In this paper we give approximation theorems for modified Stancu beta operators of differentiable functions. The Stancu beta operators were examined in [8, 1, 2, 5] and other papers

L. Rempulska

M. Skorupka

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REVUE D’ANALYSE NUMÉRIQUE ET DE THÉORIE DE L’APPROXIMATIONRev. Anal. Numér. Théor. Approx., vol. 35 (2006) no. 2, pp. 189–197ictp.acad.ro/jnaatAPPROXIMATION PROPERTIES OF MODIFIED STANCU BETAOPERATORSL. REMPULSKA∗ and M. SKORUPKA∗Abstract. In this paper we give approximation theorems for modified Stancubeta operators of differentiable functions. The Stancu beta operators were ex-amined in [8, 1, 2, 5] and other papers.MSC 2000. 41A36, 41A25.Keywords. Beta operator, degree of approximation, Voronovskaya theorem.1. INTRODUCTION1.1. In 1995 D. D. Stancu ([8]) introduced and examined approximationproperties of the following beta operators(1.1) Ln(f ;x) := 1B(nx,n+1)∫ ∞0tnx−1(1+t)nx+n+1 f(t)dt,x ∈ I = (0,∞), n ∈ N = {1, 2, ...}, of real-valued functions f bounded on I,where B is the Euler beta function ([4]) defined by the formula(1.2) B(a, b) :=∫ 10ta−1(1− t)b−1dt ≡∫ ∞0ta−1(1+t)a+b dt, a, b > 0.In [8] it was proved that if f is continuous and bounded on I, then(1.3) |Ln(f ;x)− f(x)| ≤(1 +√x(x+ 1))ω1(f ; 1√n−1)and(1.4) |Ln(f ;x)− f(x)| ≤ (3 + x(x+ 1)) ω2(f ; 1√n−1),for all x ∈ I and n ≥ 2, where ωk(f ; ·), k = 1, 2, is the modulus of continuityof the order k of f .It is known ([1], [8]) that if f is continuous and bounded on I with derivativesf ′ and f ′′, then(1.5) limn→∞n (Ln(f ;x)− f(x)) = x(x+1)2 f′′(x) at every x ∈ I∗ Institute of Mathematics, Poznań University of Technology, Piotrowo 3A, 60-965Poznań, Poland, email: {lrempuls,mariolas}@math.put.poznan.pl.190 L. Rempulska and M. Skorupka 2and the order O(1n)of approximation of r times differentiable functions f ,r ≥ 3, by Ln cannot be improved.1.2. In this paper we shall show that the approximation order of differen-tiable functions by beta operators can be improved by some modification offormula (1.1). We use the Kirov type method given for Bernstein operators in[6] (see also [7]).Similarly to [3] let p ∈ N0 = N ∪ {0},(1.6) w0(x) := 1, wp(x) := (1 + xp)−1 if p ≥ 1, x ∈ I,and let Cp be the set of all real–valued functions f defined on I, for whichwpf is uniformly continuous and bounded on I and the norm(1.7) ‖f‖p := supx∈Iwp(x) |f(x)| .We have Cp ⊂ Cq if p < q and ‖f‖q ≤ ‖f‖p for f ∈ Cp.In this paper we shall consider the functions class Cr, r ∈ N0, of all f ∈ Crhaving derivatives f (k) ∈ Cr−k, 0 ≤ k ≤ r. Clearly C0 ≡ C0.1.3. Analogously to [8] we denote by(1.8) bu,v(t) := tu−1B(u,v)(1+t)u+v , t > 0,for positive parameters u and v, where B is the beta function defined by (1.2).Let r ∈ N0. For f ∈ Cr we define the following modified Stancu betaoperators:(1.9) Ln;r(f ;x) :=∫ ∞0bnx,n+1(t)Fr(x, t) dt,x ∈ I and r ≤ n ∈ N , where(1.10) Fr(x, t) :=r∑j=0f (j)(t)j! (x− t)j , x, t ∈ I.If r = 0 and f ∈ C0, then by (1.9), (1.10) and (1.1) we have(1.11) Ln;0(f ;x) ≡ Ln(f ;x) =∫ ∞0bnx,n+1(t) f(t) dt for x ∈ I, n ∈ N.In Section 2 we shall prove that Ln;r, r ∈ N , n ≥ 2r, is a linear operatoracting from Cr into Cr. Moreover we shall prove that Ln is a linear positiveoperator acting from Cr into Cr if n ≥ 2r and r ∈ N .1.4. It is known ([1], [8]) that operators Ln(f) given by (1.1) are welldefined for functions fs(x) = xs, x ∈ I, s ∈ N , n ≥ s, and Ln(fs) arealgebraic polynomials of the order s. In [1] and [8] it was proved that(1.12) Ln(1;x) = 1, Ln(t;x) = x for x > 0, n ≥ 1,(1.13) Ln((t− x)2;x)= x(x+1)n−1 , x > 0, n ≥ 2,3 Approximation properties of modified Stancu beta operators 191and for every s ∈ N there exists a positive constant M1(s) depending only ons such that(1.14) w2s(x)Ln((t− x)2s;x)≤ M1(s)n−s for x > 0, n ≥ 2s.2. LEMMASIn this paper we shall denote by Mi(α, β), i ∈ N , suitable positive constantsdepending only on indicated parameters α and β.We shall apply the following inequalities(2.1) (wp(x))2 ≤ w2p(x), 1(wp(x))2 ≤4w2p(x) ,wp+s(x)wp(x)ws(x) ≤ 3,for x > 0 and p, s ∈ N0, which can be easily obtained from (1.6).Applying the Hölder inequality and (1.12), (2.1) and (1.14), we immediatelyobtainLemma 2.1. For every s ∈ N there exists M2(s) = const. > 0 such thatws(x)Ln(|t− x|s;x) ≤M2(s)n−s2 for x > 0, n ≥ 2s.Lemma 2.2. Let p, s ∈ N . Then there exist positive constants M3(p) andM4(p, s) such that(2.2) wp(x)Ln(1wp(t) ;x)≤M3(p) for x > 0, n ≥ 2p,and(2.3) wp+s(x)Ln(|t−x|swp(t) ;x)≤M4(p, s)n−s/2,for x > 0 and n ≥ 2(p+ s).Proof. By (1.6) we have1wp(t) = 1 + tp ≤ 2p (1 + xp + |t− x|p) , t, x ∈ I, p ∈ N,which by (1.1), (1.2) and (1.12) implies thatLn(1wp(t) ;x)≤ 2p (1 + xp + Ln (|t− x|p;x)) .Now by (1.6) and Lemma 2.1 follows (2.2).Applying the Hölder inequality and (1.14), (2.1), (2.2) and (1.6), we getwp+s(x)Ln(|t−x|swp(t) ;x)≤ 3wp(x)ws(x)Ln(|t−x|swp(t) ;x)≤≤ 6(w2s(x)Ln((t− x)2s;x))1/2 (w2p(x)Ln(1w2p(t) ;x))1/2≤M4(p, s)n−s/2 for x > 0, n ≥ 2(p+ s),and we complete the proof. 192 L. Rempulska and M. Skorupka 4Lemma 2.3. Let p ∈ N . Then there exists M5(p) = const. > 0 such thatfor every f ∈ Cp we have(2.4) ‖Ln(f)‖p ≤ M5(p) ‖f‖p for n ≥ 2p.If f ∈ C0, then ‖Ln(f)‖0 ≤ ‖f‖0 for n ≥ 1.The formula (1.1) and (2.4) show that Ln, n ≥ 2p, is a positive linear operatorfrom the space Cp into Cp, p ∈ N0.Proof. By (1.1), (1.2), (1.8) and (1.6) we have for f ∈ Cp, p ∈ N0,wp(x) |Ln(f ;x)| ≤ ‖f‖p wp(x)Ln(1wp(t) ;x), x > 0, n ≥ 2p,which by (2.2) and (1.7) yields (2.4). Lemma 2.4. Let r ∈ N . Then there exists M6(r) = const. > 0 such that(2.5) ‖Ln;r(f)‖r ≤ M6(r)r∑j=0∥∥∥f (j)∥∥∥r−j,for every f ∈ Cr and n ≥ 2r.The formulas (1.9) and (1.10) and (2.5) show that Ln;r, n ≥ 2r, is a linearoperator from the space Cr into Cr, r ∈ N .Proof. It is obvious that if f ∈ Cr, r ∈ N , then there exist ‖f (j)‖r−j for0 ≤ j ≤ r. This fact and (1.8)–(1.11) and (1.7) imply that|Ln;r(f ;x)| ≤r∑j=01j! Ln(∣∣∣f (j)(t)∣∣∣ |t− x|j ;x) ≤≤r∑j=0∥∥∥f (j)∥∥∥r−jLn(|t−x|jwr−j(t) ;x)for f ∈ Cr. Now applying Lemma 2.1 and Lemma 2.2 we obtainwr(x) |Ln;r(f ;x)| ≤ M6(r)r∑j=0∥∥∥f (j)∥∥∥r−jn−j/2,for x > 0 and n ≥ 2r, which by (1.7) yields (2.5). 3. THEOREMS3.1. First we shall give theorems on the order of approximation of functionf by operators Ln and Ln;r. We shall use the modulus of continuity ωk(f) ofthe order k = 1, 2 of f ∈ Cp, i.e.(3.1) ωk(f ;Cp; t) := sup0≤h≤t∥∥∥∆khf(·)∥∥∥p, t ≥ 0,where ∆1hf(x) ≡ ∆hf(x) = f(x + h) − f(x) and ∆2hf(x) ≡ ∆h(∆hf(x)) =f(x)− 2f(x+ h) + f(x+ 2h) for x > 0 and h ≥ 0.5 Approximation properties of modified Stancu beta operators 193Theorem 3.1. Suppose that f ∈ Cp, p ∈ N0, is twice differentiable functionon I and f ′ and f ′′ belong to Cp also. Then there exists M7(p) = const. > 0such that(3.2) ‖Ln(f)− f‖p+2 ≤ M7(p) ‖f ′′‖p n−1,for n ≥ 2p+ 4.Proof. For f satisfying our assumptions we havef(t) = f(x) + f ′(x)(t− x) +∫ tx∫ sxf ′′(u) duds, t, x > 0.By elementary calculations we getf(t) = f(x) + f ′(x)(t− x) +∫ tx(t− u)f ′′(u) duand next by (1.1) and (1.12) we can writeLn(f(t);x) = f(x) + Ln(∫ tx(t− u)f ′′(u) du;x)and|Ln(f(t);x)− f(x)| ≤ Ln(∣∣∣∣∫ tx(t− u)f ′′(u) du∣∣∣∣ ;x) .Using the inequality∣∣∣∣∫ tx(t− u)f ′′(u)du∣∣∣∣ ≤ ‖f ′′‖p ∣∣∣∣∫ tx|t−u|wp(u) du∣∣∣∣≤ ‖f ′′‖p(1wp(t) +1wp(x))(t− x)2 for t, x > 0,and (2.1) we getwp+2(x) |Ln(f ;x)− f(x)| ≤ ‖f ′′‖p(wp+2(x)Ln((t−x)2wp(t) ;x)+ 3w2(x)Ln((t− x)2;x))and next by (2.3), (1.6), (1.13) and (1.7) we obtain (3.2) for n ≥ 2p+ 4. Now we shall prove analogue of (1.4) for f ∈ Cp.Theorem 3.2. Let p ∈ N0. Then there exists M8(p) = const. > 0 such thatfor every f ∈ Cp and n ≥ 2p+ 4 we have(3.3) ‖Ln(f)− f‖p+2 ≤ M8(p)ω2(f ;Cp; 1√n).Proof. Similarly to [3] we apply the following Steklov function fh of f ∈ Cp:fh(x) := 4h2∫ h/20∫ h/20(2f(x+ s+ t)− 2f(x+ 2(s+ t))) ds dt, x, h > 0.It is known ([3]) that fh and derivatives f ′h, f ′′h belong to Cp if f ∈ Cp.Moreover for h > 0 we have(3.4) ‖f − fh‖p ≤ ω2(f ;Cp;h),194 L. Rempulska and M. Skorupka 6(3.5) ‖f ′′h‖p ≤ 9h−2 ω2(f ;Cp;h).From the above properties of fh and linearity of Ln we deduce that|Ln(f(t);x)− f(x)| ≤ |Ln(f(t)− fh(t);x)|+ |Ln(fh(t);x)− fh(x)|+ |fh(x)− f(x)|,for x > 0, h > 0 and n ≥ 2p+ 2. Applying (2.4) and (3.4) we get(3.6) wp(x) |Ln(f(t)− fh(t);x)| ≤ M5(p)‖f − fh‖p ≤ M5(p)ω2(f ;Cp;h).Theorem 3.1 and (3.5) imply thatwp+2(x) |Ln(fh(t);x)− fh(x)| ≤ M7(p) ‖f ′′h‖p n−1(3.7)≤ M7(p)h−2 n−1 ω2(f ;Cp;h).Using (1.6), (2.1), (3.6), (3.7) and (3.4), we get(3.8) wp+2(x) |Ln(f ;x)− f(x)| ≤ M9(p)ω2(f ;Cp;h)(1 + h−2n−1)for x, h > 0 and n ≥ 2p + 4. Putting h = 1√nin (3.8), we obtain the desiredinequality (3.3) by (1.7). Now we shall give an analogue of (1.3) for operators Ln;r.Theorem 3.3. Let r ∈ N . Then there exists M10(r) = const. > 0 such thatfor every f ∈ Cr we have(3.9) ‖Ln;r(f)− f‖r+1 ≤ M10(r)n−r/2 ω1(f (r);C0; 1√n), n ≥ 2r + 2.Proof. The formulas (1.8)–(1.12) imply that(3.10) Ln;r(f ;x)− f(x) = Ln (Fr(x, t)− f(x);x) ,for every f ∈ Cr, x ∈ I and n ≥ 2r + 2.Similarly to [6] and [7] we use the following modified Taylor formula off ∈ Cr at a fixed point t > 0:f(x) =r∑j=0f (j)(t)j! (x− t)j + (x−t)r(r−1)! Ir(x, t), x > 0,where(3.11) Ir(x, t) :=∫ 10(1− u)r−1(f (r)(t+ u(x− t))− f (r)(t))du.Hence we havef(x)− Fr(x, t) = (x−t)r(r−1)! Ir(x, t), t, x > 0,which used to (3.10) implies that|Ln;r(f ;x)− f(x)| ≤ Ln (|Fr(x, t)− f(x)|;x)≤ 1(r−1)! Ln(|t− x|r |Ir(x, t)|;x)7 Approximation properties of modified Stancu beta operators 195for x > 0 and n ≥ 2r + 2. Applying elementary properties of ω1 defined by(3.1), we get from (3.11)|Ir(x, t)| ≤∫ 10(1− u)r−1ω1(f (r);C0;u|x− t|)du≤ ω1(f (r);C0; |t− x|) ∫ 10(1− u)r−1du≤ 1r ω1(f (r);C0; 1√n) (√n|t− x|+ 1).From the above and by (1.6) and Lemma 2.1, we obtainwr+1(x) |Ln;r(f ;x)− f(x)| ≤≤ 1r! ω1(f (r);C0; 1√n) (√nwr+1(x)Ln(|t− x|r+1;x)+ wr(x)Ln (|t− x|r;x|))≤ M10(r)n−r/2 ω1(f (r);C0; 1√n),for x > 0 and n ≥ 2r + 2, which by (1.7) gives (3.9). From Theorem 3.2 and Theorem 3.3 we can derive the following corollaries.Corollary 3.4. If f ∈ Cp, p ∈ N0, thenlimn→∞(Ln(f ;x)− f(x)) = 0 at every x > 0.The above convergence is uniform on every interval [a, b], 0 < a < b <∞.Corollary 3.5. If f ∈ Cr, r ∈ N , thenlimn→∞nr/2 ‖Ln;r(f)− f‖r+1 = 0.Corollary 3.6. The order of approximation of f ∈ Cr, r ≥ 2, by operatorsLn;r(f) is better than by Ln(f).3.2. In this section we shall prove the Voronovskaya type theorem foroperators Ln;r, i.e. we shall give an analogue of (1.5).Theorem 3.7. Suppose that r ∈ N and f is a function belonging to Cr andhaving derivatives f (r+1) and f (r+2) continuous and bounded on I. Then forevery x > 0 we haveLn;r(f ;x)− f(x) = (−1)rf (r+1)(x)Ln((t−x)r+1;x)(r+1)!(3.12)+ (−1)r(r+1)f (r+2)(x)Ln((t−x)r+2;x)(r+2)!+ ox(n(r+2)/2)as n→∞.Proof. Fix function f and x > 0. Then for every derivative f (j), 0 ≤ j ≤ r,we can write the Taylor formula at x:f (j)(t) =r+2−j∑i=0f (j+i)(x)i! (t− x)i + ϕj(t, x)(t− x)r+2−j , t > 0,196 L. Rempulska and M. Skorupka 8where ϕj(t) ≡ ϕj(t, x) is function belonging to C0 and limt→xϕj(t) = ϕj(x) = 0.Hence Fr defined by (1.10) can be written in the formFr(x, t) =r∑j=0(−1)jj!r+2−j∑i=0f (j+i)(x)i! (t− x)j+i + Φr(t)(t− x)r+2, t > 0,with the function(3.13) Φr(t) :=r∑j=0(−1)jj! ϕj(t).By elementary calculations we getFr(x, t) =r∑j=0(−1)jr+2∑s=j(sj)f (s)(x)(t−x)ss! + Φr(t)(t− x)r+2(3.14)=r∑s=0f (s)(x)(t−x)ss!s∑j=0(sj)(−1)j+ f(r+1)(x)(t−x)r+1(r+1)!r∑j=0(r+1j)(−1)j+ f(r+2)(x)(t−x)r+2(r+2)!r∑j=0(r+2j)(−1)j + Φr(t)(t− x)r+2.Applying equalities for m ∈ N0:m∑j=0(mj)(−1)j ={1 if m = 0,0 if m ≥ 1,m∑j=0(m+1j)(−1)j = (−1)mandm∑j=0(m+2j)(−1)j = (m+ 1)(−1)j ,we derive from (1.9)–(1.11) and (3.14) the following formula:Ln;r(f ;x) = f(x) +(−1)rf (r+1)(x) Ln((t−x)r+1;x)(r+1)!(3.15)+ (−1)r(r+1)f (r+2)(x) Ln((t−x)r+2;x)(r+2)!+ Ln(Φr(t)(t− x)r+2;x), for n ≥ 2r + 2.The properties of ϕj , 0 ≤ j ≤ r, imply that Φr given by (3.13) is a functionbelonging to C0 and limt→xΦr(t) = Φr(x) = 0. Hence there exists Ln(Φ2r(t);x)9 Approximation properties of modified Stancu beta operators 197for x > 0 and n ≥ 1 and by Corollary 3.4 we have(3.16) limn→∞Ln(Φ2r(t);x)= Φ2r(x) = 0.Applying the Hölder inequality, we get∣∣∣Ln (Φr(t)(t− x)r+2;x)∣∣∣ ≤ (Ln (Φ2r(t);x))1/2 (Ln ((t− x)2r+4;x))1/2for n ≥ 2r + 4, which by (3.16) and (1.12) gives(3.17) Ln(Φr(t)(t− x)r+2;x)= ox(n−(r+2)/2)as n→∞.Using (3.17) to (3.15), we obtain (3.12). REFERENCES[1] Abel, U., Asymptotic approximation with Stancu beta operators, Rev. Anal. Numér.Théor. Approx., 27, no. 1, pp. 5–13, 1998.[2] Abel, U. and Gupta, V., Rate of convergence for Stancu beta operators for functionsof bounded variation, Rev. Anal. Numér. Théor. Approx., 33, no. 1, pp. 3–9, 2004.[3] Becker, M., Global approximation theorems for Szász-Mirakyan and Baskakov opera-tors in polynomial weight spaces, Indiana Univ. Math. J., 27, no. 1, pp. 127–142, 1978.[4] Fichtenholz, G. M., Calculus, vol. 2, Warsaw 1964.[5] Gupta, V., Abel, U. and Ivan, M., Rate of convergence of beta operators of secondkind for functions with derivatives of bounded variation, Inter. Jour. of Math. and Math.Sciences (in print).[6] Kirov, G. H., A generalization of the Bernstein polynomials, Math. Balkanica, 2, no. 2,pp. 147–153, 1992.[7] Rempulska, L. and Walczak, Z., Modified Szász–Mirakyan operators, Math. Balkan-ica, 18, pp. 53–63, 2004.[8] Stancu, D. D., On the beta approximating operators of second kind, Rev. Anal. Numer.Theor. Approx., 24, nos. 1–2, pp. 231–239, 1995.Received by the editors: January 8, 2005.

Approximation properties of modified Stancu beta operators

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Approximation properties of modified Stancu beta operators

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