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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.46 no.1 Bahía Blanca Jan./June 2005
Simultaneous approximation with linear combination of integral Baskakov type operators
Kareem J. Thamer, May A. Al-Shibeeb, A.I. Ibrahem
Abstract:
The aim of the present paper is to study some direct results in simultaneous approximation for the linear combination of integral Baskakov type operators.
1991 Mathematics Subject Classification. 41A28, 41A36.
Key words and phrases. Linear positive operators; Linear combinations; Simultaneous approximation.
1 INTRODUCTION
Agrawal and Thamer [1] introduced a new sequence of linear positive operators called integral Baskakov – type operators to approximate unbounded continuous functions on and it is defined as follow
Let , .
Then,
(1.1) ,
where is the kernel of Lupas operators . We may also write (1.1) as :
where being the Dirac delta function.
The space is normed by .
The operator (1.1) was used to study the degree of approximation in simultaneous approximation by Agrawal and Thamer [1]. It turned out that the order of approximation by the operator (1.1) is, at best, , howsoever smooth the function may be. Thus, if we want to have a better order of approximation, we have to slacken the positivity condition. This is achieved by considering some carefully chosen linear combination introduced by May [6] and Rathore [7] of the operator (1.1). The linear combination is defined as follows:
Let be arbitrary but fixed distinct positive integers. Then, following Agrawal and Sinha [3] , the linear combination of , is given by
where is the Vandermonde determinant obtained by replacing the operator column of the above determinant by the entries 1. We have
where
The object of the present paper is to investigate the degree of approximation of the operator . First we establish a Voranovskaja type asymptotic formula and then obtain an error estimate in terms of the local modulus of continuity for the operator .
2 AUXILIARY RESULTS
Throughout our work, denotes the set of natural numbers, integers, and an open interval containing .
LEMMA 2.1 [4]. If for (the set of nonnegative integers), the order moment of Lupas operators is defined by
Hence, , and there holds the recurrence relation
Consequently
(i) is a polynomial in of degree at most .
(ii) For every , where denotes the integral part of .
LEMMA 2.2 [1]. Let the function be defined as
Then,
and
Hence,
(i) is a polynomial in of degree
(ii) For every .
(iii) The coefficients of in and are given by and .
LEMMA 2.3 [5]. There exist polynomials independent of and such that
LEMMA 2.4 [6].If are defined as in (1.4), then
LEMMA 2.5 [8].Let be times differentiable on such that for some as . Then for , we have
LEMMA 2.6 [2]. For and sufficiently large, there holds
where is a certain polynomial in of degree .
3 MAIN RESULTS
In this section we shall state and prove the main results.
Theorem 3.1.Let and be bounded on every finite subinterval of admitting a derivative of order at a fixed point . Let as for some , then we have
and
where are certain polynomials in .
Further, the Limits (3.1) and (3.2) hold uniformly in , if exists and is continuous on .
Proof. By the Taylor expansion, we have
where .
Thus, using Lemma 2.5, we have for sufficiently large
where
It's clear that
Let , where
Thus, by (1.4),
Now, in view of Lemma 2.4, we have
where is a constant depending only on and .
Next, by Lemma 2.4 and Lemma 2.6,we get
Thus
Now we must prove that . For this, it is sufficient to prove that
Using Lemma 2.3, we get
where , and then applying the Schwarz inequality we get:
Since , for a given such that , whenever , and for there exists a constant such that .
Hence, as , we have
Now, by Lemma 2.2, we get
By Lemma 2.1, we have
Since is arbitrary, it follows that . The assertion (3.2) follows along similar lines by using Lemma 2.4 for in place of .
The last assertion follows, due to the uniform continuity of (enabling to became independent of ) and the uniform of term in the estimate of (because, in fact, it is a polynomial in .
The next result provides an estimate of degree approximation in .
Theorem 3.2. Let and be bounded on every finite subinterval of . Let If exists and is continuous on , then for sufficiently large
where denotes the modulus continuity of on , denotes the sup-norm on .
Proof: For by the hypothesis we have
where lies between is the characteristic function of the set .Operating on this equality by and breaking the right hand side into three parts say, corresponding to the three terms on the right hand side of (3.3) as in the proof of Theorem 3.1, we have
To estimate , we have for every
Since
Using (3.4) and Lemma 2.3, we have
Putting , then applying Schwarz inequality for summation and for integral and Lemmas 2.1 and 2.2 as in the proof of theorem 3.1, we get
Choosing , it follows that
where term holds uniformly in .
For and , we can choose a in such a way that . Hence
Now, for we can find a positive constant such that , where is any integer.
Hence, by Schwarz inequality, Lemmas 2.1 and 2.2 we have
The required result follows on combining the estimates of and . e o
Acknowledgement. The authors are thankful to the referee for making substantial improvements in the paper.
References:
[1] P.N. Agrawal and Kareem J. Thamer, Approximation of unbounded functions by a new sequence of linear positive operators, J. Math. Anal. App. 225(1998), 660-672. [ Links ]
[2] P.N. Agrawal and Kareem J. Thamer, Degree of approximation by a new sequencf linear operators, Kyungpook Math. J., 41(1) (2001), 65-73. [ Links ]
[3] P.N. Agrawal and T.A.K. Sinha, A saturation theorem for a combination of modified Lupas operators in Lp-spaces, Bull. Inst. Math. Academia Sinica 24 (1996), 159-165. [ Links ]
[4] H.S. Kasana, P.N. Agrawal and V. Gupta, Inverse and Saturation theorems for linear combination of modified Baskakov operators, Approx. Theory Appl. 7(2)(1991), 65-82. [ Links ]
[5] H.S. Kasana, G. Prasad, P.N. Agrawal and A. Sahai, On modified Szasz operators, Proc. Int. Conf. Math. Anal. And its Appl. Kuwait (1985), 29-41, Pergamon Press, Oxford (1988). [ Links ]
[6] C.P. May, Saturation and Inverse theorem for combinations of a class of exponential type operators, Canad. J. Math. 28(1976), 1224-1250. [ Links ]
[7] R.K.S. Rathore, Linear Combinations of Linear Positive Operators and Generating Relations in Special Functions, Ph.D. Thesis I.I.T. Delhi (India) (1973). [ Links ]
[8] A. Sahai and G. Prasad, On simultaneous approximation by modified Lupas operators, J. Approx. Theory 45 (1985), 122-128. [ Links ]
Kareem J. Thamer
Department of Mathematics,
College of Education-Amran,
Sana'a University,
Maeen Post Office, Box (13475), Sana'a – Republic of Yemen.
k_alabdullah2005@yahoo.com
May A. Al-Shibeeb
Rayed, P.O.Box (46379),
Post Code 11532, Saudia Arabia King Dom.
maey9999@hotmail.com
A.I. Ibrahem
Department of Mathematics,
College of Science,
Basrah University,
Basrah – Iraq.
Recibido: 26 de diciembre de 2002
Aceptado: 25 de agosto de 2005