Approximation properties of (p, q)-bernstein type operators

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1 Acta Uiv. Saietiae, Mathematica, 8, DOI: /ausm Aroximatio roerties of, -Berstei tye oerators Zoltá Fita Deartmet of Mathematics, Babeş-Bolyai Uiversity, Romaia fzolta@math.ubbcluj.ro Abstract. We itroduce a ew geeralizatio of the -Berstei oerators ivolvig, -itegers, ad we establish some direct aroximatio results. Further, we defie the limit, -Berstei oerator, ad we obtai its estimatio for the rate of covergece. Fially, we itroduce the, -Katorovich tye oerators, ad we give a uatitative estimatio. 1 Itroductio The alicatios of -calculus i the field of aroximatio theory have led to the discovery of ew geeralizatios of the Berstei oerators. The first geeralizatio ivolvig -itegers was obtaied by Luaş [7 i Te years later Phillis [12 gave aother geeralizatio of the Berstei oerators itroducig the so-called -Berstei oerators. I comariso with Phillis geeralizatio, the Luaş geeralizatio gives ratioal fuctios rather tha olyomials. Nowadays, -Berstei oerators form a area of a itesive research. A survey of the obtaied results ad refereces i this area durig the first decade of study ca be foud i [11. After that several well-ow ositive liear oerators ad other ew oerators have bee geeralized to their -variats, ad their aroximatio behavior have bee studied see e.g. [1 ad [ Mathematics Subject Classificatio: 41A25, 41A36 Key words ad hrases:, -itegers,, -Berstei oerators, limit, -Berstei oerator,, -Katorovich oerators, rate of covergece, modulus of cotiuity 222

2 Aroximatio roerties of, -Berstei tye oerators 223 The, -calculus is a further ew geeralizatio of the -calculus, its basic defiitios ad some roerties may be foud i the aers [6, [13, [14, [15. The, -itegers [, are defied by [, =, where = 0, 1, 2,... ad 0 < < 1. For = 1, we recover the well-ow -itegers see [5. Obviously [, = 1 [ /. 1 The, -factorials [,! are defied by [1, [2,... [,, if 1 [,! = 1, if = 0, ad the, -biomial coefficiets are give by [ [,! = [,![,!, 0., Further, we set a ba b... 1 a 1 b, if 1 a b, = 1, if = 0. By simle comutatios, usig 1, we get [,! = 1/2 [ /!, 2 ad, = { 1 1 1}/2 / 3 where a b, = 1/2 a b /, 4 a ba b... a 1 b, if 1 a b = 1, if = 0

3 224 Z. Fita i the case whe 0 < < 1. The goal of the aer is to itroduce a ew geeralizatio of the -Berstei oerators ivolvig, -itegers. These, -Berstei oerators aroximate each cotiuous fuctio uiformly o [0, 1, ad some direct aroximatio results are established with the aid of the modulus of cotiuity give by ωf; δ = su{ fx fy : x, y [0, 1, x y δ}, δ > 0, 5 where f C[0, 1. Further, we defie the limit, -Berstei oerator ad we estimate the rate of covergece by the modulus of cotiuiuty 5. The cocet of limit -Berstei oerator was itroduced by Il isii ad Ostrovsa [4, ad its rate of covergece was established by Wag ad Meg i [16. Fially, we defie a, -Katorovich variat of the, -Berstei oerators, ad we give a uatitative estimatio usig 5. 2, -Berstei oerators For 0 < < 1, f C[0, 1, x [0, 1 ad = 1, 2,..., we defie the, -Berstei olyomials as follows: B,, f; x = [ { 1 1}/2 x 1 x, f, [,. 6 [, For = 1 ad 0 < < 1, we recover the -Berstei olyomials see [12: B, f; x = x 1 x f [ [. 7 Theorem 1 If the seueces ad satisfy 0 < < 1 for = 1, 2,..., ad 1, 1, 1 as, the B,, f; x fx 2ω f; 21 x 2 x1 x 1/2 [ / for all f C[0, 1 ad x [0, 1. Proof. By 6, 3-4 ad 1, we have B,, f; x = x 1 x / f [ /. 8 / [ /

4 Aroximatio roerties of, -Berstei tye oerators 225 Hece, i view of [12, 13, we obtai By 8 ad [12, 14, we get x = B,/ t; x B,, t; x = B,, 1; x = B,/ 1; x = 1. 9 Aalogously, by 8 ad [12, 15, we get x 1 x [ / /. 10 / [ / [ B,, t 2 ; x = x 1 x [2 / 2 / / [ 2 / B,/ t 2 ; x = x 2 x1 x. 11 [ / O the other had, it is ow for 5 that ωf; λδ 1 λωf; δ, 12 where λ 0 ad δ > 0. The, by 8, [12, 13, Hölder s ieuality ad 9-11, we obtai B,, f; x fx [ ωf; δ ωf; δ = ωf; δ { x 1 x f f; x 1 x ω [ x 1 x 1 δ 1 [ / [ / [ / [ / [ / [ / x fx 1 δ 1 x 1 x 2 1/2 x { 1 δ 1 B,, t 2 ; x 2xB,, t; x x 2 B,, 1; x 1/2} [ / [ / x

5 226 Z. Fita Choosig δ = theorem. ωf; δ = ωf; δ {1 δ 1 x 2 x1 x {1 δ 1 21 x 2 21 x 2 x1 x } 1/2 2 [ x 2 x 2 / } x1 x 1/2. [ / 1/2 [, we arrive at the statemet of our / Theorem 2 If the seueces ad satisfy 0 < < 1 for = 1, 2,..., ad 1, 1, 1 as, the B,, f; x B,/ f; x ωf; 1 for all f C[0, 1 ad x [0, 1. Proof. Because [ / [ [ / / [ 1 1 for = 0, 1,...,, we / fid from 8, 7 ad [12, 13, that B,, f; x B,/ f; x [ x 1 x x 1 x ω f f; [ / [ / ωf; 1 B,/ 1; x = ωf; 1, [ / [ / [/ f [ / [ [ / which is the reuired estimatio. Remar 1 There exist seueces ad with the roerties eumerated i Theorem 1: = 1 1 ad 1 2 = 1 1 1, = 1, 2,... We also metio, if 0 < < 1 for = 1, 2,..., 1 ad 1 as, the [ / ad [ / [1 1 as. / Remar 2 I [9 ad [10 are itroduced two differet geeralizatios of the -Berstei olyomials 7 ivolvig, -itegers. The first oe does ot reserve eve the costat fuctios, ad the secod oe is a /-Berstei olyomial. Our, -Berstei olyomials defied by 6 are differet from the above metioed geeralizatios. The advatage of 6 is that it allows us to itroduce the limit, -Berstei oerator.

6 Aroximatio roerties of, -Berstei tye oerators Limit, -Berstei oerator For 0, 1, Il isii ad Ostrovsa roved i [4 that for each f C[0, 1, the seuece B, f; x coverges to B, f; x as uiformly for x [0, 1, where f1 x 1 B, f; x = 1 s x, if 0 x < 1 [! s=0 f1, if x = 1 is the limit -Berstei oerator. Wag ad Meg [16 roved for all f C[0, 1 ad x [0, 1 that 4 B, f; x B, f; x 2 1 l 1 ωf;. 1 For 0 < < 1, the limit, -Berstei oerator B,, : C[0, 1 C[0, 1 is defied as follows: f 1/2 x s s x B,, f; x = [,! s, if x [0, 1 s=0 f1, if x = Theorem 3 Let, 0, 1 be give such that 2 < <. The, for every f C[0, 1, x [0, 1 ad = 1, 2,..., we have B,, f; x B,, f; x 4 62 l ω f;. Proof. Due to 13 ad 2, we have We set B,, f; x = w, ; x = f x 1 [/! x 1 x ad w, ; x = s=0 1 x 1 [! s x s x. s=0

7 228 Z. Fita The, i view of 9 ad [16,. 154, 2.3, we obtai w, ; x = Usig 8, 14 ad 15, we fid B,, f; x B,, f; x = w, {f ; x =1 w, ; x = [ / [ / } f {w, ; x w, ; x } {f f } w, ; x {f f } w, ; x f =1 w, [ / f [ / ; x w, ; x f f w, ; x f f =: I 1 I 2 I The estimatio of I 1 : by 1, we have [ / = [ / [, [ / [, [ / = for = 0, 1,...,. Hece, by 15, I 1 w, ; x ω f; [ / [ / ω f;. 17

8 Aroximatio roerties of, -Berstei tye oerators 229 The estimatio of I 2 : for = 0, 1,...,, we have 1. Hece, by 12, But f f ωf; ωf; = ω f; / 1 / ω f; = = / 2 3, because 2 < < ad = 0, 1,...,. The, by 18, we obtai I 2 w, ; x w, ; x 3 ω f; = 3 ω f; w, ; x w, ; x. Taig ito accout the estimatio w, ; x w, ; x 2 1 l 1 1, where 0 < < 1 see [16,. 156, 2.9, we fid that I 2 62 l ω f;. 19 The estimatio of I 3 : for 1, we have 1. Hece, by 12 ad 2 < <, we get f f ωf; ωf; 1 2 / ω f; = 1 3ω f;. ω f;

9 230 Z. Fita The, by 15, I 3 3ω f; w, ; x 3ω f; =1. 20 Combiig ad 19-20, we obtai the statemet of the theorem. 4, -Katorovich oerators Our, -Katorovich oerators are defied as follows: K,, f; x = [ 1, [ { 1 1}/2 x 1 x,, [1, [1, 1 [, [1, fu d R /u, 21 where f C[0, 1, x [0, 1, = 1, 2,..., ad the Riema tye -itegral of f over the iterval [a, b 0 a < b; 0 < < 1 is give by see [2, [8 b a fu d R u = 1 b a j fa b a j. 22 j=0 Remar 3 I [15 the, -itegral of f over the iterval [0, a is defied as a 0 fu d, u = a j=0 j a j1 f j j1, where 0 < < 1. But 1 a / [0, a for 0 < < 1 i the sum the case j = 0, thus the fuctio f is ot defied at 1 a. For this reaso we use the Riema tye /-itegral i 21. Theorem 4 If the seueces ad satisfy 0 < < 1 for = 1, 2,..., ad 1, 1, 1 as, the K,, f; x fx 2ωf; δ x

10 Aroximatio roerties of, -Berstei tye oerators 231 for all f C[0, 1 ad x [0, 1, where { δ x = [ / 21 [ 1 / x 2 3 [ 1 x [ 1 / [ 1 2. Proof. By 21, 3-4 ad 1, we have K,, f; x = [ 1 / 1 [ 2 [ } [ 1 / [ 1 2 x 1 x / By simle comutatios, usig 22, we obtai ad [1 / [1 / [ / [1 / [1 / [1 / u d R [ / / u = [1 / 1 d R / u = [ 1 / [ / [ 1 / = [ 1 / [1 / [1 / u 2 d R [ / / u = [1 / [1 / [1 / / [ / [1 / fu R / u. 23 [ 1 /, 24 [ / [ 1 / [ 1 / [ 2 / [ / 25 2 [ 1 / [ [ 1 / / 2 2 [ 1 / 2 2 [ / I what follows, taig ito accout 23-26, the roof is similar to the roof of Theorem 1, therefore we omit the details. Refereces [1 A. Aral, V. Guta, R. P. Agarwal, Alicatios of -Calculus i Oerator Theory, Sriger, New Yor, [2 H. Gauchma, Itegral ieualities i -calculus, Comut. Math. Al., ,

11 232 Z. Fita [3 V. Guta, R. P. Agarwal, Covergece Estimates i Aroximatio Theory, Sriger, New Yor, [4 A. Il isii, S. Ostrovsa, Covergece of geeralized Berstei olyomials, J. Arox. Theory., , [5 V. Kac, P. Cheug, Quatum Calculus, Sriger, New Yor, [6 K. Kha, D. K. Lobiyal, Bézier curves based o Luaş, -aalogue of Berstei olyomials i CAGD, arxiv: [cs.gr. [7 A. Luaş, A -aalogue of the Berstei oerator, Semiar o Numerical ad Statistical Calculus, , [8 S. Mariović, P. Rajović, M. Staović, The ieualities for some tye of -itegrals, Comut. Math. Al., , [9 M. Mursalee, K. J. Asari, A. Kha, O, -aalogue of Berstei oerators, Al. Math. Comut., , [10 M. Mursalee, K. J. Asari, A. Kha, Some aroximatio results for Berstei-Katorovich oerators based o, -calculus, arxiv: v4[math.ca 15 Ja [11 S. Ostrovsa, The first decade of the -Berstei olyomials: results ad ersectives, J. Math. Aal. Arox. Theory, , [12 G. M. Phillis, Berstei olyomials based o the -itegers, A. Numer. Math., , [13 V. Sahai, S. Yadav, Reresetatios of two arameter uatum algebras ad, -secial fuctios, J. Math. Aal. Al., , [14 P. N. Sadjag, O the, -Gamma ad the, -Beta fuctios, arxiv: v1. 22 Ju [15 P. N. Sadjag, O the fudametal theorem of, -calculus ad some, -Taylor formulas, arxiv: [math.qa 22 Aug [16 H. Wag, F. Meg, The rate of covergece of -Berstei olyomials for 0 < < 1, J. Arox. Theory, , Received: Aril 15, 2016