ON THE (p, q) STANCU GENERALIZATION OF A GENUINE BASKAKOV- DURRMEYER TYPE OPERATORS

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1 International Journal of Analysis and Applications ISSN Volume 15 Number DOI: 1894/ ON THE p q STANCU GENERALIZATION OF A GENUINE BASKAKOV- DURRMEYER TYPE OPERATORS İSMET YÜKSEL ÜLKÜ DİNLEMEZ KANTAR AND BİROL ALTIN Abstract In this paper we introduce a Stancu generalization of a genuine Baskakov- Durrmeyer type operators via p q integer We investigate approimation properties of these operators Furthermore we study on the linear positive operators in a weighted space of functions and obtain the rate of these convergence using weighted modulus of continuity 1 Introduction In the field of approimation theory the quantum calculus has been studied for a long time The generalization of p q calculus was introduced by Sahai and Yadav in [15 Recently a series of papers giving p q generalizations a sequence of linear positive operators have been published in [ Our aim is to give Stancu type generalization via p q integer defined by Agrawal and Thamer as follows B n f n 1 b nk k1 b nk 1 tf t dt f 1 q 11 n k 1 k b nk k 1 nk in [5 We refer reader to [ for uneplained terminologies and notations Preliminaries and notations Let s give a table of some basic formulas motivated from q calculus used in p q calculus as the following Table1 p q calculus Relation with q calculus pn q n p q p n 1 [n q/p! [1 [! p n [nq/p! a b n a b ap bqapn 1 bq n 1 a b n n pn a b q/p d f fp fq d q f f fq Table 1 Recall that the beta function introduced [14 in q calculus is defined by B q n k KA n t k 1 1 t nk q d q t 1 Received 14 th April 17; accepted 5 th June 17; published 1 st November 17 1 Mathematics Subject Classification 41A5 41A6 Key words and phrases Baskakov-Durrmeyer operators; weighted approimation; rates of approimation p q calculus c 17 Authors retain the copyrights of their papers and all open access articles are distributed under the terms of the Creative Commons Attribution License 18

2 BASKAKOV- DURRMEYER TYPE OPERATORS 19 KA n An 1 1 n 1 A 1 s q A > 1 A A q In the formula KA n q nn 1/ and KA 1 for n N Inspiring the formula 1 we introduce p q-beta functions B n k as a generalization of B q n k A > and n k N\ } defined by B k n p n q k t k 1 1 t nk If p 1 is replaced in then the formula is reduced to 1 d t Genuine type Stancu generalization via p q integer Let s start to give a Stancu type p q generalization of these operators in 11 For β and < q < p 1 these operators are defined as follows; B nf [n 1 f b nk p q t k1 b nk p q p n 1 k q kk 1 b nk 1 p q tf β [ n k 1 k [n t d t β p n 1 1 t k 1 t nk If one replaces p q 1 and β in 1 then the operators B n are reduced to the operators B n in 11 Similar type operators studied in [ To obtain our main results we need calculating the values of Korovkin monomial functions Lemma 1 The following equalities are satisfied for e m t t m m 1 and n > B n1 1 B nt [n pq β[n β B nt [n [n1 p q 4 β [n [n 1 p n 4 [ [n q β [n [n 1 [n pnq n β [n pq β [n Proof By the definition p q beta functions in we obtain the estimate b nk 1 p q; tt m d t [ n k k 1 t km 1 1 t nk 1 d t [k m 1! [n m! [k 1! [n 1!p n m 1 q km

3 14 YÜKSEL DİNLEMEZ KANTAR AND ALTIN If we apply the operators in 1 to the equality for m we get B n1 [n 1 k1 b nk p q; p n 1 k q kk 1 b nk 1 p q; td t pn 1 k [n 1 b nk p q; pn 1 q kk 1 [n 1 p n 1 q k k1 p n 1 [n k 1! p n q k k p [k k! [n 1! 1 nk p n q k bnk p q; p k 1 And the proof of i is finished With the direct computation we obtain ii as follows: Bnt [n 1 b nk p q; p n 1 k q kk 1 For iii B k1 b nk 1 p q; t 1 β pn β k1 β B [n [n t d t β k b nk p q; pn 1 q kk 1 [k n1 pq β[n k β nt [n 1 [n pq β[n β k1 [n p n q k1 p n1 q k bn1k p q; p b nk p q; p n 1 k q kk 1 b nk 1 p q; t p n 1 β t d t β

4 BASKAKOV- DURRMEYER TYPE OPERATORS 141 Using the equality we have B nt [n β [k q k s [s p s [k s s k k [k p [k 1 q k 1 [ p n nk 1 q k 5k [n [n b nk p q; } [n [k p n nk 4 q k k b nk p q; β [n k1 Bn1 β Then B nt [n [n1 p q 4 β [n [n k p n 4 [ [n q β [n [n 1 k p n1 q k bn1k p q; Bn1 β p n q k bnk p q; p [n pqβ [n And so we have completed the proof of iii Now we consider that B[ denotes the set of all bounded functions from [ to R B[ is a normed space with the norm f B sup f : [ } and C B [ denotes the subspace of all continuous functions in B[ We denote first modulus of continuity on finite interval [ b b > ω [b f δ sup f h f 4 <hδ [b The Peetre s K functional is defined by K f δ inf f g B δ g B : g W } δ > W g C B [ : g g C B [ } By [6 p 177 Theorem 4 there eists a positive constant C such that K f δ Cω f δ ω f δ sup sup <h δ [ f h f h f 5 The weighted Korovkin- type theorems were proved by Gadzhiev [8 We give the Gadzhiev s results in weighted spaces B ρ [ denotes the set of all functions f from [ to R satisfying growth condition f N f ρ ρ 1 and N f is a } constant depending only on f B ρ [ f is a normed space with the norm f ρ sup ρ : R C ρ [ denotes the subspace of all continuous functions in B ρ [ and Cρ[ denotes the subspace of all functions f C ρ [ for f which lim eists finitely ρ

5 14 YÜKSEL DİNLEMEZ KANTAR AND ALTIN We define a genuine type generalization to the operators Bn given in 1 as follows B p p nq nnf σ β p n q n n; > [n nq nnf : pnqn β f [n pnqn β σ β p n q n n; p nq nn p n q n [n pnq n β [n pnq n [n p nq n [n pnq n β Notice that these operators are defined from the space Cρ[ into B ρ [ To satisfy hypothesis of Korovkin s Theorem we assume that lim n pn n and lim n qn n are real numbers when lim p n 1 and lim q n 1 for < q n < p n 1 and n > On the other hand since the operators n n [ p nq nnf are defined as f on the interval [n pnqn β it is enough to eamine the approimation properties of these operators at the interval [n pnqn β The following lemma can be obtained with the help of Lemma 1 Lemma The operators m 1 p nq nn1 1 p nq nnt p nq nn satisfy the following equalities for > [n pnqn β and e mt t m p nq nnt [n1 pnqn [n pnqn qn [n 1 pnqn [n pnqn [n1 pnqn [n pnqn p n n [ pnqn [n pnqn qn[n pnqn β[n 1 pnqn [n pnqn qn[n pnqn β[n 1 pnqn [n pnqn β pn n [ pnqn [n pnqn qn[n pnqn β [n 1 pnqn We need computing the second moment before giving our main results Lemma We have the following inequality p nq nnt p n p nqn for > [n pnqn β Proof By Lemma we write the second moment as [n1pnqn [n pnqn p nq nnt qn [n 1 pnqn [n 1 pnqn q n 1β1 1 qn[n pnqn β [n1 pnqn [n pnqn p n n [ pnqn [n pnqn qn[n pnqn β[n 1 pnqn [n pnqn qn[n pnqn β[n 1 pnqn [n pnqn β pn n [ pnqn [n pnqn q n[n pnqn β [n 1 pnqn [n1 pnqn [n pnqn qn [n 1 pnqn [n pnqn Considering the following equalities 1 p n n [ pnqn [n pnqn qn[n pnqn β[n 1 pnqn [n pnqn β [n 1 pnq n p n [n pnq n q n n [ pnq n [n pnq n p n [n pnq n q n n [ pnq n [n 1 pnq n p n [n pnq n q n 1 n [1 pnq n

6 BASKAKOV- DURRMEYER TYPE OPERATORS 14 we get p nq nnt p n p nq n q n [ p nq n [n pnq n p n n [ pnqn [n pnqn qn[n pnqn β[n 1 pnqn p n p nq n And the proof of the Lemma is now finished Thus we are ready to give direct results q n 1β1 q n[n pnqn β [n pnqn β 1 Lemma 4 We have the inequality for every [ and f C B [ p nq nnf f δp nq nn f B p δp nq nn : n p nqn qn 1β1 1 qn[n pnqn β Proof Using Taylor s epansion ft f t f and the Lemma we have the following equality p nq nnf f p nq nn t t t uf udu t uf udu; On the other hand combining the inequality t t uf udu f t B and Lemma we get as desired p nq nnf f p nq nn t t uf udu f B p nq nnt f B p n p nqn q n 1β1 q n[n pnqn β Theorem 1 We have the following inequality p nq nnf f Cω f δp nq nn p δp nq nn n p nqn qn 1β1 1 qn[n pnqn β 1 Proof For any g W we obtain the inequality p nq nnf f p nq nnf g f g p nq nng g

7 144 YÜKSEL DİNLEMEZ KANTAR AND ALTIN Then Lemma 4 we have p nq nnf f f g B δp nq nn g B Taking infimum over g W on the right side of the above inequality and using the inequality 5 we reach the desired result Theorem For every f Cρ[ we have the following the limit lim p n nq nnf f ρ ρ ρ Proof From Lemma it is obvious that p nq nne e and p nq nne 1 e 1 We have ρ p p nq nne e nq nnt 1 p n p nq n qn sup 1β1 1 qn[n pnqn β [ 1 Then we get lim n p n p nqn qn 1β1 qn[n pnqn β p nq nne e ρ Thus from A D Gadzhiev s Theorem in [8 we obtain the proof of Theorem Lemma 5 We assume that [n pnqn β p nq nnf; f N C[b p δp nq nn n p nqn q n < b Then we have the following inequality 1 b δp nq nnb ω [b1 f; δp nq nnb 1β1 1 for every f C qn[n pnqn β ρ[ [ Proof Let [n pnqn β b and t > b 1 Since t > 1 we have ft f K f t K f 1 b t 6 [ Let [n pnqn β b t < b 1 and δ > Then we have t ft f 1 ω [b1 f δ 7 δ Due to 6 and 7 we can write ft f K f 1 b t After using Cauchy- Schwarz s inequality we get p nq nnf; f K f 1 b Bp nq nn t [ ω [b1 f; δ 1 1 δ 1 p nq nn t 1/ K f 1 b δ p nq nn ω [b1 f; δ t ω [b1 f δ δ [ 1 1 δ 1/ p δ nq nn }

8 BASKAKOV- DURRMEYER TYPE OPERATORS 145 Considering Lemma and choosing δ : δp nq nnb and N mak f } We reach the proof of Lemma 5 Theorem For every f Cρ[ and γ > we have the limit lim sup p nq nnf f n 1 γ Proof For γ > f Cρ[ and b 1 using 4 the following inequality is satisfied p nq nnf f sup 1 γ p nq nnf f p nq nnf f 1 γ sup b 1 γ p nq nnf f p nq nnf f sup C[b b 1 p nq nnf f p nq C[b nnf f ρ sup <b Using Lemma 5 and Theorem we get the proof of Theorem References [1 A Aral and V Gupta Generalized q Baskakov operators Math Slovaca [ A Aral V Gupta and Agarwal R P Applications of q-calculus in operator theory Springer New York 1 [ A Aral and V Gupta p q Type beta functions of second kind Adv Oper Theory [4 T Acar S A Mohiuddine and M Mursaleen Approimation by p q Baskakov-Durrmeyer-Stancu Operators Comple Anal Oper Theory DOI 117/s [5 P N Agrawal and K J Approimation of unbounded functions by a new sequence of linear positive operators J Math Anal Appl [6 R A Devore and G G Lorentz Constructive Approimation Springer Berlin 199 [7 Ü Dinlemez İ Yüksel and B Altın A note on the approimation by the q-hybrid summation integral type operators Taiwanese J Math [8 A D Gadzhiev Theorems of the type of P P Korovkin type theorems Math Zametki ; English Translation Math Notes 5/ [9 V Gupta p q-szász Mirakyan Baskakov Operators Comple Anal Oper Theory DOI 117/s [1 M Mursaleen K J Ansari and A Khan On p q-analogue of Bernstein Operators Appl Math Comput [11 M Mursaleen K J Ansari and A Khan Some Approimation Results by p q-analogue of Bernstein-Stancu operators Appl Math Comput [Corrigendum: Appl Math Comput [1 M Mursaleen Md Nasiruzzaman A Khan and K J Ansari Some approimation results on Bleimann-Butzer-Hahn operators defined by p q integers Filomat [1 T Acar A Aral and S A Mohiuddine On Kantorovich modification of p q Baskakov operators J Inequal Appl 16: pp [14 A De Sole and V G Kac On integral representations of q gamma and q beta functions Atti Accad Naz Lincei Cl Sci Fis Mat Natur Rend Lincei 9 Mat Appl [15 V Sahai and S Yadav Representations of two parameter quantum algebras and p q special functions J Math Anal Appl [16 D K Verma V Gupta and P N Agrawal Some approimation properties of Baskakov-Durrmeyer-Stancu operators Appl Math Comput Gazi University Faculty of Sciences Department of Mathematics Teknikokullar 65 Ankara Turkey Corresponding author: iyuksel@gaziedutr