ON A COMPOUND APPROXIMATION OPERATOR OF D.D. STANCU TYPE

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1 REVUE D ANALYSE NUMÉRIQUE ET DE THÉORIE DE L APPROXIMATION Toe 35, N o 1, 2006, pp ON A COMPOUND APPROXIMATION OPERATOR OF DD STANCU TYPE MARIA CRĂCIUN Abstract In this note we consider a linear and positive copound approxiation operator o DD Stancu type depending o several paraeters; we give the expressions o this operator on the test unctions, the conditions under which this operator converges to a given continuous unction, an estiate o the order o approxiation using the oduli o continuity and an integral representation o the reainder Also, by using Stancu s ethod we ind an expression or the reainder using divided dierences o second order or a special case o this operator MSC A36, 41A80 Keywords Copound linear and positive approxiation operators, representation o reainder 1 INTRODUCTION In 18], 19], 24], 25], 23], 27] DD Stancu introduced and studied various classes o copound approxiation operators: 1 S,r,s x = sr p sr, x s p s,j x +jr, C 0,1], r and s are nonnegative integer paraeters satisying the condition 2sr <, while p n, x = n p xp n 1 x p n1 and p x = x in 19], 18], 25], p x = x, ] = xx + x + 1 in 24], p x = xx + + β 1, ] in 23] Cao 2] considered copound operators o Stancu type with p x = x deined on a siplex Following the interesting ideas o DD Stancu ro the previous entioned papers in 5] we considered a general class o copound operators o DD Stancu type: 2 L Q,r1,,r s r 1 r s x = p Q r 1 r s, x s p jxp s j1 x p s1 F r1,,rs,,j This wor has been supported by the MEdC-ANCS under grant no 3233/ T Popoviciu Institute o Nuerical Analysis, PO Box 68-1, Cluj-Napoca, Roania, e-ail: craciun@ictpacadro

2 34 Maria Crăciun 2 p xp n 1 x p Q n, x = n p n1, p 0 is a sequence o binoial type which is the basic sequence or the delta operator Q, F r1,,rs,,j = + + +r1+r 2++r j +r1+r 3++r j+1 +r2+r 3++r j+1 +rs j+1++r s 1+r s + + and r 1,,r s are s non-negative integer paraeters, independent o the nuber and such that 0 r 1 r s and r r s < We ention that Q is a delta operator i it is a shit invariant operator QE a = E a Q, or any a, E a is the shit operator and Qx = const 0, and p 0 is the basic sequence or the delta operator Q i it satisies: p 0 x = 1, p n 0 = 0, n 1, Qp n = np n 1 We have proved the ollowing result: Lea 1 5] I L Q,r 1,,r s is the approxiation operator deined by 2 then L Q,r 1,,r s e i = e i, or i = 0,1 and L Q,r1,,r s e 2 x = x 2 + x1 x A Q,r 1,,r s, A Q,r 1,,r s = 1 r 2 1 r s 2 d Q r 1 r s +r 1++r 2 s+ 2 2 s 1 sd Q s 1 s r u r v ], u,v=1 u v and d Q = 1 1 Q 2 p 21 p 1, Q is the Pincherle derivative o Q, Q = QX XQ 2 DEFINITION AND CONVERGENCE OF THE OPERATOR L,r1,,r s One rearable operator ro the class entioned above is the operator obtained by taing the Stancu polynoials p n, x = n x, ] 1 x n, ] x, ] is the basic 1 n, ] sequence or the delta operator in relation 2: L 3,r1,,r s x = = 1 1 r1 rs, ] r 1 r s s x j, ] 1 x s j, ] F r1,,rs 1 s, ],,j r1 r s x, ] 1 x r1 rs, ] It is easy to see that this operator is linear and i > 0 it is positive It interpolates the unction at both ends o the interval 0,1] In order to prove the convergence o this operator we need the values o this operator on the test unctions e i x = x i, i = 0,1,2 According to the results ro 5] we can state Theore 2 The values o the operator L,r1,,r s on the test unctions are: L,r1,,r s e i x = e i or i = 0,1 and L,r1,,r s e 2 x = x 2 + x1 x A,r1,,r s,

3 3 On a copound approxiation operator o DD Stancu type 35 4 A,r1,,r s = 1 r 2 1 r s 1+ r1 rs 1+ + r rs s u,v=1 u v r u r v ] I r 1 = = r s = r in the relation 3 then this operator reduces to the operator studied by DD Stancu and JW Drane in 24] and the expression 4 in this case is A,r,s = sr2 1+s+ sr1+ sr 2 1+ In the case s = 0 we obtain the well-nown classical Stancu operator introduced in 14] and studied by any atheaticians When = 0 the corresponding operator was introduced by DD Stancu in 18]; the sae author ound or the reainder o the corresponding approxiation orula a convex cobination o second order divided dierences in 19] By applying the Bohan-Korovin convergence criterion we can state the ollowing result: Theore 3 I C0,1] and the paraeter depend on such that = 0 as, then the sequence L,r1,,r s converges uniorly to on 0,1] In the ollowing we obtain soe estiates o the order o approxiation o a unction C0,1] by eans o the operator L,r1,,r s Using an inequality established by O Shisha and B Mond in 13] we can write: x L,r1,,r s x δ 2 L Using theore 2 we obtain that L,r1,,r s t x 2 ;x ],r1,,r s t x 2 ;x ω 1 ;δ = x1 x A,r1,,r s Taing into account that x1 x 1 4 A, x 0,1] and replacing δ with,r1,,r s, we obtain that L,r1,,r s 5 4 ω 1 ; A,r1,,r s Using a result o HH Gonsa and RK Kovacheva 6] we can give the ollowing evaluation with the second order odulus o continuity, or every x 0, 1] and δ 0,1/2]: x L,r1,,r s This relation iplies: L,r1,,r s x x1 x 2δ 2 16 ω 2 A,r1,,r s ] ω 2 ;δ ; A,r1,,r s 3 REPRESENTATIONS FOR THE REMAINDER We consider the ollowing approxiation orula 5 x = L,r 1,,r s x + R,r 1,,r s x

4 36 Maria Crăciun 4 Fro Theore 2 it results that the degree o exactness o this orula is 1, so or every unction C 2 0,1] we can apply the Peano s theore and we obtain the ollowing representation: R,r 1,,r s x = 1 0 G,r 1,,r s t;x t dt, G,r 1,,r s t;x is the Peano ernel deined by: G,r 1,,r s t;x = R,r 1,,r s ϕ x t, ϕ x t = x t + = 1 2 x t + x t ] Since the expression G,r 1,,r s t;x is negative, we can apply the ean value theore to the integral and we obtain that there exists ξ 0,1] such that 1 R,r 1,,r s x = ξ G,r 1,,r s t;x dt Taing x = x 2 in the previous relation we obtain that 1 G,r 1,,r s t;x dt = 1 2 R,r 1,,r s e 2 x = xx 1 2 A,r1,,r s 0 So, or every unction C 2 0,1], we obtain a Cauchy-type or or the reainder in the approxiation orula 5 R,r 1,,r s x = xx 1,r 1,,r s ξ, 0 2 A ξ 0,1] and A,r 1,,r s is deined by 4 We ention that DD Stancu ound also representations o the reainder by convex cobinations o second-order divided dierences in the case r 1 = r 2 = = r s For = 0 he obtained the ollowing representation or the reainder: R,r,s x = xx 1 2 C,r,s x, sr 1 C,r,s x = sr p sr 1, x sr + sr 2 s 1 p sr 1, x s p s 1,j x ] p s,j x +jr, +1+jr ; ] +jr, +r+jr ; Using Stancu s ethod in the ollowing we intend to ind an expression or the reainder using divided dierences in the case = 0 and s = 2, naely or the operator L,r1,r 2 x = r 1 r 2 p, x = x 1 x p r1 r 2, x + x1 x +r 1 1 x 2 + +r2 ] + x 2 } +r 1+r 2,

5 5 On a copound approxiation operator o DD Stancu type 37 Because L,r1,r 2 e 0 x = 1 we have R,r1,r 2 x = = x L,r1,r 2 x = r 1 r 2 p r1 r 2, x x +r 1 1 x 2 x + x1 x + x +r2 ] + x 2 x } +r 1+r 2 In case x coincides with one o the nodes, the deinition o the divided dierences requires the dierentiability o at x; this will be iplicitly assued urther on I we use the ollowing relations x = x ;], x +r i = x +r 1 +r i ;], i = 1,2, x +r 1+r 2 = x +r 1+r 2 +r 1+r 2 ;], x = r 1 r 2 x 1 x + r 1 + r 2 we can write x r 1 = r 1 r 2 x 1 x + r 2 x r 1 1 x r 2 = r 1 r 2 x 1 x + r 1 x r 2 1 x r 1 r 2 = r 1 r 2 x 1 x + r 1 + r 2 1 x, P,r1,r 2 x = 1 and Q,r1,r 2 x = x1 x R,r1,r 2 x = P,r1,r 2 x + Q,r1,r 2 r 1 r 2 p r1 r 2, x r 1 r 2 x 1 x] 1 x 2 ;] +x1 x r 1 r 2 2 +r i ;] +x 2 +r1+r2 ;]} p r1 r 2, x 1 xr 1 + r 2 ;] + r 2 x r 1 1 x +r1 ;] + r 1 x r 2 1 x +r2 ;] + xr 1 + r 2 +r1+r2 ;] } Using the cobinatorial identities r 1 r 2 r 1 r 2 = r1 r 2 r 1 r 2 1 and

6 38 Maria Crăciun 6 r 1 r 2 = r1 r 2 r 1 r 2 1 we can write P,r1,r 2 x = r1 r2 1 r 1 r x 2 ;] +x1 x r 1 r 2 =1 r1 r 2 1 x +1 1 x r1 r2 2 +r i ;] +x 2 +r1+r2 ;]} r1 r x 1 x r1 r x 2 ;] +x1 x 2 +r i ;] +x 2 +r1+r2 ;]} By changing the suation index in the second su and using the recurrence relation o divided dierences we obtain a;] b;] = a b a,b;] r 1 r 2 1 P,r1,r 2 x = xx 1 r1 r2 2 +x1 x 2 +r i, +ri+1 p r1 r 2 1, x 1 x 2, +1 ;] ;] +x 2 +r1+r2, +r1+r2+1 ; ]} Cobining the ters that contains 1 xr i and xr i, respectively, in Q,r1,r 2, we have: Q,r1,r 2 x = = x1 x + x 2 r 1 r 2 p r1 r 2, x 1 x 2 r i +r i ;] +r1+r2 ;]} r i ;] +ri ;] I we use again the recurrence relation or divided dierences we can write Q,r1,r 2 x = xx x r 1 r 2 2 r 2 i p r1 r 2, x, +ri ;] +x 2 r 2 i +r i, +r1+r2 ;] }

7 7 On a copound approxiation operator o DD Stancu type 39 So, using divided dierences we obtained the ollowing or or the reainder 6 R,r1,r 2 x = = xx 1 2 +x1 x r 1 r 2 + r 1 r 2 2 +r i r 1 r 2 1, +ri+1 p r1 r 2, x 2 r 2 i p r1 r 2 1, x 1 x 2, +1 ;] ;] +x 2 +r1+r2, +r1+r2+1 ; ] 1 x, +ri ;] + x +ri, +r1+r2 ;] Taing into account that R,r1,r 2 e 2 x = xx 1 2 r1 r 2 + r r 2 2 one observes that the previous expression can be represented in the ollowing or: D,r1,r 2 x = = 1 r 1 r 2+r 2 1 +r2 2 + x1 x r 1 r 2 + R,r1,r 2 x = R,r1,r 2 e 2 xd,r1,r 2 r 1 r 2 2 +r i p r1 r 2, x, +ri+1 2 r 2 i r 1 r 2 1 p r1 r 2 1, x ;] + x 2 +r1+r2, +r1+r2+1 ; ] } 1 x 2, +1 ;] 1 x, +ri ;] +x } +ri, +r1+r2 ;] It is easy to see that all the coeicients o this linear unctional are positive and that their su is D,r1,r 2 e 2 x = 1 or any x in 0,1], so it is a convex cobination o the second-order divided dierences Fro the above representation it results that i is a nonlinear convex resp concave unction on 0,1] then we have L,r1,r 2 > resp L,r1,r 2 < Because the degree o the corresponding approxiation orula is one and R,r1,r 2 0 whenever is a convex unction, by using a theore o T Popoviciu 10] we can state that there exist three distinct points ξ 1, ξ 2, ξ 3 on 0,1] such that R,r1,r 2 x = R,r1,r 2 e 2 x ξ 1,ξ 2,ξ 3 ;] REFERENCES 1] Altoare, F and Capiti, M, Korovin-type approxiation theory and its applications Appendix A by Michael Pannenberg and Appendix B by Ferdinand Becho de Gruyter Studies in Matheatics, 17 Walter de Gruyter & Co, Berlin, ] Cao, F Modulus o continuity, K-unctional and Stancu operator on a siple Indian J Pure Appl Math, 35, no 12, , ] Crăciun, M, Approxiation operators constructed by eans o Sheer sequences, Rev Anal Nuér Théor Appro 30, no 2, pp , 2001

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