Done, Yesim

Ibikli, Ertan

Sucu, Sezgin

Bulgarian Digital Mathematics Library at IMI-BAS

M a t h e m a t i c aB a l k a n i c a—————————NewSeries Vol. 28, 2014, Fasc. 1–2Approximation Properties of Schurer-StancuType PolynomialsSezgin Sucu 1, Yes¸im Do¨ne 2, Ertan I˙bikli 3Presented by Virginia KiryakovaConsidering four non-negative parameters α1, α2, β1, β2 which satisfy0 ≤ α2 ≤ α1 ≤ β1 ≤ β2and a given non-negative integer p, the generalized Schurer-Stancu type operatorsSn,p,α,β : C [0, 1 + p] −→ C[α2n+ β2,n+ α2n+ β2]Sn,p,α,β (f ;x) :=(n+ β2n)n+p n+p∑k=0f(k + α1n+ β1)Ckn+p(x− α2n+ β2)k (n+ α2n+ β2− x)n+p−kare constructed. Theorem on convergence and the degree of approximation are established bymodulus of continuity.MSC 2010: 41A25, 41A35KeyWords: Schurer-Stancu type operators, modulus of continuity, rate of convergence,Korovkin theorem1. IntroductionBernstein polynomials were introduced by Bernstein [3]. These polyno-mials have been studied intensively and defined variously generalized forms.In 1962, F. Schurer [5] introduced and studied the linear positive opera-tors called that Bernstein-Schurer polynomials,Bn,p : C [0, 1 + p] −→ C [0, 1]defined byBn,p (f ;x) :=n+p∑k=0f(kn)Ckn+pxk (1− x)n+p−k . (1)38 S. Sucu, Y. Do¨ne, E. I˙bikliD. D. Stancu [7] constructed the linear positive operators Bα,βn (f ;x) associatedwith a real function f ∈ C[0, 1], have the property that limn→∞Bα,βn (f ;x) = f(x)uniformly on [0, 1] in 1969. These operators are known Bernstein-Stancu poly-nomials.Later, D. Ba˘rbosu [2] constructed the linear positive operators as a gen-eralization of Bernstein operatorsPn,p,α,β : C [0, 1 + p] −→ C [0, 1]are defined byPn,p,α,β (f ;x) :=n+p∑k=0f(k + αn+ β)Ckn+pxk (1− x)n+p−k (2)and investigated approximation properties of these operators which are knownSchurer-Stancu polynomials.Then, A. D. Gadjiev and A. M. Ghorbanalizadeh [4] introduced a newconstruction of Bernstein-Stancu type polynomials with shifted knots. Theselinear positive operatorsSn,α,β : C [0, 1] −→ C[α2n+ β2,n+ α2n+ β2]are defined bySn,α,β (f ;x) : =(n+ β2n)n×n∑k=0f(k + α1n+ β1)Ckn(x− α2n+ β2)k (n+ α2n+ β2− x)n−k(3)where α2n+β2 ≤ x ≤ n+α2n+β2 and for αk, βk (k = 1, 2) are non-negative real num-bers provided 0 ≤ α2 ≤ α1 ≤ β1 ≤ β2. Moreover, A. D. Gadjiev and A. M.Ghorbanalizadeh investigated of convergence and the degree of approximationof polynomials Sn,α,β (f ;x) in the space of continuous functions.Let p be a non-negative integer and letSn,p,α,β : C [0, 1 + p] −→ C[α2n+ β2,n+ α2n+ β2]be defined for any f ∈ C [0, 1 + p] and n ∈ N, bySn,p,α,β (f ;x) :=(n+ β2n)n+p×n+p∑k=0f(k + α1n+ β1)Ckn+p(x− α2n+ β2)k (n+ α2n+ β2− x)n+p−k(4)Approximation Properties of Schurer-Stancu . . . 39where α2n+β2 ≤ x ≤ n+α2n+β2 and for αk, βk (k = 1, 2) are non-negative real numbersprovided 0 ≤ α2 ≤ α1 ≤ β1 ≤ β2. It is clear that for p = 0 we obtain Bernstein-Stancu type polynomials (3), for α2 = β2 = 0 we get Schurer-Stancu polynomials(2) and if α1 = α2 = β1 = β2 = 0 Bernstein-Schurer polynomials (1) areobtained.This paper is about generalization of Schurer-Stancu type polynomials(4) with shifted knots symbolized by Sn,p,α,β (f ;x). Convergence of the operators(4) is investigated with the help of the well known Bohman-Korovkin theorem.The degree of convergence of Sn,p,α,β (f ;x) is established by using the modulusof continuity. Some numerical examples are given with the help of Maple13.2. Approximation properties of Sn,p,α,β operatorsLemma 1. Generalized Schurer-Stancu type operators, defined by (4),are linear and positive.The assertions follow from the definition of operators.Lemma 2. For ∀x ∈ [0, 1] and ∀n ∈ N the generalized Schurer-Stancutype operators (4) have the following propertiesSn,p,α,β(1;x) = 1 (5)Sn,p,α,β(t;x) =(n+ β2n+ β1)(n+ pn)x+−pα2 + n(α1 − α2)n(n+ β1)(6)Sn,p,α,β(t2;x) =(n+ β2n+ β1)2 (n+ p)(n+ p− 1)n2(x− α2n+ β2)2+n+ β2(n+ β1)2(n+ pn)(x− α2n+ β2)+2α1n+ β2(n+ β1)2(n+ pn)(x− α2n+ β2)+α21(n+ β1)2 . (7)P r o o f. Let bep˜n,p,k,α,β(x) =(n+ β2n)n+pCkn+p(x− α2n+ β2)k (n+ α2n+ β2− x)n+p−k.Using the definition (4), we easily getSn,p,α,β(1;x) =n+p∑k=0p˜n,p,k,α,β(x) = 1,40 S. Sucu, Y. Do¨ne, E. I˙bikliwhere we use binomial expansion. Like this way, it is obvious thatSn,p,α,β(t;x) =n+p∑k=0(k + α1n+ β1)p˜n,p,k,α,β(x)=n+ pn+ β1n+p∑k=0kn+ pp˜n,p,k,α,β(x) +α1n+ β1n+p∑k=0p˜n,p,k,α,β(x)=(n+ pn+ β1)(n+ β2n)(x− α2n+ β2)+α1n+ β1.Then, we can conclude thatSn,p,α,β(t;x) =(n+ β2n+ β1)(n+ pn)x+−pα2 + n(α1 − α2)n(n+ β1),i.e. (6) holds. After simple calculation, we can writeSn,p,α,β(t2;x) =n+p∑k=0(k + α1n+ β1)2p˜n,p,k,α,β(x)=1(n+ β1)2{n+p∑k=0k2p˜n,p,k,α,β(x)+2α1n+p∑k=0kp˜n,p,k,α,β(x) + α21n+p∑k=0p˜n,p,k,α,β(x)}=1(n+ β1)2{(n+ β2n)2(n+ p)(n+ p− 1)(x− α2n+ β2)2+(n+ β2n)(n+ p)(x− α2n+ β2)+2α1(n+ β2n)(n+ p)(x− α2n+ β2)+ α21}.Taking into account of the above equality, we getSn,p,α,β(t2;x) =(n+ β2n+ β1)2 (n+ p)(n+ p− 1)n2(x− α2n+ β2)2+n+ β2(n+ β1)2(n+ pn)(x− α2n+ β2)+ 2α1n+ β2(n+ β1)2(n+ pn)(x− α2n+ β2)+α21(n+ β1)2 ,Approximation Properties of Schurer-Stancu . . . 41i.e. (7) holds and the proof ends.Theorem 3. Let f be a continuous function on [0, 1 + p]. Thenlimn→∞ ‖Sn,p,α,β(f ;x)− f(x)‖C[α2n+β2,n+α2n+β2] = 0 .P r o o f. Taking into account of the equalities in Lemma 2, for ν = 0, 1, 2we can writelimn→∞ maxx∈[α2n+β2,n+α2n+β2] |Sn,p,α,β(tν ;x)− xν | = 0. (8)DenotingSn,p(f ;x) = Sn,p,α,β(f ;x) ; x ∈[α2n+β2, n+α2n+β2]f(x) ; x ∈[0, α2n+β2)∪(n+α2n+β2, 1] ,we easily get‖Sn,pf − f‖C[0,1] = maxx∈[α2n+β2,n+α2n+β2] |Sn,p,α,β(f ;x)− f(x)| . (9)On the other hand, using (8) and (9) we immediately getlimn→∞ ‖Sn,p(tν ;x)− xν‖C[0,1] = 0 ν = 0, 1, 2.Applying the Korovkin’s theorem to the sequence of linear positive operatorsSn,p we obtainlimn→∞ ‖Sn,p(f ;x)− f(x)‖C[0,1] = 0for every continuous function f . Therefore (9) giveslimn→∞ maxx∈[α2n+β2,n+α2n+β2] |Sn,p,α,β(f ;x)− f(x)| = 0and the proof is completed.42 S. Sucu, Y. Do¨ne, E. I˙bikli3. The rate of convergence of Sn,p,α,β operatorsQualitative result known as a sequence of operators approximate theidentity operator was obtained in Section 2 with the help of well-known Korovkintheorem. The other important problem in approximation theory is quantitativeestimate which is about how quickly do the operators converge the identityoperator. The aim of the present section is a study of quantitative result usingmodulus of continuity.In order to obtain the degree of convergence, we are going to define themodulus of continuity of the function f as follows:ω(f ; δ) = sup {|f(x)− f(y)| : x, y ∈ [0, 1] , |x− y| ≤ δ} (10)for any positive number δ [1].Before starting our main theorem, we give lemma and auxiliary resultconcerning the order of approximation.Lemma 4. The operators (4) verifySn,p,α,β((t− x)2 ;x) = 1(n+ β1)2{(n+ β2)2 (n+ p)n2×(x− α2n+ β2)(n+ α2n+ β2− x)+((n+ pnβ2 − β1)x+(α1 − n+ pnα2))2+(p+ 2β2n+ pn− 2β1)px2+(α1 − n+ pnα2)2px}. (11)P r o o f. The linearity of Sn,p,α,β leads us toSn,p,α,β((t− x)2 ;x) = Sn,p,α,β(t2;x)− 2xSn,p,α,β(t;x) + x2Sn,p,α,β(1;x) .Taking into account of the equality in Lemma 2 , we obtainApproximation Properties of Schurer-Stancu . . . 43Sn,p,α,β((t− x)2 ;x) =(n+ β2n+ β1)2 (n+ p)(n+ p− 1)n2(x− α2n+ β2)2+(n+ β2)(n+ β1)2 (n+ pn)(x− α2n+ β2)+ 2α1(n+ β2)(n+ β1)2 (n+ pn)(x− α2n+ β2)+α21(n+ β1)2− 2x[(n+ β2n+ β1)(n+ pn)x+−pα2 + n(α1 − α2)n(n+ β1)]+ x2=(1n+ β1)2×{(n+ β2)2 (n+ p)(n+ p− 1)n2(x− α2n+ β2)2+ (n+ β2) (n+ pn)(x− α2n+ β2)+2α1 (n+ β2) (n+ pn)(x− α2n+ β2)+ α21− 2x[(n+ β1) (n+ β2)(n+ pn)x+ (n+ β1)−pα2 + n(α1 − α2)n]+x2 (n+ β1)2}=1(n+ β1)2×{(n+ β2)2 (n+ p)n2(x− α2n+ β2)(n+ α2n+ β2− x)+((n+ pnβ2 − β1)x+(α1 − n+ pnα2))2+(p+ 2β2n+ pn− 2β1)px2 +(α1 − n+ pnα2)2px}.Thus, we get the desired result.It is well known the following result obtained by O. Shisha and B. Mond[6].Theorem 5. Let (Ln)n∈N, Ln : C [a, b]→ B [a, b] be a sequence of linearpositive operators, reproducing the constant functions. For any f ∈ C [a, b],44 S. Sucu, Y. Do¨ne, E. I˙biklix ∈ [a, b] and δ ∈ [0, b− a], the following|Ln(f ;x)− f(x)| ≤{1 + δ−1√Ln((t− x)2 ;x)}ω(f ; δ)holds.Theorem 6. For any f ∈ C [0, 1 + p] and each x ∈[α2n+β2, n+α2n+β2]generalized Schurer-Stancu type operators (4) verify|Sn,p,α,β(f ;x)− f(x)| ≤ 2ω(f ;√δn,p,α,β,x) (12)whereδn,p,α,β,x =1(n+ β1)2{(n+ β2)2 (n+ p)n2(x− α2n+ β2)(n+ α2n+ β2− x)+((n+ pnβ2 − β1)x+(α1 − n+ pnα2))2+(p+ 2β2n+ pn− 2β1)px2+(α1 − n+ pnα2)2px}(13)and δn,p,α,β,x tends to zero as n→∞.P r o o f. The main step in the proof of this theorem consists in the fol-lowing inequality given in Theorem 5|Sn,p,α,β(f ;x)− f(x)| ≤{1 + δ−1√Sn,p,α,β((t− x)2 ;x)}ω(f ; δ) . (14)Applying Lemma 4 and (14) we get|Sn,p,α,β(f ;x)− f(x)| ≤(1 +1δ1n+ β1×{(n+ β2)2 (n+ p)n2(x− α2n+ β2)(n+ α2n+ β2− x)+((n+ pnβ2 − β1)x+(α1 − n+ pnα2))2+(p+ 2β2n+ pn− 2β1)px2+(α1 − n+ pnα2)2px}1/2)ω(f ; δ).Approximation Properties of Schurer-Stancu . . . 45In the above inequality by choosingδ =√δn,p,α,β,x =1n+ β1{(n+ β2)2 (n+ p)n2(x− α2n+ β2)(n+ α2n+ β2− x)+((n+ pnβ2 − β1)x+(α1 − n+ pnα2))2+(p+ 2β2n+ pn− 2β1)px2 +(α1 − n+ pnα2)2px}1/2,we arrive to (12) and the proof ends.Remark 7. Theorem 6 expresses the order of local approximation off by Sn,p,α,βf . For getting the order of global approximation, we must take in(13) the maximum of δn,p,α,β,x when x ∈[α2n+β2, n+α2n+β2],|Sn,p,α,β(f ;x)− f(x)| ≤ 2ω(f ;√δn,p,α,β)whereδn,p,α,β =1(n+ β1)2{n+ p4+((β2 − β1) n+ α2n+ β2+ (α1 − α2))2+(p+ 2n+ β2n(β2 − β1) + 2 pnβ2 +pn2β22)p(n+ α2n+ β2)2+(2n+ β2n(α1 − α2)− 2 1nα2(β2 − β1)− 2 pnα2 − 2 pn2α2β2)×pn+ α2n+ β2+(pn2α22 − 21nα2(α1 − α2))p}.Clearly, this maximum depends on the relations between αk, βk, p (k = 1, 2).Example 8. Let be f(x) = sin(2pi(x+ 1)32 ), p = 1, α1 = 0.2, α2 = 0.1,β1 = 0.3 and β2 = 0.4. If it is chosen n = 20 and n = 100, approximationto function f by Sn,p,α,β generalized Schurer-Stancu type operators has beenshown in Figure 1. Furthermore, we give its algorithm after the figure.46 S. Sucu, Y. Do¨ne, E. I˙bikliFigure 1: Approximation to f(x) = sin(2pi(x + 1)32 ) by generalized Schurer-Stancu type polynomialswith(plots):> p:=1:> alpha1:=0.2:> alpha2:=0.1:> beta1:=0.3:> beta2:=0.4:> f:=x-> sin(2*Pi*(x+1)ˆ(3/2))> S:=(n,p,alpha1,alpha2,beta1,beta2)-> ((n+beta2)/n)ˆ(n+p)*sum(f((k+alpha1)/(n+beta1))*binomial(n+p,k)*(x-(alpha2/(n+beta2)))ˆk*((n+alpha2)/(n+beta2)-x)ˆ(n+p-k),k=0..n+p):> p1:=plot(f(x),x=0..1,color=black):> p2:=plot(evalf(S(20,p,alpha1,alpha2,beta1,beta2)),x=(alpha2/(20+beta2))..(20+alpha2)/(20+beta2),color=red, linestyle=2):> p3:=plot(evalf(S(100,p,alpha1,alpha2,beta1,beta2)),x=(alpha2/(100+beta2))..(100+alpha2)/(100+beta2),color=green, linestyle=4):> display([p1,p2,p3]);Approximation Properties of Schurer-Stancu . . . 47Example 9. In the following Table 1 we calculate the error bound offunction f(x) = sin(pix) with the help of modulus of continuity for α1 = 2,α2 = 1, β1 = 3, β2 = 4 and p = 1 by using Maple13.n Error estimate by Sn,p,α,β operators to the function f (x) = sin(pix)10 1.246331908102 0.353109854103 0.100809812104 0.031463164105 0.009936086106 0.003141640107 0.000993460108 0.000314159109 0.0000993451010 0.000031415Table 1. The error bound of function f(x) = sin(pix) .References[1] F. A l t o m a r e, M. C a m p i t i. Korovkin Type Approximation Theory andIts Application, de Gruyter Studies in Mathematics, Vol. 17, Walter deGruyter and Co., Berlin, 1994.[2] D. B a˘ r b o s u. Schurer-Stancu type operators. // Studia Univ. Babes-BolyaiMath., 48, No 3, (2003), 31–35.[3] S. N. B e r n s t e i n. De´monstration du the´ore´me de Weierstrass fonde´e surle calcul des probabilite´s. // Commun. Soc. Math. Kharkow, 13, No 2,(1912), 1–2.[4] A. D. G a d j i e v, A. M. G h o r b a n a l i z a d e h, Approximation propertiesof a new type Bernstein-Stancu polynomials of one and two variables. //Appl. Math. Comput. 216, (2010), 890-901.[5] F. S c h u r e r. Linear positive operators in approximation theory. Math. Inst.Techn. Univ. Delft report, 1962.[6] O. S h i s h a, B. M o n d. The degree of convergence of linear positive oper-ators. // Proc. Nat. Acad. Sci. U.S.A., 60, (1968), 1196–1200.[7] D. D. S t a n c u. Asupraunei generaliza˘ri a polinoamelor lui Bernstein. //Studia Univ. Babes-Bolyai Math. (Cluj), 14, No 2 (1969), 31–45.48 S. Sucu, Y. Do¨ne, E. I˙bikli1 Department of Mathematics, Received: May 7, 2012Faculty of Science, Ankara University,Tandog˘an TR-06100,Ankara, TURKEYe-mail: ssucu@ankara.edu.tr2 Department of Statistic,Faculty of Science, Ankara University,Tandog˘an TR-06100,Ankara, TURKEYe-mail: yesim.done@ankara.edu.tr3 Department of Mathematics,Faculty of Science, Ankara University,Tandog˘an TR-06100,Ankara, TURKEYe-mail: ibikli@ankara.edu.tr

Approximation Properties of Schurer-Stancu Type Polynomials

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Approximation Properties of Schurer-Stancu Type Polynomials

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