A Study on the Bezier Variant of Certain Bernstein Durrmeyer Operator

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(1)Journal of Computer and Mathematical Sciences, Vol.6(3),135-142, March 2015 (An International Research Journal), www.compmath-journal.org. ISSN 0976-5727 (Print) ISSN 2319-8133 (Online). A Study on the Bezier Variant of Certain Bernstein Durrmeyer Operator Yogita Parmar, V. K. Gupta and S. K. Tiwari School of Studies in Mathematics, Vikram University, Ujjain, M. P., INDIA. (Received on: March 2, 2015) ABSTRACT In this paper, we introduce a Bezier Variant of a new type of Bernstein Durrmeyer operator, which was introduced by Gupta3. We estimate the rate of convergence by using the decomposition technique of function of bounded variation and applying the optimum bound. In this paper it is observed that the analysis for our Bezeir variant of new Bernstein Durrmeyer operators is different from the usual Bernstein Durrmeyer operators studied by Zeng and Chen10. Keywords: Lebesgue integrable functions, Bernstein polynomials, Bezier variant, Functions of bounded variation.. 1. INTRODUCTION For a function defined on the interval [0,1], the operator introduced by Durrmeyer1 are defined by . , 1 ∑ , , , 0, 1. (1). where , 1 . Gupta3 introduced a different Durrmeyer type modification of the Bernstein polynomials and estimated the rate of convergence for functions of bounded variation. The operators introduced by Gupta3 are defined by . , ∑ , , , where. 0,1. March, 2015 | Journal of Computer and Mathematical Sciences | www.compmath-journal.org. (2).

(2) 136. Yogita Parmar, et al., J. Comp. & Math. Sci. Vol.6 (3), 135-142 (2015). " % #! & , 1& % #! and % 1 ." 1. It is easily verified that the values of , used in (1) and (2) are same. Also it is easily verified that . ' , 1,. . . ( , 1 . and , 0 By considering the integral modification of Bernstein polynomials in the form (2) some approximation properties become simpler in the analysis. So it is significant to study further on the different integral modification of Bernstein polynomials introduced by Gupta3. For ) * 1, we now define the Bezier variant of the operators (2), to approximate Lebesgue integral functions on the interval [0,1] as . + , ,+ , ∑ ,,. 0,1. (3). where + + ,, -, -,& and . -, ' ,. , .. where k ≤ n and 0 otherwise. Some important properties of -, are as follows:. (i) -, -,& , , # 0,1,2,3 …, ′ (ii) -, , , # 1,2,3 …, " (iii) -, , 22, # 1,2,3 …, (iv) -, 3 -, 3 -,4 3 5 3 -, 3 0, 0 6 6 1.. For every natural number #, -, increases strictly from 0 to 1 on [0,1]. Alternatively we may rewrite the operators (3) as . ,+ , 7,+ , . 0881. March, 2015 | Journal of Computer and Mathematical Sciences | www.compmath-journal.org. (4).

(3) Yogita Parmar, et al., J. Comp. & Math. Sci. Vol.6 (3), 135-142 (2015). where. 137. . + 7,+ , ' ,, , . . It is easily verified that ,+ , are linear positive operators ,+ 1, 1 and for ) = 1 the operators , , 9 , , i.e. the operators (3) reduce to the operators + (2). For further properties of ,, , we refer the reader to3.. Guo2 studied the rate of convergence for bounded variation functions for Bernstein Durrmeyer operators. Zeng and Chen10 were the first to estimate the rate of convergence for the Bezier-variant of Bernstein Durrmeyer operators. Several other Bezier variants of some summation-integral type operators were studied in4,6 and9 etc. It is well known that Bezier basis functions play an important role in computer aided design. Moreover the recent work on different Bernstein Bezier type operators considered the advantage of the operators ,+ , Bernstein Durrmeyer operators considered in10 is that one does not require the result of the type Lemma 3 and Lemma 4 of10. Consequently some approximation formulae become simpler. In Further for ) = 1, these operators provide improved estimates over the main result of2 and3. In the present paper, we estimate the rate of point wise convergence of the operators ,+ , at those points 0, 1 at which one sided limits and exist. 2. AUXILIARY RESULTS In this section we give certain results, which are necessary to prove the main result. Lemma 2.13 If m is sufficiently large , then 21 1 8 ; 4 , 8 : :. Lemma 2.24 For every 0 8 # 8 :, 0,1 and for all : <, we have 1 ;, 8 =2>:1 Lemma 2.3 For all 0,1, there holds ) + ,;, 8 ). ;, 8 =2>:1 . Proof. Using the well known inequality ?@+ + ? 8 )|@ |, 0 8 @, 8 1, ) * 1 and by Lemma 2.2, we obtain March, 2015 | Journal of Computer and Mathematical Sciences | www.compmath-journal.org.

(4) 138. Yogita Parmar, et al., J. Comp. & Math. Sci. Vol.6 (3), 135-142 (2015). ,;, 8 ). ;, 8 +. ). =2>:1 . Lemma 2.4 Let 0,1 and 7;,+ , be the kernel defined by (4). Then for m sufficiently large, we have B;,+ , C D 7;,+ , 8 E. and. 4+."" , ;"EF. . 1 B;,+ , G D H 7;,+ , 8. 4+."" , ;H"F. 0 8 C 8 ,. (5). 8 G 8 1,. (6). Proof. We first prove (5), as follows E. E. ( 7;,+ , 8 ( 7;,+ , . . 4 C4. 1 8 4 , C4 ;,+ );, 4 , 8 C4 2)1 8 , C4 by Lemma 2.1. The proof of (6) is similar. 3. MAIN RESULT. In this section we prove the following main theorem. Theorem 3.1 Let f be a function of bounded Variation on the interval [0,1] and suppose ) ≥ 1. Then for every 0,1 and m sufficiently large, we have I;,+ , J. 1 ) 1 ). KI ) 1 ). ; "&"/√. 5) | |. 8 ' :1 =2>:1 . . M. ""/√. March, 2015 | Journal of Computer and Mathematical Sciences | www.compmath-journal.org. N" ,.

(5) Yogita Parmar, et al., J. Comp. & Math. Sci. Vol.6 (3), 135-142 (2015). 139. , RS 0 8 6 T 0, RS N" Q , RS 6 8 1 and UVWN" is the total variation of N" on [a, b]. where. Proof. Clearly. I;,+ , X. First, we have. 1 ) 1 ) YI. ) 1 + 8 ?;,+ N" , ? Z;,+ [N , . Z | | 4. +&. . ". ;,+ [N , ( ;,+ , ( 7;,+ , ". . . ". ( 7;,+ , 2 ( 7;,+ , . . ". 1 2 ( 7;,+ , . . ". 1 2 \( 7;,+ , ( 7;,+ , ] . . ". 1 2 \1 ( 7;,+ , ] . . 1 2 ( 7;,+ , ". Using lemma 2.2, Lemma 2.3 and the fact that . . ' ;,. ( ;, , .. we have. ". ;. . +. ;,+ [N , 1 2 ' ,;, ( ;, ;. +. . ". 1 2 ' ,;, ' ;,. . .. March, 2015 | Journal of Computer and Mathematical Sciences | www.compmath-journal.org. (7).

(6) 140. Yogita Parmar, et al., J. Comp. & Math. Sci. Vol.6 (3), 135-142 (2015) . + 1 2 ' ;,. -;,. .. Since ;. +&. ' ,;,. .. 1,. therefore we have. ;. ;. )1 2 + +& ;,+ [N , . 2 ' ;,. -;,^ ' ,;,. ) 1 ) 1 . . By the mean value theorem, it follows +&. +& +& + -;,.& ) 1 ;,. _;,. , ,;,. -;,. where + + + 6 _;,. 6 -;,. -;,.&. Hence. ;. )1 + + _;,. a I;,+ [N , . I 8 2 ' ;,. `-;,. ) 1 .. ;. + + -;,.& a 8 2 ' ;,. `-;,. . ;. 4 , 8 2) ' ;,. .. where we have used the inequality + @+ 6 ) @, 0 8 @, 8 1 and ) * 1. Applying Lemma 2.2, we get +. Z;,+ [N , +&Z . 4+. =4b;"". , 0,1. (8). Next we estimate ;,+ N" , . By a Lebesgue-Stieltjes integral representation, we have . ;,+ N" , ( 7;,+ , N" . `c c c a 7;,+ , N" d F e f f4 fg , say,. where k l0, /√:m , k4 l /√: , 1 /√:m /√:, 1.. (9) and. kg l 1 . March, 2015 | Journal of Computer and Mathematical Sciences | www.compmath-journal.org.

(7) Yogita Parmar, et al., J. Comp. & Math. Sci. Vol.6 (3), 135-142 (2015). 141. We first estimate E . Writing C /√: and using Lebesgue-Stieltjes integration by parts, we have E. f ( N" `B;,+ , a . E. N" C B;,+ , C ( B;,+ , N" . Since |N" C | 8 U"E&N" , it follows that. |f | 8 U"E&N" B;,+ , C B;,+ , U"oN" E. By using (5) of Lemma 2.4, we get |f | 8 U"E&N" . 4+."" E 4+."". "oF U"oN" ; +"EF. Integration by parts the last term we have after simple computation |f | 8. E " N 2). 1 U"N" Uo ". 2( q p 4 : g . Now replacing the variable y in the last integral by /√, we obtain |f | 8. 4+." IU"N" . ;". " ∑; U" r N" I 8 √s. t+ ∑; U" r N" ;"" " √s. (10). Using a similar method and (6) of Lemma 2.4, we get |fg | 8. t+ ∑; U"&"/√N" ;"" ". (11). Finally we estimatef4 . For l /√:, 1 /√:m, we have "&"/√; N" , √;. |N" | |N" N" | 8 U""/ and therefore "&"/√. |f4 | 8 U""/√ V. "&"/√. N" ""/√. `B;,+ , a. Since W B; , 8 1, for all @, u 0,1, therefore "&"/√; N" √;. |f4 | 8 U""/. (12). Collecting the estimates (9) - (12), we have v+. "&"/√. ?;,+ N" , ? 8 ;"" ∑; U""/√. N" . March, 2015 | Journal of Computer and Mathematical Sciences | www.compmath-journal.org. (13).

(8) 142. Yogita Parmar, et al., J. Comp. & Math. Sci. Vol.6 (3), 135-142 (2015). Combining the estimates of (7), (8) and (9), our theorem follows.. For ) = 1, we obtain the following corollary, which is an improved estimate over the main result of2, and3. Corollary 3.2 Let f be a function of bounded variation on the interval [0,1]. Then for every 0,1 and n sufficiently large, we have 1 I; , I 2 | |. 8 =4b;"". v. ;"". ∑; U. "&"/√ N" . ""/√. REFERENCES 1. Durremeyer J. L., Une formule d’inversion de la transformee de Laplace: Application a la Theorie des Moments, These de 3e cycle, Faculte des Sciences de l’ Universite de Paris (1967). 2. Guo S., On the rate of convergence of Durrmeyer operator for function of bounded variation, J. Approx. theory, 51, 183-192 (1987). 3. Gupta V., A note on the rate of convergence of Durrmeyer type operators for functions of bounded variation. Soochow J. Math., 23(1), 115-118 (1997). 4. Gupta V., Rate of convergence on Baskakov Beta Bezier operators for functions of bounded Variation, Int. J. Math. and Sci., 32(8), 471-479 (2002). 5. Gupta V., Rate of approximation by new sequence of linear positive operators, Comput. Math. Appl., 45(12), 1895-1904 (2003). 6. Gupta V., Degree of approximation to function of bounded variation by Bezier variant of MKZ operators, J. Math. Appl., 289(1), 292-300 (2004). 7. Gupta M.K., A note on the Bezier variant of certain Bernstein Durrmeyer operators, J. of Inequalities in Pure and Applied Mathematics, 74(6), 1-7 (2005). 8. Zeng X.M., Bounds for Bernstein basis functions and Meyer – Koing – Zeller basic functions, J. Math. Anal. Appl., 219, 364-376 (1998). 9. Zeng X.M., On the rate of convergence of the generalized Szasz type operators for functions of bounded variation, J. Math. Anal. Appl., 226, 309-325 (1998). 10. Zeng X.M. and Chen W., On the rate of convergence of the generalized Durrmeyer type operators for functions of bounded variation, J. Approx. Theory, 102, 1-12 (2000).. March, 2015 | Journal of Computer and Mathematical Sciences | www.compmath-journal.org.

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