In this paper we will determinate the first absolute moments for Bernstein, Szász-Mirakjan, Bleimann-Butzer-Hahn, Meyer-König and Zeller operators. For the Szász-Mirakjan operators we give some properties with the absolute moment of high order

Ovidiu T. Pop

Petru I. Braica

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REVUE D’ANALYSE NUMÉRIQUE ET DE THÉORIE DE L’APPROXIMATIONRev. Anal. Numér. Théor. Approx., vol. 39 (2010) no. 2, pp. 156–163ictp.acad.ro/jnaatTHE FIRST ABSOLUTE MOMENT FOR SOME OPERATORSOVIDIU T. POP† and PETRU I. BRAICA∗Abstract. In this paper we will determinate the first absolute moments forBernstein, Szász-Mirakjan, Bleimann-Butzer-Hahn, Meyer-König and Zeller op-erators. For the Szász-Mirakjan operators we give some properties with theabsolute moment of high order.MSC 2000. 41A10, 41A10, 41A36.Keywords. Bernstein’s polynomials, Szász-Mirakjan operators, Bleimann-Butzer-Hahn operators, Meyer-König and Zeller operators, absolute momentof high order.1. PRELIMINARIESLet N be the set of positive integers and N0 = N ∪ {0}. For any m ∈ N, letBm : C([0, 1])→ C([0, 1]) be the Bernstein operators, defined by(1.1) (Bmf)(x) =m∑k=0pm,k(x)f(km),where pm,k(x) are the fundamental Bernstein’s polynomials, given by(1.2) pm,k(x) =(mk)xk(1− x)m−k,for any x ∈ [0, 1] and any k ∈ {0, 1, . . . ,m} (see [3] or [11]).For any m ∈ N, let Sm : C2([0,+∞)) → C([0,+∞)) be the Szász-Mirakjanoperators (see [4], [7], [11] or [12]), defined by(1.3) (Smf)(x) = e−mx∞∑k=0(mx)kk! f(km),for any x ∈ [0,+∞).The operators Zm : B([0, 1])→ C([0, 1]) defined by(1.4) (Zmf)(x) = (1− x)m+1∞∑k=0(m+kk)xkf(km+k),†National College “Mihai Eminescu”, 5 Mihai Eminescu Street, 440014 Satu Mare, Ro-mania, e-mail: ovidiutiberiu@yahoo.com.∗ Secondary School “Grigore Moisil”, 1 Mileniului Street, 440037 Satu Mare, Romania,e-mail: petrubr@yahoo.com.2 The first absolute moment for some operators 157for any x ∈ [0, 1) and (Zmf)(1) = f(1), m ∈ N, are called Meyer-König andZeller operators (see [6] or [11]).For m ∈ N, let the operators Lm : CB([0,+∞))→ CB([0,+∞)), defined by(1.5) (Lmf)(x) = (1 + x)−mm∑k=0(mk)xkf(km+1−k),for any x ∈ [0,∞). These operators are called Bleimann-Butzer-Hahn opera-tors (see [2] or [11]).In what follows, let I ⊂ R be an interval. We recall that the functionsϕx, ψx :I→R are defined byϕx(t) = |t− x| , ψx(t) = t− x,for any (x, t) ∈ I × I. For a sequence of operators (Lm)m≥1, define Tm,i by(1.6) (Tm,iLm)(x) = mi(Lmψix)(x),for any x ∈ I, any m ∈ N and any i ∈ N0.2. MAIN RESULTSIn [5] is proved the following result contained in Theorem 2.1.Theorem 2.1. Let m ∈ N and 0 ≤ x ≤ 1. If i = [mx], then(2.1) (Bmϕx) (x) = 2x(1− x)pm−1,i (x) .Theorem 2.2. Let m ∈ N and 0 ≤ x. If i = [mx], then(2.2) (Smϕx) (x) = 2xe−mx (mx)ii! .Proof. From i = [mx] it results i ≤ mx < i + 1, equivalent withim≤ x <i+ 1m. We get(Smϕx)(x) = e−mx∞∑k=0(mx)kk!∣∣ km − x∣∣= 2e−mxi∑k=0(mx)kk!(x− km)+ e−mx∞∑k=0(mx)kk!(km − x)= 2e−mx[xi∑k=0(mx)kk! − xi∑k=1(mx)k−1(k−1)!]+ (Sme1) (x)− x (Sme0) (x)= 2xe−mx (mx)ii! ,so (2.2) is obtained. 158 Ovidiu T. Pop and Petru I. Braica 3Theorem 2.3. Let m ∈ N and 0 ≤ x < m. If i =[(m+1)xx+1], then(2.3) (Lmϕx)(x) = x(1 + x)−m (2(mi )xi − xm) .In the case when m ∈ N and x ≥ m, then(2.4) (Lmϕx)(x) = xm+1(1 + x)−m.Proof. From i =[(m+1)xx+1]it results i ≤ (m+1)xx+1 < i + 1, equivalent withxi+ i ≤ mx+ x < xi+ x+ i+ 1, from where im−i+1 ≤ x <i+1m−i . Taking thatx < m into account, one obtains i =[(m+1)xx+1]≤ (m+1)xx+1 < m.We get(Lmϕx)(x) = (1 + x)−mm∑k=0(mk)xk∣∣∣ km−k+1 − x∣∣∣= 2 (1 + x)−mi∑k=0(mk)xk(x− km−k+1)+ (1 + x)−mm∑k=0(mk)xk(km−k+1 − x)= 2 (1+x)−m(xi∑k=1(mk)xk −i∑k=1(mk)xk km−k+1)+(Lme1) (x)−x (Lme0) (x)= 2 (1 + x)−m(xi∑k=1(mk)xk − xi∑k=1(mk−1)xk−1)− x(x1+x)m= 2x(1 + x)−m(mi)xi − x(x1+x)m= x (1 + x)−m(2(mi)xi − xm),so (2.3) is obtained. If x ≥ m, we have that km−k+1 ≤ x, for any k ∈{0, 1, . . . ,m} and then(Lmϕx)(x) = (1 + x)−mm∑k=0(mk)xk(x− km−k+1)= x (Lme0) (x)− (Lme1) (x) = xm+1(1 + x)−m,so (2.4) holds. Theorem 2.4. Let m ∈ N and 0 ≤ x < 1. If i =[mx1−x], then(2.5) (Zmϕx)(x) = 2xi+1(1− x)m+1(m+ii).4 The first absolute moment for some operators 159Proof. From i =[mx1−x]it results i ≤ mx1−x < i + 1, equivalent withim+i ≤x < i+1m+i+1 . We get(Zmϕx)(x) = (1− x)m+1i∑k=0(m+kk)xk∣∣∣x− km+k ∣∣∣= 2(1− x)m+1i∑k=0(m+kk)xk(x− km+k)+ (1− x)m+1∞∑k=0(m+kk)xk(km+k − x)= 2(1− x)m+1(xi∑k=0(m+kk)xk −i∑k=1(m+kk)xk km+k)++ (Zme1)(x)− x(Zme0)(x)= 2x(1− x)m+1(1 +i∑k=1((m+kk)xk −(m+k−1k−1)xk−1))= 2xi+1(1− x)m+1(m+ii),so (2.5) is obtained. It is known the result contained in Lemma 2.5.Lemma 2.5. If m,n ∈ N0,m 6= 0, then (Bmψnx) (x), with x ∈ [0, 1] is apolynomial in variable x.Lemma 2.6. If m,n ∈ N and n is even, then(2.6) (Bmϕnx) (x) = x(1− x)qm,n(x),for any x ∈ [0, 1], where qm,n(x) is a polynomial in variable x and qm,n(x) > 0,for any x ∈ [0, 1].Proof. For m = 1, we get(B1ϕnx) (x) =1∑k=0(k − x)np1,k(x)= xnp1,0(x) + (1− x)np1,1(x) = x(1− x)q1,n(x),160 Ovidiu T. Pop and Petru I. Braica 5where q1,n(x) = xn−1 + (1− x)n−1.For m ≥ 2, we get(Bmϕnx) (x) =m∑k=0(km − x)npm,k(x)= xn(1− x)m +m−1∑k=1(km − x)npm,k(x) + (1− x)nxm= x(1− x)qm,n,whereqm,n(x) = xn−1(1− x)m−1+m−1∑k=1(km − x)n (mk)xk−1(1− x)m−k−1 + (1− x)n−1xm−1.For any x ∈ [0, 1], every summand from qm,n(x) is positive. But for k = 1 andk = m − 1, which means(1m − x)n (m1)(1 − x)m−2, respectively(m−1m − x)n ( mm−1)xm−2 cannot be simultaneously null.Taking the above remark into account, it results(Bmϕnx) (x) = x(1− x)qm,n(x)is a polynomial in variable x and qm,n(x) > 0, for any x ∈ [0, 1]. Corollary 2.7. If m,n ∈ N, with n even, then(2.7)(Tm,nBm) (x) = mn (Bmψnx) (x) = mn (Bmϕnx) (x) = mnx(1− x)qm,n(x).For the Szász-Mirakjan operators, we get(2.8) (Tm,sSm) (x) = ms (Smψsx) (x) = mse−mx∞∑k=0(mx)kk!(km − x)s,where m ∈ N, s ∈ N0 and x ∈ [0,+∞).Lemma 2.8. If m, s ∈ N , with s ≥ 2, then the following(2.9) ((Tm,sSm)(x))′ = ms−2∑i=0(si)(Tm,iSm) (x)holds, for any x ∈ [0,+∞).6 The first absolute moment for some operators 161Proof. From (2.8), we obtain((Tm,sSm)(x))′ =(mse−mx∞∑k=0(mx)kk!(km − x)s)′= −ms+1e−mx∞∑k=0(mx)kk!(km − x)s+ms+1e−mx∞∑k=0(mx)k−1(k−1)!(km − x)s−− smse−mx∞∑k=0(mx)kk!(km − x)s−1= −m (Tm,sSm) (x)−ms (Tm,s−1Sm) (x) +ms+1e−mx∞∑k=0(mx)kk!(k+1m − x)s= −m (Tm,sSm) (x)−ms (Tm,s−1Sm) (x)+ms+1e−mx∞∑k=0(mx)kk!((km − x)+ 1m)s= −m (Tm,sSm) (x)−ms (Tm,s−1Sm) (x)+ms+1e−mx∞∑k=0(mx)kk!s∑i=0(si) (1m)s−i ( km − x)i= −m (Tm,sSm) (x)−ms (Tm,s−1Sm) (x)+s∑i=0(si)mi+1e−mx∞∑k=0(mx)kk!(km− x)i= −m (Tm,sSm) (x)−ms (Tm,s−1Sm) (x) +ms∑i=0(si)(Tm,iSm) (x)from where, the relation (2.9) follows. Lemma 2.9. If m, s ∈ N, with s ≥ 2, then(2.10) (Tm,sSm)(x) = ms−2∑i=0(si) ∫ x0(Tm,iSm)(t)dtholds, for any x ∈ [0,∞).Proof. It yields immediately, taking that (Tm,sSm)(0) = 0 into account andintegrate the relation (2.9). For any m ∈ N and any x ∈ [0,∞), it is well known that (Tm,0Sm)(x) = 1,(Tm,1Sm)(x) = 0, then the following holds:Corollary 2.10. If m, s ∈ N , with s ≥ 4, then(2.11) (Tm,sSm)(x) = mx+ms−2∑i=2(si) ∫ x0(Tm,iSm)(t)dtfor any x ∈ [0,∞).162 Ovidiu T. Pop and Petru I. Braica 7Let m ∈ N and x ∈ [0,∞). We obtain(Tm,0Sm)(x) =(Sme0)(x) = 1,(Tm,1Sm)(x) =m(Smψx)(x) = m((Sme1)(x)− x(Sme0)(x)) = 0,(Tm,2Sm)(x) =m(20) ∫ x0(Tm,0Sm)(t)dt = mx,(Tm,3Sm)(x) =m((30) ∫ x0(Tm,0Sm)(t)dt+(31) ∫ x0(Tm,1Sm)(t)dt)= mx,(Tm,4Sm)(x) =mx+m(42) ∫ x0(Tm,2Sm)(t)dt = mx+ 3m2x2.Lemma 2.11. If m, s ∈ N, with s > 3, then exist a(s)2 , a(s)3 , . . . , a(s)[s/2] ≥ 0,a(s)[s/2] 6= 0, such that(2.12) (Tm,sSm)(x) = mx+m[s/2]∑k=2a(s)k xk,for any x ∈ [0,∞), where a(s)k depends of m and k ∈{2, 3, . . . ,[s2]}.Proof. We prove by mathematical induction.For s = 4 we get (Tm,4Sm)(x) = mx+ 3m2x2, so that a(4)2 = 3m > 0.We assume that(Tm,jSm)(x) = mx+m[j/2]∑k=2a(j)k xk,for a(j)2 , aj3, . . . , a(j)[j/2] ≥ 0, a(j)[j/2] 6= 0, for any j ∈ {4, 5, . . . , s}. Taking relation(2.11) into account, we obtain(Tm,s+1Sm)(x) == mx+m((s+12) ∫ x0(Tm,2Sm)(t)dt+(s+13) ∫ x0(Tm,3Sm)(t)dt+s−1∑i=4(s+1i) ∫ x0(mt+m[i/2]∑k=2a(i)k tk)dt)= mx+m((s+12)mx22 +(s+13)mx22 +s−1∑i=4(s+1i)(mx22 +m[i/2]∑k=2a(i)kk+1xk+1))= mx+ 12(m(s+12)+m(s+13)+ . . .+m(s+1s−1))mx2+(m(s+14)a(4)23 +m(s+15)a(5)23 + . . .+m(s+1s−1)a(s−1)23)mx3+8 The first absolute moment for some operators 163+ . . .+m · a(s+1)[ s+12]x[s+12]where:• if s is even,then for k ∈{2, 3, . . . ,[s−32]}we get k + 1 <[s−22]+ 1 =[s−12]+ 1 =[s+12]anda(s+1)[s+12] = m(s+1s−2)a(s−2)[s−22][s−22]+1+m(s+1s−1) a(s−1)[ s−12 ][s−12]+1> 0;• if s is odd,then for k ∈ {2, 3 . . . ,[s−22]} we get k + 1 <[s−12]+ 1 =[s+12]anda(s+1)[s+12] = m(s+1s−1)a(s−1)[s−12][s−12]+1> 0.And now, the proof is done. REFERENCES[1] M. Beker and R.J. Nessel, A global approximation theorem for Meyer-König andZeller operators, Math. Zeitschr., 160, pp. 195–206, 1978.[2] G. Bleimann, P.L. Butzer and L.A. Hahn, Bernstein-type operator approximatingcontinous function on the semi-axis, Indag. Math., 42, pp. 255–262, 1980.[3] S.N. Bernstein, Démonstration du théorème de Weierstrass fondée sur le calcul deprobabilité, Commun. Soc. Math. Kharkow (2), 13, pp. 1–2, 1912–1913.[4] J. Favard, Sur les multiplicateurs d’interpolation, J. Math. Pures Appl., (9), 23,pp. 219–247, 1994.[5] J. Meier, Zur Approximation durch gewöhnliche und modifizierte Bernstein-Operatorenunter besonderer Berücksichtigung quantitaver Aussagen und asymtotischer Formeln,Staatsexamensarbeit, Universität Duisburg, 1982.[6] W. Meyer-König and K. Zeller, Bernsteinsche Potenzreihen, Studia Math., 19,pp. 89–94, 1960.[7] G.M. Mirakjan, Approximation of continuous functions with the aid of polynomials,Dokl. Acad. Nauk SSSR, 31, pp. 201–205, 1941 (in Russian).[8] M.W. Müller, Die Folge der Gammaoperatoren, Dissertation, Stuttgart, 1967.[9] O.T. Pop, About the first order modulus of smoothness, Carpathian J. Math., 20,pp. 101–108, 2004.[10] F. Schurer and F.W. Steutel, On the degree of approximation of functions inC1([0, 1[) by Bernstein polynomials, T. H. Report 75-WSK-07, Eindhoven Universityof Technology, 1975.[11] D.D. Stancu, Gh. Coman, O. Agratini and R. Tr̂ımbiţaş, Numerical analysis andapproximation theory, I, Cluj University Press, Cluj-Napoca, 2001, (in Romanian).[12] O. Szász, Generalization of S. N. Bernstein’s polynomials to the infinite interval, J.Research, National Bureau of Standards, 45, pp. 239–245, 1950.Received by the editors: December 15, 2009.

The first absolute moment for some operators

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The first absolute moment for some operators

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