Some Approximation Results For (p, q)-lupaş-schurer Operators

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1 Filomat 3:1 018, Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: Some Approimation Results For p, q-lupaş-schurer Operators K. Kanat a, M. Sofyalıoğlu a a Polatlı Faculty of Science and Arts, Gazi University, 06900, Anara, Turey Abstract. In this paper, we introduce Lupaş-Schurer operators based on p, q-integers. Then, we deal with the approimation properties for p, q-lupaş-schurer operators based on Korovin type approimation theorem. Moreover, we compute rate of convergence by using modulus of continuity, with the help of functions of Lipschitz class and Peetre s K-functionals. 1. Introduction In 191, Bernstein defined the following sequences of linear and positive operators B n : C0, 1 C0, 1 B n f ; f n n 1 n, 1 where n N and f C0, 1. In 1987, Lupaş 3 introduced q-calculus for Bernstein operators. He defined q-analogue of the Bernstein operators in the following form L n,q f ; f q n n q q 1 1 n q. n j1 {1 + qj 1 } The operators L n,q f ; generate positive linear operators for all q > 0. As we see above, there are two inds of q-analogue of Bernstein operators i.e Phillips and Lupaş. In 015, Mursaleen, Ansari and Khan 4 first introduced the concept of p, q-calculus in approimation theory. They defined a generalisation of q-bernstein operator and called it as p, q-bernstein operators. The form of the p, q-bernstein operators are as follows B n, f ; 1 p nn 1 n f p n n p 1 {p j q j }. 3 n 1 j0 010 Mathematics Subject Classification. Primary 41A10; Secondary 41A5, 41A36 Keywords. p, q-integers; Lupaş operators; Korovin type approimation theorem; modulus of continuity; functions of Lipschitz class; Peetre s K-functionals Received: 7 February 017; Accepted: 7 September 017 Communicated by Dragan Djordjević addresses: adiranat@gazi.edu.tr K. Kanat, melesofyalioglu@gazi.edu.tr M. Sofyalıoğlu

2 K. Kanat, M. Sofyalıoğlu / Filomat 3:1 018, In 017, p, q-analogue of Lupaş Bernstein operators are defined by Khalid et al 13. For each p > 0 and q > 0, L n : C0, 1 C0, 1 p f n n n p n n 1 1 n L n f ; n j1 {pj q j 1 } are p, q-analogue of Lupaş Bernstein operators. Note that p, q-analogue of Lupaş Bernstein operators generate positive linear operators for all p > 0 and q > 0. Consequently, the p, q-counterparts introduced by Mursaleen et al 4 is generalisation of q-analogue of Bernstein operators given by Phillips 19 whereas Khalid et al 13 generalised q-lupaş Bernstein operators. The novelty of p, q-calculus in computer aided geometric design CAGD given by Khalid et al 13 will help readers to understand the application. Another advantage of using parameter p has been shown in 9. Besides this, we also refer to the reader some recent papers on p, q-calculus in approimation theory: e.g. 1, 6, 7, 8, 10, 11, 1, 16, 17 and 18. Before proceeding further, we recall significant definitions and notations on the concept of p, q-calculus. For any non-negative p anq q, the p, q-integers of the number n is defined by n : p n 1 + p n q + + pq n + q n 1 p n q n p q if p q 1 np n 1 if p q 1 n if p q 1 n q if p 1 where n q denotes q-integers for n 0, 1,,.... The p, q-binomial epansion is defined as where n a + by n : n n!!n! p n n 1 a n b n y, 5 are the p, q-binomial coefficients. By using 5 we obtain and + y n + yp + qyp + q y... p n 1 + q n 1 y 1 n 1 p qp q... p n 1 q n 1. More information about p, q-calculus can be read from 4 and 15., 4. Construction of The Operator Schurer type generalization of linear positive operators has been studied in several years. In this part, we construct the class of the p, q-analogue of Lupaş Schurer operators. Definition.1. We consider for each p > 0, q > 0 and for any m N, 0, 1 and f C0, 1 + l, fied l N + {0}. We construct the p, q-analogue of Lupaş Schurer operators by p f m+l m + l m p m+l m+l 1 1 m+l L f ;. 6 j1 {pj q j 1 }

3 K. Kanat, M. Sofyalıoğlu / Filomat 3:1 018, Note that if we tae p q 1 p, q-lupaş-schurer operators reduce to be Schurer-Bernstein operators which are defined in the article of Schurer 14 in 196. We have the following lemma to give some equalities for the operators 6. Lemma.. Let L.;. is given by 6. The following equalities hold. L L 1; 1, L t; m + l m, 8 L t ; pm+l 1 m + l m L t ; m + l m 1 t ; pm+l 1 m + l Proof. i Firstly, we begin with n So, we have from 1 n + m + q m + l 9 m p1 + q, 10 q m + l m + l + 1 m p1 + q m 11 p n n 1 1 n n p n n 1 1 n 1 p1 + qp 1 + q... p n q n 1 n {p j q j 1 }. 1 j1 p n n 1 1 n n j1 {pj q j 1 } In the 13 we choose n : m + l. Then we get m + l p m+l m+l 1 1 m+l j1 {pj q j 1 } Now, let us write L 1; L 1; m + l p m+l m+l 1 1 m+l j1 {pj q j 1 }. 15

4 By using 14 and 15 we obtain L 1; 1. ii Secondly, we write L t; as follows L t; 1 direct calculations yield, L p m+l m K. Kanat, M. Sofyalıoğlu / Filomat 3:1 018, m + l p m+l m m+l! p m+l m+l! m m t; m + l p m+l m+l 1 1 m+l j1 {pj q j 1 }!m+l! p m+l m+l 1 j1 {pj q j 1 } 1!m+l! p m+l m+l 1 j1 {pj q j 1 } p m+l 1 m+l! m!m+l 1! p m+l 1m+l q +1 p m+l 1 m+l m 1 m+l 1 m+l +1 1 m+l 1 j1 {pj q j 1 } p m+l 1m+l q m+l 1 j1 {pj q j 1 } p m+l 1 m + l p m+l 1 m m + l p m+l 1 m p m+l 1m+l q +1 1 m+l 1 1 j0 {p j 1 + q j } p m+l 1m+l q m+l 1 1 j1 {p j 1 + q j } p m+l 1m+l q +1 j0 {p j q j+1 } 1 1 m+l 1. Now, suppose that L u t; u + 1 u u+1, or equivalently, u 1 m + l m m + l m m + l m m+l 1 u u + 1 m+l 1 u u + 1 u u + 1 L 1;. p m+l 1m+l 1 p m+l 1 j0 j0 {p j+1 + q j+1 u} p m+l 1m+l { p j + q j q p u} q p u q p u

5 K. Kanat, M. Sofyalıoğlu / Filomat 3:1 018, Finally, we obtain L t; m + l m. iii Thirdly, let us write L t ; L t ; 1 p m+l m p m+l m p m+l m p m+l 1 +1 m m + l p m+l m+l 1 1 m+l j1 {pj q j 1 } m+l!!m+l! p m+l m+l 1 1 m+l j1 {pj q j 1 } m+l! 1!m+l! p m+l m+l 1 1 m+l j1 {pj q j 1 } m+l!!m+l 1! p m+l 1m+l q m+l 1 j1 {pj q j 1 }. Tae + 1 p + q, then we get L t ; m + l m p m+l 1 p +q m +q m + l m m+l!!m+l 1! p m+l 1m+l q +1 j1 {pj q j 1 } p m+l 1 p p m+l m+l 1 p m+l 1m+l q m+l 1 1 j0 {p j 1 + q j } p m+l 1m+l q +1 1 m+l 1 1 j0 {p j 1 + q j }.

6 K. Kanat, M. Sofyalıoğlu / Filomat 3:1 018, 17 9 p m+l 1m+l q L t ; pm+l 1 m+l 1 m + l p1 m m+l 1 j1 {p j 1 + q j } m+l 1 1 p1 + qm + l m m+l pm+l 1 m + l m m + l p m+l 1 L 1; + q m + l m p1 + q pm+l 1 m + l m L 1; + q m + l m p1 + q pm+l 1 m + l L m p m+l m+l 3 q +1 1 m+l 1 j0 {p j 1 + q j } m+l m+l m + l p m+l m+l 3 1 j {p j 1 + q j } 1 p 1 q p 1 m+l m + l p m+l m+l 3 q p 1 1 j {p j + q j q p 1 } 1; + q m + l m p1 + q L 1; pm+l 1 m + l + q m + l 1 m m p1 + q. As a result, we obtain L t ; pm+l 1 m + l m + q m + l. m p1 + q iv In order to show the equality of the first central moment L t ;, we will use the linearity of the operator L : L t ; Lt; L1; m + l m m + l 1. m

7 K. Kanat, M. Sofyalıoğlu / Filomat 3:1 018, v For the second central moment L t ;, we will again use the linearity of the operator L : L t ; L t ; L t; + L 1; pm+l 1 m + l m pm+l 1 m + l m As a consequence, the proof is completed. 3. Main Results It is obvious that operator L f ; is linear and positive. + q m + l m + l + m p1 + q m q m + l + m + l + 1 m p1 + q m. 1 Remar For q 0, 1 and p q, 1, lim n n 0 or p q. In order to reach convergence results of our operator L f ;, we tae the sequences q m 0, 1 and p m q m, 1 such that lim m p m 1, lim m q m 1, lim m p m m 1 and lim m q m m 1. Thus, we have lim m m pm,q m. Here, we can give the following theorem which guarantees the approimation process based on Korovin s type approimation theorem. Theorem 3.. Let L f ; satisfy the conditions in Remar 3.1 for 0 < q m < p m 1. Then for each monotone increasing function f C 0, l + 1, L f ; converges uniformly to f on 0, 1. Proof. To prove this theorem, it is sufficient by the Korovin theorem to show that lim L t ; C0,l+1 0, 0, 1,. m i By Lemma., Equation 7, it is clear that lim L 1; 1 C0,l+1 lim m sup L m 0,1 ii By Lemma., Equation 8, we have lim L t; C0,l+1 lim m sup L m 0,1 lim m sup 0. lim m 0,1 1; 1 0. t; m + l m m + l m 1 iii Using Lemma., Equation 9, we can write lim L t ; C0,l+1 lim L t ; m 0. sup m 0,1 p m+l 1 m + l lim m m q m + l + 1 m p1 + q Thus, because of the positivity and linearity of L f ;, the proof is completed by the classical Korovin approimation theorem.

8 Currently, we will give the following lemmas. K. Kanat, M. Sofyalıoğlu / Filomat 3:1 018, Lemma 3.3. Let the function f be a monotone increasing function then L f ; is a linear and positive operator. Proof. The proof is obvious. So we will omit it. Lemma 3.4. Hölder Inequality Let 1 α + 1 β 1. Then L f ; satisfies the following inequality L α 1 α 1 f ; L f ; L β β ;. 4. Rate of Convergence In this section, we will give the approimation of order of the operator L f ; by means of modulus of continuity, with the help of functions of Lipschitz class and Peetre s K-functionals. Let f C0, l + 1. The modulus of continuity of f, denoted by w f, δ, is defined w f, δ sup f y f. 16 y δ,y 0,1 Then it is nown that lim w f, δ 0 for f C0, l + 1; and also, for any δ > 0 and each t, 0, 1, we have δ 0 + f t f t w f, δ δ Firstly, we will give the rate of convergence of L f ; by means of modulus of continuity. Theorem 4.1. Let p : p m and q : qm, 0 < qm < p m 1, be sequences satisfying the conditions given in Remar 3.1. Then for all f C 0, l + 1, L f ; f C0,l+1 ω f ; δ m, where δ m L t ; 18 p m+l 1 m + l q m + l + m + l m p1 + q m. m Proof. In order to prove this theorem, we will use the linearity and positivity of the operator L f ;, we have L f ; f L f t f ; f t f ; q;. L Then applying 17, we have L f ; f L t w f, δ m + 1 ; δ m w f, δ m L δ t ; + w f, δ m m w f, δ m L δ t ;. 19 m

9 K. Kanat, M. Sofyalıoğlu / Filomat 3:1 018, L f ; f C0,l+1 sup L f ; f 0,1 w f, δ m L δ t ;. m Choose p m+l 1 m + l q m + l δ m + m + l m p1 + q m. m Thus, we get the desired result L f ; f C0,l+1 ω f ; δ m then the proof is completed. Now, we will give the rate of convergence of L f ; with the help of functions of Lipschitz class. We recall that a function f Lip M α on 0, l + 1 if the inequality f t f M t α ; t, 0, 1 0 holds. Theorem 4.. Let f Lip M α. Let p : p m and q : qm, 0 < qm < p m 1, then we have L f ; f Mδα m, where δ m L t ; 1 p m+l 1 m + l q m + l + m + l m p1 + q m. m Proof. Let f Lip M α and 0 < α 1. Since L f ; is linear and monotone, by using 0, we have L f ; f L f t f ; ML t α ;. If we tae p α, q α and apply Hölder inequality, then we obtain L f ; f M { L t ; } α Mδ α m immediately. If we choose the proof is completed. p m+l 1 m + l q m + l δ m + m + l m p1 + q m, m Lastly, we will give the rate of convergence of our operator L f ; by means of Peetre-K functionals. First of all, we give the following lemma:

10 Lemma 4.3. For f C0, 1 + l, we have K. Kanat, M. Sofyalıoğlu / Filomat 3:1 018, L f ; f. Proof. By using the definition of 6, we obtain p f m+l m + l L f ; m f p m+l m + l m the desired result. f L 1; f p m+l m+l 1 j1 {pj q j 1 } 1 m+l p m+l m+l 1 1 m+l j1 {pj q j 1 } And then, we recall the properties of Peetre s K-functionals. C 0, l + 1 is the space of the functions f, for which f, f and f are continuous on 0, l + 1. We write the norm of function f in the space C 0, l + 1 f C 0,l+1 f C0,l+1+ f C0,l+1 + f C0,l+1. Now, we define classical Peetre s K-functional as follows: { C0,l+1 K f, δ : inf f + δ C } C 0,1 0,l+1 and second modulus of smoothness of the function is defined by ω f, δ : sup sup 0<h<δ,+h 0,l+1 f + h f + h + f, where δ > 0. By 0, it is nown that for A > 0 K f, δ Aω f, δ. Theorem 4.4. Let f C0, l + 1 and 0 < q m < p m 1. Then we have for all n N, there eists a positive constant M such that, L f ; f Mω f, α m + ω f, β m, where α m L t ; + m + l 1 3 m and β m m + l 1. m 4

11 K. Kanat, M. Sofyalıoğlu / Filomat 3:1 018, Proof. Define an auiliary operator L f ; : C0, l + 1 C0, 1 by m + l L f ; L f ; f + f. 5 From Lemma., we have m L 1; 1, m + l L t ; L t ; + m m + l 1 m + l + + m m 0. 6 This means that the operators L f ; are linear. For a given function C 0, l + 1, we have by the Taylor epansion that t t + t + t u udu, t 0, 1. 7 Applying L operator to both sides of the equation 7, we get So, L ; L + t + t + L t ; + L t u udu t t u udu. t L ; L t ; + L t u udu. Using 5 and 6 we obtain t L ; L t u udu t L t u udu m + l + u m m+l m m + l u udu m udu. 8 Moreover, t t u udu t t u u du t t u du t 9

12 and m+l m m + l u udu m K. Kanat, M. Sofyalıoğlu / Filomat 3:1 018, m+l m m + l u du m m + l m + l m When we rewrite 9 and 30 in the absolute value of 8, we obtain L ; L t ; + m + l 1 m L t ; + m + l 1 m α m, m + m + l m where α m L t ; + m + l 1. m Currently, we will find a bound for the auiliary operator L f ;. In the light of the Lemma 4.3 we obtain L f ; L Accordingly, m + l f ; f + f m m + l f ; + f + f L 3 f. L f ; f L m m + l f ; f + f f L ; m L f ; f + L ; m + l + f f m m + l 4 f + α m + ω f, 1 m 4 f + α m + ω f, β m, 31 where β m m + l 1. m 3 Finally, for all C 0, l + 1 tae the infimum of the equation 31. We get L f ; f 4K f, α m + ω f, β m. 33

13 K. Kanat, M. Sofyalıoğlu / Filomat 3:1 018, As a result, using the property of Peetre s K-functional, we obtain L f ; f Mω f, α m + ω f, β m. 34 Thus the proof is completed. 5. Conclusion In this paper, we introduced p, q-analogue of Lupaş-Schurer operators by using p, q-integers. p, q- analogue of Lupaş-Schurer operators has an advantage to generate positive linear operators for all p > 0 and q > 0 whereas p, q-analogue of Bernstein-Schurer operators 5 generates positive linear operators only if 0 < q < p 1. We obtained some approimation properties of the constructed operators and dealed with the rate of convergence by using modulus of continuity, with the help of functions of Lipschitz class and Peetre s K-functionals. References 1 T. Acar, A. Aral and S. A. Mohiuddine, On Kantorovich modifications of p, q-basaov operators, J. Inequal. Appl., :98. S.N. Bernstein, Demonstration du theoreme de Weierstrass fondee sur le caucul des probabilities, Communications of the Kharov Mathematical Society, , A. Lupaş, A q analogue of the Bernstein operator, in Seminar on Numerical and Statistical Calculus Cluj-Napoca, Univ. Babeş-Bolyai M.Mursaleen, K. J. Ansari, Asif Khan, On p, q-analogue of Bernstein Operators, Applied Mathematics and Computation, , Erratum: Appl. Math. Comput M.Mursaleen, M.Nasiruzzaman, A. Nurgali, Some approimation results on Bernstein-Schurer operators defined by p, q- integers, Journal of Inequal. Appl : M. Mursaleen, K. J. Ansari and Asif Khan, Some Approimation Results by p, q-analogue of Bernstein-Stancu Operators, Applied Mathematics and Computation 64, 015, M. Mursaleen, Md. Nasiuzzaman and A. Nurgali, Some approimation results on Bernstein-Schurer operators defined by p, q-integers, J. Ineq. Appl., : M. Mursaleen, Md. Nasiruzzaman, Asif Khan and K. J. Ansari, Some approimation results on Bleimann-Butzer-Hahn operators defined by p, q-integers, Filomat 30:3 016, , DOI 10.98/FIL M. 9 M. Mursaleen, F. Khan and Asif Khan, Approimation by p, q-lorentz polynomials on a compact dis, Comple Anal. Oper.Theory M. Mursaleen, Md. Nasiruzzaman, Faisal Khan and Asif Khan, On p, q-analogue of divided difference and Bernstein operators, J. Nonlinear Funct. Anal. 017, Article ID 5, 11 M Mursaleen, A.M. Sarsenbi and T. Khan, On p, q-analogue of two parametric Stancu-Beta operators, J. Ineq. Appl., : M. Mursaleen, Khursheed J. Ansari and Asif Khan, Some approimation results for Bernstein-Kantorovich operators based on p, q-calculus, U.P.B. Sci. Bull. Series A, Khalid Khan, D.K. Lobiyal, Bezier curves based on Lupas p, q-analogue of Bernstein functions in CAGD, Journal of Computational and Applied Mathematics Volume F. Schurer, Linear Positive Operators in Approimation Theory, Math. Inst., Techn. Univ. Delf Report Mahouton Norbert Hounonnou, Joseph Dsir Buweli Kyemba,R p, q-calculus: differentiation and integration, SUT J. Math A. Wafi, N. Rao, Bivariate-Schurer-Stancu Operators based on p, q-integers, Filomat, 017 IN EDITING. 17 A. Wafi, N. Rao, p, q-bivariate-bernstein-chlodowsy Operators, Filomat, 017 IN EDITING. 18 N. Rao, A. Wafi, Stancu-Variant of Generalized Basaov Operators, Filomat, 017, 31 9, G. M. Phillips, Bernstein polynomials based on the q integers, Ann. Numer. Math DeVore, RA., Lorentz, G.G Constructive Approimation. Springer, Berlin, 177.