k(x) vn+1,k(t)f(t)dt, x ∈ [0,∞ ) . k=0 0

Let us remark that many years ago W. Meyer-König and K. Zeller [6] have introduced, in order to approximate functions g from C [0,1 ] , the so-called Bernstein-power series Mng defined as

 ( ∞∑ ( ) { mn,k(z)g n+kk-1-, z ∈ [0,1) (Mng ) (z) = ( k=0 g(1), z = 1,

with  ( ) mn,k (z) = n+kk-1 zk(1 - z)n. Because  ( ) vn,k 1y-y = mn,k (y) , 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

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).

2. Auxiliary Results

In this section, we shall give certain definition and lemmas which will be used in the sequel.

For every n ∈ N and n > (r + 1) we have

( ∑∞ ∫ ∞ ||| vn,k(x) = 1, bn,k(t)dt = 1 { 0 | k=0 ∫ ∞ ||( kv (x) = xv (x), tb (t)dt = k-+-r-+-1. n n,k n+1,k-1 0 n- r,k+r n - r - 1
(2.1)

Lemma 2.1. Let m, r ∈ N0 (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

then

 1 + r + x (1 + 2r ) μr,n,0(x) = 1, μr,n,1(x) = -----------------, (n - r - 1 )

(2.2)

 2(2r2 +-4r +-n-+-1)x2 +-2(2r2-+-5r-+-2-+-n-)x-+-(r2-+-3r-+-2)- μr,n,2(x ) = (n - r - 1)(n - r - 2) ,

(2.3)

and there holds the recurrence relation:

pict

where  ∘ --------- φ(x) = x (1 + x). Consequently, for each x ∈ [0,∞ )

 ( -[(m+1 )∕2]) μr,n,m(x) = O n .
(2.5)

Proof. We can easily obtain (2.2) and (2.3) by using the definition of μr,n,m (x ) . 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

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),

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 ).

This leads to (2.4). The proof of (2.5) easily follow from (2.2) and (2.4). □

Lemma 2.2. If f is differentiable r times (r = 1,2,...) on [0, ∞ ), then we get

( ) ∑∞ ∫ ∞ L(r)f (x) = β (n,r) vn+r,k(x ) bn-r,k+r (t)f(t)dt, x ∈ [0,∞ ) n k=0 0
(2.6)

where

 r∏-1 n + l (n + r - 1)!(n - r - 1)! β (n, r) = ----------- = ---------------2-------. l=0 n - (l + 1) ((n - 1)!)

Proof. By using the Leibniz theorem, we obtain

pict

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

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

On integrating r times by parts, we get the required result. □

3. Voronovskaya Asymptotic Formula

Theorem 3.1. Let f integrable in [0,∞ ), admits its (r + 1) - th and (r + 2) - th derivatives, which are bounded at a fixed point x ∈ [0,∞ ) and f (r)(t) = O (tα) as t → ∞ for some α > 0, 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)

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)!)

Proof. Using Taylor's formula, we have

 (t - x)2 (t - x )2 f(r)(t) - f(r)(x ) = (t - x)f(r+1)(x ) +-------f (r+2)(x) + --------ζ(t,x ), 2 2
(3.1)

where

pict

Now, for arbitrary ɛ > 0, A > 0 there exists a δ > 0 such that

| | | | |ζ (t,x )| ≤ ɛ for |t - x| ≤ δ, x ≤ A.
(3.2)

Using the value of (2.6) in (3.1), we get

pict

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

In order to completely prove the theorem, it is sufficient to show that

nRn,r(x) → 0 as n → ∞.

Now

nRn,r(x) = Qn,r,1(x ) + Qn,r,2(x )

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

and

 ∑∞ ∫ Qn,r,2(x) = n- vn+r,k(x) | | bn- r,k+r(t)(t - x)2ξ(t,x)dt 2 k=0 |t-x|>δ

Using (3.2) and (2.3).

pict

Finally we estimate Qn,r,2(x) , using the assumption of theorem,

pict

Thus, from (3.3) and (3.4), we have

 | | 2 nl→im∞ |nRn,r(x)| ≤ 2ɛφ (x).

Since ɛ is arbitrary, therefore

lim (nR (x)) = 0. n→∞ n,r

This completes the proof. □

Theorem 3.2. Let  (r+1) f ∈ C [0,∞ ) and [0,λ ] ⊆ [0,∞ ) and let  ( (r+1) ) ω f ;. be the modulus of continuity of f(r+1 ) then for r = 0,1, 2,...

∥ ∥ ∥∥ ---1---( (r) ) (r)∥∥ ∥ β(n,r ) L n f - f ∥ C[0,λ] pict

where the norm is sup-norm over [0,λ] ,

η = 2(2r2 + 4r + n + 1)λ2 + 2 (2r2 + 5r + 2 + n)λ + (r2 + 3r + 2)

and

 ----------1----------- C (n, r) = (n - r - 1)(n - r - 2).

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

Thus

pict

Since,

 ( | |) || (r+1) (r+1) || |y---x| ( (r+1) ) f (y) - f (x) < 1 + δ .ω f ;δ .

Hence, by Schwartz's inequality

pict

Further, choosing δ = C (n,r ) 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.

References

[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