THE COMBINED SHEPARD ABEL GONCHAROV UNIVARIATE OPERATOR

REVUE D ANALYSE NUMÉRIQUE ET DE THÉORIE DE L APPROXIMATION Tome 32, N o 1, 2003, pp 11 20 THE COMBINED SHEPARD ABEL GONCHAROV UNIVARIATE OPERATOR TEODORA CĂTINAŞ Abstrct We extend the Sheprd opertor by combnng t wth the Abel- Gonchrov unvrte opertor n order to ncrese the degree of exctness nd to use some specfc functonls We study ths combned opertor nd gve some of ts propertes We ntroduce the correspondng nterpolton formul nd study ts remnder term MSC 2000 65D05 Keywords Sheprd-Abel-Gonchrov opertors, nterpolton formul, degree of exctness, remnder term 1 INTRODUCTION 11 The Sheprd unvrte opertor Recll frst some results regrdng the multvrte Sheprd opertor for the unvrte cse Let f be rel vlued functon defned on X R nd x X, = 0,, N, be some dstnct ponts The unvrte Sheprd opertor s defned by (1) (Sf) (x) = A (x) f (x ), where (2) A (x) = wth µ R (see, eg, [12) brycentrc form N j=0, j N j=0, j k x x j µ x x j µ, The bss functons A my be wrtten n A (x) = x x µ n x x k µ Ths work ws supported by CNCSIS Grnt no 8/139, 2003 Bbeş-Boly Unversty, Fculty of Mthemtcs nd Computer Scence, str M Kogălncenu 1, 3400 Cluj-Npoc, Romn, e-ml: tgule@mthubbclujro

12 Teodor Cătnş 2 It s esy to check tht nd (3) A (x v ) = δ v,, v = 0,, N, A (x) = 1 The mn propertes of the opertor S re: The nterpolton property (Sf) (x ) = f (x ), = 0,, N; The degree of exctness s gex (S) = 0 The gol when extendng the opertor S by combnng wth other opertors s to ncrese the degree of exctness nd to use other sets of functonls Let Λ := {λ = 0,, N} be set of functonls nd let P be the correspondng nterpolton opertor We consder tht Λ Λ re the subsets ssocted to the functonls λ, = 0,, N We hve N Λ = Λ nd Λ Λj, exceptng the cse Λ = {λ }, = 0,, N, when Λ Λj =, for j We ssocte the nterpolton opertor P to ech subset Λ, = 0,, N The opertor S P defned by (4) (S P f) (x) = A (x) (P f) (x) s the combned opertor of S nd P (see, eg, [12) Remrk 1 As noted n [12, f P, = 0,, N, re lner opertors, then S P s lner opertor Remrk 2 [12 Let P, = 0,, N, be some rbtrry lner opertors If gex (P ) = r, = 0,, N, then gex (S P ) = r m := mn {r 0,, r N } Assume tht Λ s set of Brkhoff type functonl, e, Λ B = { λ kj λ kj f = f (j) (x k ), j I k, k = 1,, N }, where I k {0, 1,, r k }, for r k N Denote r M = mx {r 1,, r N } Remrk 3 [12 If µ > r M then λ kj (S P f) = λ kj (f), j I k, k = 0,, N, where P s the nterpolton opertor correspondng to the set Λ B

3 The combned Sheprd Abel Gonchrov unvrte opertor 13 (5) In the proof of ths result the followng reltons re used: A (v) (x k ) = 0, v I k, k = 0,, N, k, A (v) (x ) = 0, v I, v 1, A (j) (x ) = 1 12 The Abel Gonchrov unvrte opertor Let n N,, b R, < b, nd f : [, b R be functon hvng the frst n dervtves f (), = 1, 2,, n Gven the nodes x [, b, 0 n, nd the vlues f () (x ), 0 n, we consder the Abel Gonchrov nterpolton problem of fndng polynoml P n f of degree n such tht (see, eg, [8 nd [10) (6) (P n f) () (x ) = f () (x ), 0 n The determnnt of ths lner system 1 x 0 x 2 0 x n 0 0 1! 2x 1 nx n 1 1 (7) D = 0 0 2! n(n 1)x n 2 2 = 1 1! 2!, 0 0 0 s lwys nonzero nd the problem (6) hs therefore unque soluton The Abel Gonchrov nterpolton polynoml P n f cn be wrtten n the form n (P n f)(x) = g k (x)f (k) (x k ), where g k, k = 0,, n re clled Gonchrov polynomls of degree k [9, determned by the condtons { g (s) (8) k (x s) = 0, f k s, (x) = 1 g (k) k Accordng to [8, [9 nd [10, we hve: g 0 (x) = 1, g 1 (x) = x x 0, g k (x) = x = 1 k! t1 tk 1 dt 1 dt 2 dt k x 0 x 1 x k 1 [ k 1 x k g j (x) ( ) k j x k j j, k = 2,, n j=0

14 Teodor Cătnş 4 Remrk 4 [10 When ll the nodes concde, then the problem (6) s Tylor nterpolton problem nd P n f tkes the form n (x x (P n f)(x) = 0 ) k k! f (k) (x 0 ) Regrdng the degree of exctness, we obtn the followng result Theorem 1 The Abel Gonchrov opertor P n hs the degree of exctness n, e, dex(p n ) = n Proof It s esly seen tht for the test functons e (x) = x, x [, b, we hve (P n e )(x) = e (x), = 0,, n, whle (P n e n1 )(x) = g 0 (x)x n1 0 g 1 (x)(n1)x n 1 g n (x)(n1) 2x n e n1 (x) We obtn the followng result regrdng the remnder R n f of the Abel Gonchrov nterpolton formul wth Theorem 2 If f H n1 [, b then (R n f)(x) = f = P n f R n f (9) ϕ n (x, s) = (x s)n ϕ n (x, s)f (n1) (s)ds, n g k (x) (x k s) n k (n k)! Proof From Theorem 1 we hve tht gex(p n ) = n By Peno s theorem we obtn wth For ll x [, b we hve (R n f)(x) = ϕ n (, s) = R n [ ( s) n ϕ n (x, s)f (n1) (s)ds, ϕ n (x, s) = (x s)n [ = ( s)n ( s) n P n n [ (xk s) g k (x) n (k), whch, fter some mmedte mnpultons, mples (9)

5 The combned Sheprd Abel Gonchrov unvrte opertor 15 2 THE COMBINED SHEPARD ABEL GONCHAROV UNIVARIATE OPERATOR In ths secton we shll ssume tht there exsts f () (x ), = 0,, N, on the set of N 1 prwse dstnct ponts x [, b, 0 N Let us consder the set of lner functonls of Abel Gonchrov type: Λ AG (f) := { λ (f) : λ (f) = f () (x ), = 0,, N } We ttch to ech node x, = 0,, N, set of nodes X,n, n N, n N, = 0,, N, defned by (10) X,n = {x, x 1,, x n } = {x υ : υ = 0,, n}, = 0,, N, where x Nk1 = x k, k = 0,, n We ssocte to ech set of nodes X,n, = 0,, N, the Abel Gonchrov nterpolton opertor, denoted P n, = 0,, N, correspondng to the set of functonls Λ AG The opertors P n, = 0,, N, exst nd re unque becuse the ponts of the sets X,n, = 0,, N, re prwse dstnct so the determnnt of the nterpolton system of the form (7) s lwys dfferent from zero We hve (11) (P n f) (k) (x k ) = f (k) (x k ), k n, 0 N Remrk 5 The set of lner functonl of Abel Gonchrov type, Λ AG, s ncluded n the set of lner functonl of Brkhoff type We notce tht n cse of the Abel Gonchrov nterpolton we hve the dvntge tht the determnnt of the nterpolton system of the form (7) s lwys dfferent from zero, thus the nterpolton polynoml lwys exsts nd s unque We consder the Abel Gonchrov polynomls of degree n, ssocted to the sets of nodes X,n, = 0,, N, nd the sets of lner functonls of Abel Gonchrov type gven by n (12) (P n f)(x) = g j (x)f (j ) (x j ), = 0,, N, j= wth the Gonchrov polynomls gven by g 0 (x) = 1, g 1 (x) = x x, [ g k (x) = 1 k! x k k 1 j=0 g j (x) ( k j ) x k j j, k 1 Theorem 3 The Abel Gonchrov opertors P n, = 0,, N, hve the degree of exctness n, e, (13) dex(p n ) = n, = 0,, N

16 Teodor Cătnş 6 The proof s obtned smlrly to tht of Theorem 1 Remrk 6 If we consder the sets X,n, = 0,, N, of the form (10) such tht ech of them hs n 1 elements, n N, then We denote by S AG n dex(p n ) = n, = 0,, N the Sheprd opertor of Abel Gonchrov type, gven by (S AG n f)(x) = A (x)(p n f)(x), where A, = 0,, N, re gven by (2) nd P n, = 0,, N, re gven by (12) We cll Sn AG the combned Sheprd Abel Gonchrov opertor Prtculr cse When ll the nodes concde, x 0 = = x N, we obtn the Sheprd opertor of Tylor type: n (Tnf)(x) = (P n (x x f)(x) = ) j j! f (j) (x ), = 0,, N, j=0 nd the combned Sheprd Tylor opertor gven by (ST n f)(x) = A (x)(tnf)(x) The mn propertes of the Sheprd Tylor opertor ST n re: For µ > n (ST n f) (j) (x ) = f (j) (x ), j = 0,, n, = 0,, N; gex(st n ) = n Theorem 4 The opertor S AG n s lner Proof For rbtrry h 1, h 2 : [, b R nd α, β R, one gets S AG n (αh 1 βh 2 )(x) = n = A (x) g j (x)(αh 1 βh 2 ) (j ) (x j ) = α = αs AG n j= n A (x) g j (x)h(j ) 1 (x j ) β j= (h 1 )(x) βsn AG (h 2 )(x), n A (x) g j (x)h 2 (j ) (x j ) whch shows the lnerty of S AG n j=

7 The combned Sheprd Abel Gonchrov unvrte opertor 17 Theorem 5 If µ > N, the opertor S AG n (14) (S AG n f) (k) (x k ) = f (k) (x k ), 0 k N hs the nterpolton property: Proof Tkng nto ccount tht µ > N, we hve k (Sn AG f) (k) ( (x k ) = k (ν) ν) A (x k )(P n f) (k ν) (x k ) By (5) nd (11) we obtn (14) ν=0 Theorem 6 Sn AG f = f, for ll f P n, where P n s the set of polynomls of degree t most n Proof From Theorem 2 we hve gex(sn AG ) = mn { gex(p n ) = 0,, N }, nd, tkng nto ccount (13), we obtn gex(s AG n ) = n Remrk 7 If the condtons of Remrk 6 re verfed, e, gex(p n ) = n, = 0,, N, then by Theorem 2 we hve gex(s AG n ) = mn {0,,N} n The Sheprd Abel Gonchrov nterpolton formul s f = Sn AG f Rn AG f, where Rn AG f denotes the remnder When ll the nodes concde we hve the Sheprd Tylor nterpolton formul nd the followng result s known Theorem 7 [4 If f H n1 [, b nd x 0 = x 1 = = x N, then where wth ϕ n (x, s) = 1 (R AG n f)(x) = { (x s) n ϕ n (x, s)f (n1) (s)ds, A (x) [ (x s) 0 n (x x ) (x x) } Theorem 8 If f H n1 [, b then (R AG n f)(x) = (15) ϕ n (x, s) = (x s)n ϕ n (x, s)f (n1) (s)ds, n A (x) g j (x) (x j s) n j (n j)! j=

18 Teodor Cătnş 8 Proof Theorem 6 mples gex(sn AG ) = n Applyng the Peno s theorem, we obtn wth ϕ n (, s) = R AG n For ll x [, b we hve (R AG n f)(x) = ϕ n (x, s) = (x s)n [ ( s) n = ( s)n ϕ n (x, s)f (n1) (s)ds, j= A ( )P n [ ( s) n n [ A (x) g j (x) (xj s) n (j ), nd fnlly (15) Prtculr cse We consder n = 1 We hve the correspondng Sheprd Abel Gonchrov opertor gven by (S AG 1 f)(x) = The nterpolton formul s A (x)(p 1 f) [ = A (x) g 0 (x)f(x ) g 1 (x)f (x 1 ) f = S AG 1 f R AG 1 f, where R1 AG f s the remnder, whch ccordng wth Theorem 8 hs the followng form: wth (R AG 1 f)(x) = ϕ 1 (x, s)f (s)ds, [ ϕ 1 (x, s) = (x s) A (x) g 0 (x)(x s) g 1 (x)(x 1 s) 0 Exmple 1 Consder f : [0, 4 R, f(x) = 3 sn πx 4, nd the nodes x =, = 0,, 4 Fgure 1 shows the Sheprd pproxmton correspondng to these dt, whle Fgure 2 shows the Sheprd Abel Gonchrov pproxmton The fgures were drwn usng Mtlb

9 The combned Sheprd Abel Gonchrov unvrte opertor 19 3 25 2 15 1 05 0 0 05 1 15 2 25 3 35 4 Fg 1: Sheprd nterpolton 35 3 25 2 15 1 05 0 05 0 05 1 15 2 25 3 35 4 Fg 2: Sheprd-Abel-Gonchrov nterpolton REFERENCES [1 Agrwl, A nd Wong, P J Y, Error Inequltes n Polynoml Interpolton nd ther Applctons, Kluwer Acdemc Publshers, Dordrecht, 1993 [2 Cheney, W nd Lght, W, A Course n Approxmton Theory, Brooks/Cole Publshng Compny, Pcfc Grove, 2000 [3 Comn, Gh, The remnder of certn Sheprd type nterpolton formuls, Stud Unv Bbeş Boly, Mthemtc, XXXII, no 4, pp 24 32, 1987 [4 Comn, Gh, Sheprd-Tylor nterpolton, Itnernt Semnr on Functonl Equtons, Approxmton nd Convexty, Cluj-Npoc, pp 5 14, 1988 [5 Comn, Gh nd Ţâmbule, L, A Sheprd-Tylor pproxmton formul, Stud Unv Bbeş Boly, Mthemtc, XXXIII, no 3, pp 65 73, 1988 [6 Comn, Gh nd Trîmbţş, R, Combned Sheprd unvrte opertors, Est Jurnl on Approxmtons, 7, no 4, pp 471 483, 2001

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