In this note we consider a linear and positive compound approximation operator of D.D. Stancu type depending of several parameters; we give the expressions of this operator on the test functions, the conditions under which this operator converges to a given continuous function, an estimate of the order of approximation using the moduli of continuity and an integral representation of the remainder. Also, by using Stancu's method we find an expression for the remainder using divided differences of second order for a special case of this operator

Maria Crăciun

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REVUE D’ANALYSE NUMÉRIQUE ET DE THÉORIE DE L’APPROXIMATIONRev. Anal. Numér. Théor. Approx., vol. 35 (2006) no. 1, pp. 33–40ictp.acad.ro/jnaatON A COMPOUND APPROXIMATION OPERATOROF D.D. STANCU TYPE∗MARIA CRĂCIUN†Abstract. In this note we consider a linear and positive compound approximationoperator of D.D. Stancu type depending of several parameters; we give the expressionsof this operator on the test functions, the conditions under which this operator convergesto a given continuous function, an estimate of the order of approximation using themoduli of continuity and an integral representation of the remainder. Also, by usingStancu’s method we find an expression for the remainder using divided differences ofsecond order for a special case of this operator.MSC 2000. 41A36, 41A80.Keywords. Compound linear and positive approximation operators, representation ofremainder.1. INTRODUCTIONIn [18], [19], [24], [25], [23], [27] D.D. Stancu introduced and studied various classesof compound approximation operators:(1) (Sm,r,sf) (x) =m−sr∑k=0pm−sr,k (x)s∑j=0ps,j (x) f(k+jrm),where f ∈ C [0, 1] , r and s are nonnegative integer parameters satisfying the condition2sr < m, while pn,k (x) =(nk)pk(x)pn−k(1−x)pn(1) and pk (x) = xk in [19], [18], [25], pk (x) =x[k,−α] = x (x+ α) ... (x+ (k − 1)α) in [24], pk (x) = x (x+ α+ kβ)[k−1,−α] in [23].Cao [2] considered compound operators of Stancu type with pk (x) = xk definedon a simplex.Following the interesting ideas of D.D. Stancu from the previous mentioned papersin [5] we considered a general class of compound operators of D.D. Stancu type:(2)(LQm,r1,...,rsf)(x) =m−r1...−rs∑k=0pQm−r1...−rs,k (x)s∑j=0pj(x)ps−j(1−x)ps(1) Fr1,...,rsm,k,j (f)∗This work has been supported by the MEdC-ANCS under grant no. 3233/17.10.2005.†“T. Popoviciu” Institute of Numerical Analysis, P.O. Box 68-1, Cluj-Napoca, Romania, e-mail:craciun@ictp.acad.ro.34 Maria Crăciun 2where pQn,k (x) =(nk)pk(x)pn−k(1−x)pn(1) , (pk)k≥0 is a sequence of binomial type which isthe basic sequence for the delta operator Q,F r1,...,rsm,k,j (f) =f(k+r1+r2+...+rjm)+ f(k+r2+r3+...+rj+1m)+ f(k+r1+r3+...+rj+1m)+ ...+ f(k+rs−j+1+...+rs−1+rsm)and r1, ..., rs are s non-negative integer parameters, independent of the number mand such that 0 ≤ r1 ≤ ... ≤ rs and r1 + ... + rs < m. We mention that Q is a deltaoperator if it is a shift invariant operator (QEa = EaQ, for any a, where Ea is theshift operator) and Qx = const 6= 0, and (pk)k≥0 is the basic sequence for the deltaoperator Q if it satisfies: p0 (x) = 1, pn (0) = 0, n ≥ 1, Qpn = npn−1.We have proved the following result:Lemma 1. [5] If LQm,r1,...,rs is the approximation operator defined by (2) thenLQm,r1,...,rsei = ei, for i = 0, 1 and(LQm,r1,...,rse2)(x) = x2 + x (1− x)AQm,r1,...,rs ,whereAQm,r1,...,rs =1m2[(m− r1 − ...− rs)2dQm−r1...−rs+r21+...+r2s+ 2s−1(sdQs − 1) s∑u,v=1u6=vrurv],and dQm = 1− m−1m(Q′)−2pm−2(1)pm(1) , Q′ is the Pincherle derivative of Q, Q′ = QX−XQ.2. DEFINITION AND CONVERGENCE OF THE OPERATOR LOααm,r1,...,rsfOne remarkable operator from the class mentioned above is the operator obtainedby taking the Stancu polynomials pαn,k (x) =(nk)x[k,−α](1−x)[n−k,−α]1[n,−α] (x[k,−α] is the basicsequence for the delta operator Oαα ) in relation (2):(LOααm,r1,...,rsf)(x) =(3)= 11[m−r1−...−rs,−α]m−r1...−rs∑k=0(m−r1−...−rsk)x[k,−α] (1− x)[m−r1−...−rs−k,−α]·s∑j=0x[j,−α] (1− x)[s−j,−α]1[s,−α]F r1,...,rsm,k,j (f) .It is easy to see that this operator is linear and if α > 0 it is positive.It interpolates the function f at both ends of the interval [0, 1].In order to prove the convergence of this operator we need the values of this oper-ator on the test functions ei (x) = xi, i = 0, 1, 2. According to the results from [5] wecan stateTheorem 2. The values of the operator LOααm,r1,...,rsf on the test functions are:(LOααm,r1,...,rsei)(x) = ei (x) , for i = 0, 1 and(LOααm,r1,...,rse2)(x) = x2 + x (1− x)AOααm,r1,...,rs ,3 On a compound approximation operator of D.D. Stancu type 35where(4)AOααm,r1,...,rs = 1m2[(m− r1 − ...− rs) 1+α(m−r1−...−rs)1+α + r21 + ...+ r2s + 2α1+αs∑u,v=1u6=vrurv].If r1 = ... = rs = r in the relation (3) then this operator reduces to the operatorstudied by D.D. Stancu and J.W. Drane in [24] and the expression (4) in this case isAOααm,r,s = sr2(1+αs)+(m−sr)(1+α(m−sr))m2(1+α) .In the case s = 0 we obtain the well-known classical Stancu operator introduced in[14] and studied by many mathematicians.When α = 0 the corresponding operator was introduced by D.D. Stancu in [18];the same author found for the remainder of the corresponding approximation formulaa convex combination of second order divided differences in [19].By applying the Bohman-Korovkin convergence criterion we can state the followingresult:Theorem 3. If f ∈ C[0, 1] and the parameter α depend on m such that α =α (m) → 0 as m → ∞, then the sequence(LOααm,r1,...,rsf)converges uniformly to f on[0, 1].In the following we obtain some estimates of the order of approximation of a func-tion f ∈ C[0, 1] by means of the operator LOααm,r1,...,rsf.Using an inequality established by O. Shisha and B. Mond in [13] we can write:∣∣∣f (x)− (LOααm,r1,...,rsf) (x)∣∣∣ ≤ [1 + 1δ2LOααm,r1,...,rs ((t− x)2 ;x)]ω1 (f ; δ) .Using theorem 2 we obtain that LOααm,r1,...,rs((t− x)2 ;x)= x (1− x)AOααm,r1,...,rs . Tak-ing into account that x (1− x) ≤ 14 , ∀x ∈ [0, 1] and replacing δ with√AOααm,r1,...,rs , weobtain that ∥∥∥f − LOααm,r1,...,rsf∥∥∥ ≤ 54ω1(f ;√AOααm,r1,...,rs) .Using a result of H.H. Gonska and R.K. Kovacheva [6] we can give the followingevaluation with the second order modulus of continuity, for every x ∈ [0, 1] andδ ∈ [0, 1/2]:∣∣∣f (x)− (LOααm,r1,...,rsf) (x)∣∣∣ ≤ 32 [1 + x(1−x)2δ2 AOααm,r1,...,rs]ω2 (f ; δ) .This relation implies:∥∥∥f − LOααm,r1,...,rsf∥∥∥ ≤ 2716ω2(f ;√AOααm,r1,...,rs) .3. REPRESENTATIONS FOR THE REMAINDERWe consider the following approximation formula(5) f (x) =(L∇ααm,r1,...,rsf)(x) +(R∇ααm,r1,...,rsf)(x) .36 Maria Crăciun 4From Theorem 2 it results that the degree of exactness of this formula is 1, so forevery function f ∈ C2 [0, 1] we can apply the Peano’s theorem and we obtain thefollowing representation:(R∇ααm,r1,...,rsf)(x) =∫ 10Gαm,r1,...,rs(t;x)f′′ (t) dt,where Gαm,r1,...,rs(t;x) is the Peano kernel defined by:Gαm,r1,...,rs(t;x) =(R∇ααm,r1,...,rsϕx)(t) , ϕx (t) = (x− t)+ =12 [x− t+ |x− t|] .Since the expression Gαm,r1,...,rs(t;x) is negative, we can apply the mean value theoremto the integral and we obtain that there exists ξ ∈ [0, 1] such that(R∇ααm,r1,...,rsf)(x) = f ′′ (ξ)∫ 10Gαm,r1,...,rs (t;x) dt.Taking f (x) = x2 in the previous relation we obtain that∫ 10Gαm,r1,...,rs (t;x) dt =12(R∇ααm,r1,...,rse2)(x) = x(x−1)2 AOααm,r1,...,rs .So, for every function f ∈ C2 [0, 1] , we obtain a Cauchy-type form for the remainderin the approximation formula (5)(R∇ααm,r1,...,rsf)(x) = x(x−1)2 A∇ααm,r1,...,rsf′′ (ξ) ,where ξ ∈ [0, 1] and A∇ααm,r1,...,rs is defined by (4).We mention that D.D. Stancu found also representations of the remainder by con-vex combinations of second-order divided differences in the case r1 = r2 = ... = rs.For α = 0 he obtained the following representation for the remainder:(Rm,r,sf) (x) = x(x−1)m2 (Cm,r,sf) (x) ,where(Cm,r,sf) (x) = (m− sr)m−sr−1∑k=0pm−sr−1,k (x)s∑j=0ps,j (x)[x, k+jrm ,k+1+jrm ; f]+ sr2m−sr∑k=0pm−sr−1,k (x)s−1∑j=0ps−1,j (x)[x, k+jrm ,k+r+jrm ; f].Using Stancu’s method in the following we intend to find an expression for the re-mainder using divided differences in the case α = 0 and s = 2, namely for the operator(Lm,r1,r2f) (x) =m−r1−r2∑k=0pm−r1−r2,k (x) ·{(1− x)2 f(km)+ x (1− x)[f(k+r1m)+ f(k+r2m)]+ x2f(k+r1+r2m)},where pm,k (x) =(mk)xk (1− x)k .5 On a compound approximation operator of D.D. Stancu type 37Because (Lm,r1,r2e0) (x) = 1 we have(Rm,r1,r2f) (x) == f (x)− (Lm,r1,r2f) (x)=m−r1−r2∑k=0pm−r1−r2,k (x) ·{(1− x)2(f (x)− f(km))+ x (1− x) ··[(f (x)− f(k+r1m))+(f (x)− f(k+r2m))]+ x2(f (x)− f(k+r1+r2m))}.In case x coincides with one of the nodes, the definition of the divided differencesrequires the differentiability of f at x; this will be implicitly assumed further on.If we use the following relationsf (x)− f(km)=(x− km) [x, km ; f],f (x)− f(k+rim)=(x− k+r1m) [x, k+rim ; f], i = 1, 2,f (x)− f(k+r1+r2m)=(x− k+r1+r2m) [x, k+r1+r2m ; f],mx− k = (m− r1 − r2 − k)x− k (1− x) + (r1 + r2)x,mx− k − r1 = (m− r1 − r2 − k)x− k (1− x) + r2x− r1 (1− x) ,mx− k − r2 = (m− r1 − r2 − k)x− k (1− x) + r1x− r2 (1− x) ,mx− k − r1 − r2 = (m− r1 − r2 − k)x− k (1− x) + (r1 + r2) (1− x) ,we can write(Rm,r1,r2f) (x) = (Pm,r1,r2f) (x) + (Qm,r1,r2f) (x) ,where(Pm,r1,r2f) (x) = 1mm−r1−r2∑k=0pm−r1−r2,k (x) · [(m− r1 − r2 − k)x− k (1− x)]·{(1− x)2[x, km ; f]+x (1− x)2∑i=1[x, k+rim ; f]+x2[x, k+r1+r2m ; f]}and(Qm,r1,r2f) (x) =x(1−x)mm−r1−r2∑k=0pm−r1−r2,k (x) ·{(1− x) (r1 + r2)[x, km ; f]+((r2x− r1 (1− x))[x, k+r1m ; f]+ (r1x− r2 (1− x))[x, k+r2m ; f] )+ x (r1 + r2)[x, k+r1+r2m ; f] }.Using the combinatorial identities(m− r1 − r2 − k)(m−r1−r2k)= (m− r1 − r2)(m−r1−r2−1k)and38 Maria Crăciun 6k(m−r1−r2k)= (m− r1 − r2)(m−r1−r2−1k−1)we can write(Pm,r1,r2f) (x) =(m−r1−r2)mm−r1−r2−1∑k=0(m−r1−r2−1k)xk+1 (1− x)m−r1−r2−k·{(1− x)2[x, km ; f]+x (1− x)2∑i=1[x, k+rim ; f]+x2[x, k+r1+r2m ; f]}−m−r1−r2∑k=1(m−r1−r2−1k−1)xk (1− x)m−r1−r2−k+1·{(1− x)2[x, km ; f]+x (1− x)2∑i=1[x, k+rim ; f]+x2[x, k+r1+r2m ; f]}.By changing the summation index in the second sum and using the recurrence relationof divided differences[x, a; f ]− [x, b; f ] = (a− b) [x, a, b; f ]we obtain(Pm,r1,r2f) (x) =x(x−1)(m−r1−r2)m2m−r1−r2−1∑k=0pm−r1−r2−1,k (x)·{(1− x)2[x, km ,k+1m ; f]+x (1− x)2∑i=1[x, k+rim ,k+ri+1m ; f]+x2[x, k+r1+r2m ,k+r1+r2+1m ; f]}.Combining the terms that contains (1− x) ri and xri, respectively, in Qm,r1,r2 , wehave:(Qm,r1,r2f) (x) == x(1−x)mm−r1−r2∑k=0pm−r1−r2,k (x) ·{(1− x)2∑i=1ri([x, km ; f]−[x, k+rim ; f])+ x2∑i=1ri([x, k+rim ; f]−[x, k+r1+r2m ; f]) }.If we use again the recurrence relation for divided differences we can write(Qm,r1,r2f) (x) =x(x−1)m2m−r1−r2∑k=0pm−r1−r2,k (x) ··{(1− x)2∑i=1r2i[x, km ,k+rim ; f]+x2∑i=1r2i[x, k+rim ,k+r1+r2m ; f]}.7 On a compound approximation operator of D.D. Stancu type 39So, using divided differences we obtained the following form for the remainder(Rm,r1,r2f) (x) =(6)= x(x−1)m2{(m− r1 − r2)m−r1−r2−1∑k=0pm−r1−r2−1,k (x) ·((1− x)2[x, km ,k+1m ; f]+x (1− x)2∑i=1[x, k+rim ,k+ri+1m ; f]+x2[x, k+r1+r2m ,k+r1+r2+1m ; f])+m−r1−r2∑k=0pm−r1−r2,k (x)2∑i=1r2i((1− x)[x, km ,k+rim ; f]+ x[x, k+rim ,k+r1+r2m ; f])}.Taking into account that (Rm,r1,r2e2) (x) =x(x−1)m2(m− r1 − r2 + r21 + r22)oneobserves that the previous expression can be represented in the following form:(Rm,r1,r2f) (x) = (Rm,r1,r2e2) (x) (Dm,r1,r2f) (x) ,where(Dm,r1,r2f) (x) == 1m−r1−r2+r21+r22{(m− r1 − r2)m−r1−r2−1∑k=0pm−r1−r2−1,k (x) ·((1− x)2[x, km ,k+1m ; f]+ x (1− x)2∑i=1[x, k+rim ,k+ri+1m ; f]+ x2[x, k+r1+r2m ,k+r1+r2+1m ; f] )+m−r1−r2∑k=0pm−r1−r2,k (x)2∑i=1r2i((1− x)[x, km ,k+rim ; f]+x[x, k+rim ,k+r1+r2m ; f] )}.It is easy to see that all the coefficients of this linear functional are positive andthat their sum is (Dm,r1,r2e2) (x) = 1 for any x in [0, 1] , so it is a convex combinationof the second-order divided differences.From the above representation it results that if f is a nonlinear convex (resp.concave) function on [0, 1] then we have Lm,r1,r2f > f (resp. Lm,r1,r2f < f).Because the degree of the corresponding approximation formula is one andRm,r1,r2f 6= 0 whenever f is a convex function, by using a theorem of T. Popovi-ciu [10] we can state that there exist three distinct points ξ1, ξ2, ξ3 on [0, 1] suchthat(Rm,r1,r2f) (x) = (Rm,r1,r2e2) (x) [ξ1, ξ2, ξ3; f ] .REFERENCES[1] Altomare, F. and Campiti, M., Korovkin-type approximation theory and its applications.Appendix A by Michael Pannenberg and Appendix B by Ferdinand Beckhoff. de Gruyter Studiesin Mathematics, 17. Walter de Gruyter & Co., Berlin, 1994.[2] Cao, F. Modulus of continuity, K-functional and Stancu operator on a simplex, Indian J. PureAppl. Math., 35, no. 12, 1343–1364, 2004.[3] Crăciun, M., Approximation operators constructed by means of Sheffer sequences, Rev. Anal.Numér. Théor. Approx., 30, no. 2, pp. 135–150, 2001.40 Maria Crăciun 8[4] Crăciun, M., On compound operators constructed with binomial and Sheffer sequences, Rev.Anal. Numér. Théor. Approx., 32, no. 2, pp. 135–144, 2003.[5] Crăciun, M., On compound operators depending on s parameters, Rev. Anal. Numér. Théor.Approx., 33, no. 1, pp. 51–60, 2004.[6] Gonska, H.H. and Kovacheva, R.K., The second order modulus revisited: remarks, applica-tions, problems, Conf. Semin. Mat. Univ. Bari, 257, pp. 1–32, 1994.[7] Lupaş, A., Approximation operators of binomial type, Proc. IDoMAT 98, International Seriesof Numerical Mathematics, ISNM vol. 132, Birkhäuser Verlag, Basel, pp. 175–198, 1999.[8] Manole, C., Approximation operators of binomial type, Univ. of Cluj–Napoca, Research Sem-inars, Seminar on numerical and statistical calculus, Preprint nr. 9, 1987, 93–98.[9] Popoviciu, T., Remarques sur les polynômes binomiaux, Bul. Soc. Ştiinte Cluj, 6, 146–148,1931.[10] Popoviciu, T., Sur le reste dans certaines formules lineaires d’approximation de l’analyse,Mathematica, Cluj, 1(24), 95–142, 1959.[11] Rota, G.C., Kahaner, D. and Odlyzko, A., Finite Operator Calculus, J. Math. 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C., Compound poweroid operators of approximation, Ren-diconti del Circolo Matematico di Palermo, Suppl. 68, pp. 845–854, 2002.Received by the editors: February 15, 2006.

On a compound approximation operator of D.D. Stancu type

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On a compound approximation operator of D.D. Stancu type

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