VORONOVSKAJA-TYPE THEOREM FOR CERTAIN GBS OPERATORS. Ovidiu T. Pop National College Mihai Eminescu of Satu Mare, Romania

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1 GLASNIK MAEMAIČKI Vol , VORONOVSKAJA-YPE HEOREM FOR CERAIN GBS OPERAORS Ovidiu. Pop National College Mihai Einescu o Satu Mare, Roania Abstract. In this paper we will deonstrate a Voronovsaja-type theore and approxiation theore or GBS operator associated to a linear positive operator.. Introduction In this section, we recall soe notions and results which we will use in this article. Let N be the set o positive integers and N 0 = N {0}. For N, let B : C0, ] C0, ] the Bernstein operators, deined or any unction C0, ] by. B x = p, x, where p, x are the undaental polynoials o Bernstein, deined as ollows.2 p, x = x x, or any x 0, ] and any {0,,..., } see 7,28]. In 932, E. Voronovsaja in the paper 3], proved the result contained in the ollowing theore Matheatics Subject Classiication. 4A0, 4A25, 4A35, 4A36. Key words and phrases. Linear positive operators, GBS operators, the irst order odulus o soothness, Voronovsaja-type theore, approxiation theore. 79

2 80 O.. POP heore.. Let C0, ] be a two ties derivable unction in the point x 0, ]. hen the equality.3 li B x x x x] = x 2 holds. Let p N 0. For N, F. Schurer see 26] introduced and studied in 962 the operators B,p : C0, + p] C0, ], naed Bernstein-Schurer operators, deined or any unction C0, + p] by.4 +p B,p x = p, x, where p, x denotes the undaental Bernstein-Schurer polynoials, deined as ollows + p.5 p, x = x x +p = p +p, x or any x 0, ] and any {0,,..., + p}. For N, let the operators M n : L 0, ] C0, ] deined or any unction L 0, ] by.6 M x = + p, x 0 p, ttdt, or any x 0, ]. hese operators were introduced in 967 by J. L. Durreyer in ] and were studied in 98 by M. M. Derriennic in 9]. For N, let the operators K : L 0, ] C0, ] deined or any unction L 0, ] by.7 K x = + p, x tdt, or any x 0, ]. he operators K, N, are naed Kantorovich operators, introduced and studied in 930 by L. V. Kantorovich see 4]. For the ollowing construction see 8]. Deine the natural nuber 0 by { ax{, β]}, i β R\Z.8 0 = ax{, β}, i β Z.

3 VORONOVSKAJA-YPE HEOREM FOR CERAIN GBS OPERAORS 8 For the real nuber β, we have that.9 + β γ β or any natural nuber, 0, where { { } ax + β, {β}, i β R\Z.0 γ β = 0 + β = ax{ + β, }, i β Z. For the real nubers α, β, α 0, we note, i α β. µ α,β = + α β, i α > β. γ β For the real nubers α and β, α 0, we have that µ α,β and α + β µα,β or any natural nuber, 0 and or any {0,,..., }. For the real nubers α and β, α 0, 0 and µ α,β deined by.8-., let the operators P α,β : C 0, µ α,β ] C 0, ], deined or any unction C 0, µ α,β ] by.3 P α,β + α x = p, x, + β or any natural nuber, 0 and or any x 0, ]. hese operators are naed Stancu operators, introduced and studied in 969 by D. D. Stancu in the paper 27]. In 27], the doain o deinition o the Stancu operators is C0, ] and the nubers α and β veriy the condition 0 α β. In 980, G. Bleiann, P. L. Butzer and L. Hahn introduced in 6] a sequence o linear positive operators L, L : C B 0, C B 0,, deined or any unction C B 0, by.4 L x = + x x + or any x 0, and any N, where C B 0, = { : 0, R, bounded and continuous on 0, }. For N consider the operators S : C 2 0, C 0, deined or any unction C 2 0, by.5 S x = e x x,!,

4 82 O.. POP or any x 0,, where C 2 0, = } and is inite. { C 0, : li x x + x 2 exists he operators S are naed Mirajan-Favard-Szász operators and were introduced in 94 by G. M. Mirajan in 6]. hey were intensively studied by J. Favard in 944 in 2] and O. Szász in 950 in 29]. Let or N the operators V : C 2 0, C 0, be deined or any unction C 2 0, by + x.6 V x= + x +x, or any x 0,. he operators V are naed Basaov operators and they were introduced in 957 by V. A. Basaov in 4]. W. Meyer-König and K. Zeller have introduced in 5] a sequence o linear and positive operators. Ater a slight adjustent given by E. W. Cheney and A. Shara in 8], these operators tae the or Z : B 0, C 0,, deined or any unction B 0, by.7 Z x = + x + x, + or any N and or any x 0,. hese operators are naed the Meyer-König and Zeller operators. Observe that Z : C 0, ] C 0, ], N. In the paper 3], M. Isail and C. P. May consider the operators R. For N, R : C0, C0, is deined or any unction C0, by.8 R x = e x +x +! x e x +x + x or any x 0,. We consider I R, I an interval and we shall use the ollowing unctions sets: EI, FI which are subsets o the set o real unctions deined on I, BI = { : I R, bounded on I }, CI = { : I R, continuous on I } and C B I = BI CI. I BI, then the irst order odulus o soothness o is the unction ω; : 0, R deined or any 0 by.9 ω; = sup { x x : x, x I, x x }.

5 VORONOVSKAJA-YPE HEOREM FOR CERAIN GBS OPERAORS 83 Let I, J R intervals, EI J, FI J which are subsets o the set o real unctions deined on I J and L : EI J FI J be a linear positive operator. he operator UL : EI J FI J deined or any unction EI J, any x, y I J by.20 ULx, y = Lx, +, y, x, y is called GBS operator Generalized Boolean Su operator associated to the operator L, where and stand or the irst and second variable see 3]. I EI J and x, y I J, let the unctions x = x,, y =, y : I J R, x s, t = x, t, y s, t = s, y or any s, t I J. hen, we can consider that x, y are unctions o real variable, x : J R, x t = x, t or any t J and y : I R, y s = s, y or any s I. 2. Preliinaries For the ollowing construction and result see 9] and 2], where p = or any N or p = or any N. Let I, J be intervals with I J. For any N and {0,,..., p } N 0 consider the unctions ϕ, : J R with the property that ϕ, x 0 or any x J and the linear positive unctionals A, : EI R. Deinition 2.. For N deine the operator L : EI FJ by 2. L x = or any EI and x J. p ϕ, xa,, Proposition 2.2. he L, N operators are linear and positive on EI J. Deinition 2.3. For N, let L : EI FJ be an operator deined in 2.. For i N 0, deine,i by 2.2,i L x = i L ψ i x x = i p ϕ, xa, ψ i x, or any x I J, where or x I, ψ x : I R, ψ x t = t x or any t I. In what ollows s N 0 is even and we suppose that the operators L veriy the conditions: there exists, the sallest α s, α s+2 0, so that,j L x 2.3 li αj = B j x R, x I J, j {s, s + 2} and 2.4 α s+2 < α s + 2.

6 84 O.. POP heore 2.4. Let : I R be a unction. I x I J and is a s ties dierentiable unction in x with s continuous in x, then 2.5 li s αs L x s i x i i!,i L x ] = 0. Assue that is s ties dierentiable unction on I, with s continuous on I and there exists an interval K I J such that there exist s N and j R depending on K, so that or any s and any x K we have,j L x 2.6 j, αj where j {s, s + 2}. hen the convergence given in 2.5 is unior on K and s,i L x 2.7 s αs L x s! s + s+2 ω or any x K and s. s ; i x i i! In the ollowing we consider that 2.8,0 L x = or any x I J and any N., 2+α s α s+2 Corollary 2.5. Let : I R be a unction. Assue that is s ties dierentiable in x I J and s is continuous in x. hen 2.9 li L x = x i s = 0 and 2.0 li s αs s L x i x i i!,i L x ] = s x s! B s x i s 2. I is s ties dierentiable unction on I J, with s continuous on I J and 2.6 taes place or an interval K I J then the convergence ro 2.9 and 2.0 is unior on K. Fro 2.3 and 2.8 it results that 2. α 0 = 0 and then =.

7 VORONOVSKAJA-YPE HEOREM FOR CERAIN GBS OPERAORS 85 Corollary 2.6. Let : I R be a unction. I x I J and is continuous in x, then 2.3 li L x = x. Assue that is continuous on I and there exists an interval K I J such that there exists 0 N and 2 R depending on K, so that or any 0 and any x K we have,2 L x α2 hen the convergence given in 2.3 is unior on K and 2.5 L x x + 2 ω ; or any x K and N, 0. Proo. It results ro heore 2.4. For N, let the linear positive unctionals A, : EI I R with the property: i x, y I I, then 2.6 A, = A, F, 2.7 A, x = A, x and 2.8 A, y = A, y, or any {0,,..., p } N 0, any EI I, where we note F : I R, Ft = t, t or any t I. Now, with the help o L operators, we construct a sequence o bivariate operators. In the ollowing let 0, ]. Deinition 2.7. he operators L : EI I FJ J, N, deined or any unction EI I and any x, y J J by 2.9 L x, y = p are naed the bivariate operators o L-type. ϕ, x + ϕ, ya, heore 2.8. he L, N operators are linear and positive on EI I J J. Proo. he proo ollows iediately.

8 86 O.. POP 3. Main results heore 3.. Let : I I R be a unction. I is continuous in x, x, y, y I I J J, then 3. li L x, y = x, x + y, y. Assue that is continuous on I I and there exists an interval K I J such that there exists 0 N and 2 R depending on K, so that or any 0 and any x K we have,2 L x α2 hen the convergence given in 3. is unior on K K and 3.3 L x, y x, x+ y, y + 2 ω F; or any x, y K K and N, 0. Proo. I N, then L x, y = p p ϕ, xa, + ϕ, ya, and taing 2.6 into account, we obtain 3.4 L x, y = L Fx + L Fy. We have L x x, x + y, y] = L Fx Fx] + L Fy Fy] L Fx Fx + L Fy Fy and apply the Corollary 2.6. Rear 3.2. In general, the sequence L, where : I I R is doesn t converge to the unction. Lea 3.3. Let the GBS operators UL associated to the L operators. I N, UL : EI I FJ J have the or 3.5 UL x, y = L x x + L y x] where x, y J J and EI I. + L x y + L y y] L x, y = L x x + L y x L Fx] + L x y + L y y L Fy]

9 VORONOVSKAJA-YPE HEOREM FOR CERAIN GBS OPERAORS 87 Proo. It results ro deinition o GBS operator, 2., and 3.4. heore 3.4. Let : I I R be a unction. I x, y I I J J, the unctions x, y and F are s ties dierentiable in x and y, the unctions s x τ s, s y t s and F s are continuous in x and y, then 3.6 li s αs s i i! + { UL x, y i τ i x, x + i t i x, y F i x,il x i τ i x, y + i t i y, y F i y,i L y ]} = 0. Assue that the unctions x, y and F are s ties dierentiable on I or any x, y I, with s y τ s, s x t s and F s continuous on I or any x, y I and there exists an interval K I J such that there exist s N and j R depending on K, so that or any s and any x K we have,j L x 3.7 j αj where j {s, s+s}. hen the convergence given in 3.6 is unior on K K and 3.8 s αs UL x, y s i i! i + τ i x, y + i s! s+ s+2 ω i τ i x, x + i t i x, y F i x,il x t i y, y F i y,i L y ] s x τ s ; +ω s y t s ; + ] + ω F s ; or any x, y K and any s.

10 88 O.. POP Proo. We use the 2.5 relation ro heore 3. or the unctions x, y and F and we obtain 3.8 relation. I we note by S the let eber o 3.8 relation and taing 2.7 relation into account, we can write { ] s i S = s αs L x x i i! τ i x, x,il x s + L y i ] x i i! t i x, y,i L x s + i i! F i x ],i L } x L Fx { ] s i + L x y i i! τ i x, y,il y ] s + L y i y i i! t i y, y,il y s + i i! F i y ]},i L y L Fy s i s αs L x x i i! τ i x, x,i L x s + s αs L y i x i i! t i x, y,il x s + s αs L Fx i i! F i x ],il x s i + s αs L x y i i! τ i x, y,i L y s + L y i y i i! t i y, y,i L y s + L Fy i i! F i y,il y { s s! x s + s+2 ω τ s ; +

11 VORONOVSKAJA-YPE HEOREM FOR CERAIN GBS OPERAORS 89 s y + ω + t s ; { + ω s! s + s+2 s y + ω t s ; + ω F s ; ]} s x ω τ s ; ]} F s ;, ro where we obtain 3.8 relation. Fro 3.8 the unior convergence or 3.6 results. heore 3.5. Let : I I R be a unction. I x, y I I J J, the unctions x, y and F are s ties dierentiable in x and y, the unctions s x τ s, s y t s and F s are continuous in x and y, then 3.9 li i s = 0, and 3.0 li s αs UL x, y = x, y { UL s x, y i i x, x i! τi + i t i x, y F i x,i L x i ]} + τ i x, y + i t i y, y F i y,il y = s s! τ s x, x + s t s x, y F s x B s x s ] + τ s x, y + s t s y, y F s y B s y. Assue that the unctions x, y and F are s ties dierentiable on I or any x, y I, with s x τ s, s y t s, F s continuous on I or any x, y I and there exists an interval K I J such that there exist s N and j R depending on K so that or any s and any x K we have,j L x 3. j αj where j {s, s+2}. hen the convergence given in 3.9 and 3.0 is unior on K K.

12 90 O.. POP Proo. It results ro heore 3.4 and Corollary 2.5. Corollary 3.6. Let : I I be a unction. I x, y I I J J, the unctions x, y and F are continuous in x and y, then 3.2 li UL x, y = x, y. Assue that the unctions x, y and F are continuous on I or any x, y I and there exists an interval K I J such that there exist 0 N and 2 R depending on K, so that or any N, 0 and any x K we have,2 L x α2 hen the convergence given in 3.2 is unior on K K and UL x, y x, y ω x ; + ω y ; or any x, y K and any s. + ω F; Proo. It results ro heore 3.4 or s = 0. ] Corollary 3.7. Let : I I R be a unction. I x, y I I J J, the unctions x, y and F are two ties dierentiable in x and y, the unctions 2 x τ 2, 2 y t 2 and F are continuous in x and y, then 3.5 li 2 α2 { UL x, y x, y τ + τ x, x + t x, y F x, L x ]} x, y + t y, y F y,l y = 2 2 τ 2 x, x + 2 t 2 x, y F s B s x 2 ] + τ 2 x, y + 2 t 2 y, y F y B s y.

13 VORONOVSKAJA-YPE HEOREM FOR CERAIN GBS OPERAORS 9 Assue that the unctions x, y and F are two ties dierentiable on I or any x, y I, with 2 x τ 2, 2 y t 2 and F continuous on I or any x, y I and there exists an interval K I J such that there exist 2 N and j R depending on K so that or any 2 and any x K we have,j L 3.6 αj j where j {2, 4}. hen the convergence given in 3.5 is unior on K K. Proo. It results ro heore 3.5 or s = 2. In the ollowing, by particularization and applying heore 3.5, Corollary 3.6 and Corollary 3.7, we can obtain Voronovsaja s type theore and approxiation theore or soe nown operators. Because every application is a siple substitute in the theores o this section, we won t replace anything. In Applications , let p =, ϕ, = p,, where N, {0,,..., } and K = 0, ]. Application 3.. I I = J = 0, ], EI = FJ = C0, ], A, = where N, {0,,..., } and C0, ] then we obtain the Bernstein operators. We have 2 = 5 4, 4 = 9 6,,B x = 0, x 0, ], N and 0 = 2 = see 9]. I = 2 we obtain the GBS operators UB 2 associated to the B 2 operators, studied in the paper 2]. hese operators do not satisy 0 the assuptions o heore A ro paper 3]. here exists no satisactory choice o and 2 in Corollary 5 to express the degree o approxiation o UB 2 operators see 3]. Application 3.2. I I = J = 0, ], EI = L 0, ], FJ = C0, ], A, = + 0 p, ttdt, where N, {0,,..., } and L 0, ], then we obtain the Durreyer operators. In this case 2 = 3 2, 4 = 7 4,, M 2x x =, x 0, ], N and 0 = 2 = see 9].

14 92 O.. POP Application 3.3. I I = J = 0, ], EI = L 0, ], FJ = C0, ], A, = tdt, where N, {0,,..., } and L 0, ], then we obtain the Kantorovich operators. We have 2 =, 4 = 3 2,, K x = 2x, x 0, ], N and = 2 = 3 see 9]. Application 3.4. Let α, β R, α 0. I I = 0, µ α,β ], J = 0, ], + α EI = C0, µ α,β ], FJ = C0, ], A, =, where N, + β {0,,..., } and C0, µ α,β ], then we obtain the Stancu operators. Application 3.5. Let p N 0. I I =0, + p], J =0, ], EI=C0, + p], FJ = C0, ], K = 0, ], ϕ, = p, = p +p,, A, =, p = + p, where N, {0,,..., } and C0, + p], then we obtain the Schurer operators. In Applications and Application 3.0 let K = 0, b], b > 0. Application 3.6. I I = J = 0,, EI = FJ = C0,, ϕ, x = x + x or any x 0,, A, =, + p =, where N, {0,,..., } and C0,, then we obtain the Bleiann-Butzer-Hahn operators. In this case 2 = 4b + b 2 see 22] or 25]. In Applications let p = or any N. Application 3.7. I I = J = 0,, EI = C 2 0,, FJ = x x C0,, ϕ, x = e or any x 0,, A, =,! where N, N 0 and C 2 0,, then we obtain the Mirajan- Favard-Szász operators. We have 2 = b, 4 = 3b 2 + b,, S x = 0, x 0,, N and 0 = 2 = see 2]. Application 3.8. I I = J = 0,, EI = C 2 0,, FJ = + x C0,, ϕ, x = + x or any x 0,, + x A, = where N, N 0 and C 2 0,, then we obtain the Basaov operators. In this case 2 = b+b, 4 = 9b 4 +8b 3 +0b 2 +b,, V x = 0, x 0,, N and 0 = 2 = see 2].

15 VORONOVSKAJA-YPE HEOREM FOR CERAIN GBS OPERAORS 93 Application 3.9. I I = J = K = 0, ], EI = EJ = C0, ], + ϕ, x = x + x or any x 0,, A, =, + where N, N 0 and C0, ], then we obtain the Meyer-König and Zeller operators. We have 2 = 2 see 2]. Application 3.0. I I = J = 0,, EI = FJ = C0,, ϕ, x = e +x + x +x or any x 0,, A, =! + x where N, N 0 and C0,, then we obtain the Isail-May operators. In this case 2 = b + b 2, 4 = b 2 + b 4 +,, R x = A, x = 0, x 0,, N, 0 = and 2 = 2 see 24]. Reerences ] O. Agratini, Aproxiare prin operatori liniari, Presa Universitară Clujeană, Cluj-Napoca, 2000 in Roanian. 2] I. Badea and D. Andrica, Voronovsaja-type theores or certain non-positive linear operator, Rev. Anal. Nu. héor. Approx , ] C. Badea and C. Cottin, Korovin-type heores or Generalized Boolean Su Operators, Colloquia Matheatica Societatis János Bolyai, 58, Approxiation heory, Kecseét Hungary 990, ] V. A. Basaov, An exaple o a sequence o linear positive operators in the space o continuous unctions, Dol. Acad. Nau, SSSR, 3 957, ] M. Becer and R. J. Nessel, A global approxiation theore or Meyer-König and Zeller operators, Math. Zeitschr , ] G. Bleiann, P. L. Butzer and L. A. Hahn, Bernstein-type operator approxiating continuous unctions on the sei-axis, Indag. Math , ] S. N. Bernstein, Déonstration du théorèe de Weierstrass ondée sur le calcul de probabilités, Coun. Soc. Math. Kharow 2, , -2. 8] E. W. Cheney and A. Shara, Bernstein power series, Canadian J. Math , ] M. M. Derriennic, Sur l approxiation des onctions intégrables sur 0, ] par les polynôes de Bernstein odiiés, J. Approx. heory 3 98, ] Z. Ditzian and V. oti, Moduli o Soothness, Springer Verlag, Berlin, 987. ] J. L. Durreyer, Une orule d inversion de la transorée de Laplace: Applications à la théorie des oents, hèse de 3e cycle, Faculté des Sciences de l Université de Paris, ] J. Favard, Sur les ultiplicateurs d interpolation, J. Math. Pures Appl , ] M. Isail and C. P. May, On a aily o approxiation operators, J. Math. Anal. Appl , ] L. V. Kantorovich, Sur certain développeents suivant les polynôes de la ore de S. Bernstein, I, II, C. R. Acad. URSS 930, , ] W. Meyer-König and K. Zeller, Bernsteinsche Potenzreihen, Studia Math ,

16 94 O.. POP 6] G. M. Mirajan, Approxiation o continuous unctions with the aid o polynoials, Dol. Acad. Nau SSSR 3 94, in Russian. 7] M. W. Müller, Die Folge der Gaaoperatoren, Dissertation, Stuttgart, ] O.. Pop, New properties o the Bernstein-Stancu operators, Anal. Univ. Oradea, Fasc. Mateatica o XI 2004, ] O.. Pop, he generalization o Voronovsaja s theore or a class o linear and positive operators, Rev. Anal. Nu. héor. Approx , ] O.. Pop, About a class o linear and positive operators, Carpathian J. Math , ] O.. Pop, About soe linear and positive operators deined by ininite su, De. Math , ] O.. Pop, About operator o Bleiann, Butzer and Hahn, Anal. Univ. iişoara , ] O.. Pop, he generalization o Voronovsaja s theore or a class o bivariate operators, to appear in Stud. Univ. Babes-Bolyai Math. 24] O.. Pop, he generalization o Voronovsaja s theore or exponential operators, Creative Math. & In , ] O.. Pop, About a general property or a class o linear positive operators and applications, Rev. Anal. Nu. héor. Approx , ] F. Schurer, Linear positive operators in approxiation theory, Math. Inst. ech. Univ. Delt. Report, ] D. D. Stancu, Asupra unei generalizări a polinoaelor lui Bernstein, Studia Univ. Babeş-Bolyai, Ser. Math.-Phys , 3-45 in Roanian. 28] D. D. Stancu, Gh. Coan, O. Agratini and R. rîbiţaş, Analiză nuerică şi teoria aproxiării, I, Presa Universitară Clujeană, Cluj-Napoca, 200 in Roanian. 29] O. Szász, Generalization o. S. N. Bernstein s polynoials to the ininite interval, J. Research, National Bureau o Standards , ] A.F. ian, heory o Approxiation o Functions o Real Variable, New Yor, Macillan Co. 963, MR22# ] E. Voronovsaja, Déterination de la ore asyptotique d approxiation des onctions par les polynôes de M. Bernstein, C. R. Acad. Sci. URSS 932, O.. Pop National College Mihai Einescu 5 Mihai Einescu Street Satu Mare Roania E-ail: ovidiutiberiu@yahoo.co Received: Revised: