Print version ISSN 0041-6932
Rev. Unión Mat. Argent. vol.50 no.1 Bahía Blanca June 2009
Ali J. Mohammad and Amal K. Hassan
Abstract. In this paper, we study some direct results in simultaneous approximation for a new sequence of linear positive operators of Szãsz-Beta type operators. First, we establish the basic pointwise convergence theorem and then proceed to discuss the Voronovaskaja-type asymptotic formula. Finally, we obtain an error estimate in terms of modulus of continuity of the function being approximated.
Key words and phrases. Linear positive operators; Simultaneous approximation; Voronovaskaja-type asymptotic formula; Degree of approximation; Modulus of continuity.
In  Gupta and others studied some direct results in simultaneous approximation for the sequence:
where , and . After that, Agrawal and Thamer  applied the technique of linear combination introduced by May  and Rathore  for the sequence . Recently, Gupta and Lupas  studied some direct results for a sequence of mixed Beta-Szãsz type operator defined as .
In this paper, we introduce a new sequence of linear positive operators of Szãsz-Beta type operators to approximate a function belongs to the space , as follows:
We may also write the operator (1.1) as where , being the Dirac-delta function.
The space is normed by .
There are many sequences of linear positive operators with are approximate the space . All of them (in general) have the same order of approximation . So, to know what is the different between our sequence and the other sequences, we need to check that by using the computer. This object is outside our study in this paper.
Throughout this paper, we assume that denotes a positive constant not necessarily the same at all occurrences, and denotes the integer part of .
For the Szãsz operators are defined as , and for (the set of nonnegative integers), the -th order moment of the Szãsz operators is defined as .
LEMMA 2.1.  For , the function defined above, has the following properties:
(i) , and the recurrence relation is
(ii) is a polynomial in of degree at most ;
(iii) For every,.
From above lemma, we get
For , the -th order moment for the operators (1.1) is defined as:
LEMMA 2.2.For the function , we have , and there holds the recurrence relation:
Further, we have the following consequences of :
(i) is a polynomial in of degree exactly ;
(ii) For every .
Proof: By direct computation, we have , and . Next, we prove (2.2). For it clearly holds. For , we have
Using the relations and , we get:
By using the identity , we have
Integrating by parts, we get
from which (2.2) is immediate.
From the values of and , it is clear that the consequences (i) and (ii) hold for and . By using (2.2) and the induction on the proof of consequences (i) and (ii) follows, hence the details are omitted.
From the above lemma, we have
LEMMA 2.3. Let and be any two positive real numbers and . Then, for any , we have
Making use of Schwarz inequality for integration and then for summation and (2.3), the proof of the lemma easily follows.
LEMMA 2.4.  There exist polynomials independent of and such that
Firstly, we show that the derivative is an approximation process for .
Theorem 3.1. If for some and exists at a point , then
Further, if exists and is continuous on , then (3.1) holds uniformly in .
Proof: By Taylor's expansion of , we have
where, . Hence
Now, using Lemma 2.2 we get that is a polynomial in of degree exactly
, for all . Further , we can write it as:
Next, making use of Lemma 2.4 we have
Since as , then for a given , there exists a such that , whenever . For , there exists a constant such that .
Now, since . Hence,
Now, applying Schwartz inequality for integration and then for summation, (2.1) and (2.3) we led to
Again using Schwarz inequality for integration and then for summation, in view of (2.1) and Lemma 2.3, we have
Now, since is arbitrary, it follows that . Also, as and hence , combining the estimates of and , we obtain (3.1).
To prove the uniformity assertion, it sufficient to remark that in above proof can be chosen to be independent of and also that the other estimates holds uniformly in .
Our next theorem is a Voronovaskaja-type asymptotic formula for the operators .
THEOREM 3.2. Let for some . If exists at a point , then
Further, if exists and is continuous on the interval , then (3.3) holds uniformly on .
Proof: By the Taylor's expansion of , we get
where as .
By Lemma 2.2 and (3.2), we have
Hence in order to prove (3.3) it suffices to show that as , which follows on proceeding along the lines of proof of as in Theorem 3.1.
The uniformity assertion follows as in the proof of Theorem 3.1.
Finally, we present a theorem which gives as an estimate of the degree of approximation by for smooth functions.
THEOREM 3.3. Let for some and . If exists and is continuous on , then for sufficiently large ,
where are constants independent of and , is the modulus of continuity of on , and denotes the sup-norm on .
Proof. By Taylor's expansion of , we have
where lies between , and is the characteristic function of the interval . Now,
By using Lemma 2.2 and (3.2), we get
To estimate we proceed as follows:
Now, for , using Schwartz inequality for integration and then for summation, (2.1) and (2.3), we have
Therefore, by Lemma 2.4 and (3.4), we get
(since but fixed )
Choosing and applying (3.5), we are led to
Since , we can choose in such a way that for all . Thus, by Lemmas 2.3 and 2.4 , we obtain
For , we can find a constant such that . Hence, using Schwarz inequality for integration and then for summation ,(2.1), (2.3), it easily follows that for any , uniformly on .
Combining the estimates of , the required result is immediate.
 P.N. Agrawal and Kareem J. Thamer. Linear combinations of Szãsz-Baskakov type operators, Demonstratio Math.,32(3) (1999), 575-580. [ Links ]
 Vijay Gupta and Alexandru Lupas. Direct results for mixed Beta-Szãsztype operators, General Mathematics 13(2), (2005), 83-94. [ Links ]
 Vijay Gupta, G.S. Servastava and A. Shahai. On simultaneous approximation by Szãsz-Beta operators, Soochow J. Math. 21, (1995), 1-11. [ Links ]
 C.P. May. Saturation and inverse theorems for combinations of a class of exponential- type operators, Canad. J. Math. 28 (1976), 1224-1250. [ Links ]
 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 ]
 E. Voronovskaja. Détermination de la forme asymptotique d'ápproximation des fonctions par les polynômes de S.N. Bernstein, C.R. Adad. Sci. USSR (1932), 79-85. [ Links ]
Ali J. Mohammad
University of Basrah,
College of Education,
Dept. of Mathematics,
Amal K. Hassan
University of Basrah,
College of Science,
Dept. of Mathematics,
Recibido: 8 de noviembre de 2006
Aceptado: 11 de marzo de 2008