ixf (f(x) s(x))dg 0, i 1,2 n. (1.2) MID POINT CUBIC SPLINE INTERPOLATOR

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1 Iterat. J. Math. & Math. Sc. Vol. 10 No..1 (1987) LOCAL BEHAVIOUR OF THE DERIVATIVE OF A MID POINT CUBIC SPLINE INTERPOLATOR H.P. DIKSHIT ad S.S. RANA Departmet of Mathematcs ad Computer Sceces R.D. Uversty Jabalpur Ida (Receved Jauary 16, 1985 ad revsed form Jue 5, 1985) ABSTRACT. I the preset paper, we obta a asymptotcally precse estmate for the dervatve of the dfferece betwee the cubc splle terpolatg at the md pots of a uform partto ad the fucto terpolated. KEY WORDS AND PHRASES. Local behavour, dervatve, md pot, cubc sple terpolato MATHEMATICS SUBJECT CLASSIFICATION CODE. Prmary 41 A05, 65 DO7 1. INTRODUCTION. Cosder a partto P of [0,I] defed by P 0=x <x < < x o such that x x_ p for all Let P3 be the class of all plecewlse polyomal fuctos s defed over P such that the restrcto s of s over [x_ x] s a polyomal of degree 3 or less for 1,. The class of perodc cubc splles over P s defed by S(3,P) {s: s e P3 s e C [0,], s (j) (0) s (j) (,I), j 0,1,}. Uder certa restrctos o the choce of y, Mehr ad Sharma [I] have studed covergece propertes of the terpolat from S(3,P) matchg a gve fucto at the + yp (0=<y I), 1,,. However, the terpolato at the pots Y x-i md pots, whch correspods to the choce y 1/ s ot covered []. Assumg to a oegatlve measure dg, where g(x+p) g(x) K (costat), oe of the authors (Dksht [], Theorem ) has proved the followg whch covers the case y I/. THEOREM I. Suppose s odd ad I p (6px 4x 3 p3)dg 0 I p dg > 0 (I I) 0 0 The there exsts a uque s S(3,P), whch satsfes the terpolatory codto: xf (f(x) s(x))dg 0, 1,. (1.) x_ We observe that the case whch g has a sgle jump of at p/, (I.I) holds ad the terpolatory codto (1.) reduces to the codto: s(t ) f(t ) t (x + X_l)/,. (1.3)

2 64 H.P. DIKSHIT ad S. S. RANA It may be metoed that the dervatve of a cubc sple terpolator has bee used for smoothg of hstograms (see Boeva, Kedall ad Stefaov [3] ad Schoeberg 4 [4]). Cosderg a fucto f C ad ts uque sple terpolat s S(3,P) matchg at the kots < x >=0 Roseblatt [5] has obtaed asymptotcally precse estmate for s f I the preset paper, we obta a smlar precse estmate for the cubc sple terpolatg at the md pots betwee the successve kots.. ERROR BOUNDS. Let f C 4 ad be perodc wth perod I. Let the umber of mesh pots of P be eve. I ths secto, we shall estmate (s f) where s s the cubc sple satsfyg the terpolatory codto (1.3). Let M ad F deote the traspose s"(x ) of [M I, M M_l] ad [F I, F, F_l] respectvely, wth M ad F 1p- +j I =+ (-) l j j j where aj f(tj+l) f(tj) + f(tj_l). For coveece, we cosder the rest of ths secto, the class S*(3,P) of sples s e S(3,P) for whch s" (0) O. Thus t folows from the proof of Theorem that where -I x - coeffcet matrx C (cj) cj 6_lj + j + +lj (.1) C M F (.) s gve by I order to estmate e, we frst determe a upper boud for e"(x). For ths we otce that the equato (.) yelds CE F" (.3) where E ad F" are the traspose of the matrces [E E E_ I] ad [Fy, F F"_l] respectvely, wth m e"(x) ad F F B. ( 4) 1 where 8 f"(x_ I) + f"(x) + f"(x+l). C s of course vertble (see [], p. 108) ad we frst obta the followg prelmary results for determg the elemets of C-. LEMMA.1. For gve real umbers a ad b wth b ->a, let D(a,b) (dj) be a x matrx wth ad 8 (b a + a dj (l-a) 6-lj + b 6j + a 6+lj (.5) The 8 D(a,b) (b+g) +l (b- )+l (.6) Proof of Lemma.1. It s easly see that D (a,b) satsfes the dfferece equato: D (a,b) b D_ l(a b) + a(l-a) D_(a,b) 0 wth D_l(a,b) O, Do(a,b) I, D l(a,b) b. The lemma follows from the

3 DERIVATIVE OF A MID POINT CUBIC SPLINE INTERPOLATOR 65 above dfferece equato by usg the ducto hypothess. LEMMA.. Suppose b I/ ad Q( b) s the matrx obtaed from D (I/ b) by replacg I/ by ts frst row. The q - (b+r) Q(a, b) b (I r + ar (I r- (.7) where r (-/q) -(b-(b /4) 1/). Proof of Lemma.. It follows from the defto of Q(a,b) that IQ (e b) 4b ID_l(I/ b) -ID (/ b) ( 8) - wth IQo(e,b) ad IQl(e,b) b. The result of Lemma. follows from (.8) by a applcato of Lemma.1. -I LEMMA.3. The coeffcet matrx of (.) s vertble ad f C (Sj), the 8.. ca just be approxmated asymptotcally as by r lj-l ( + r) (.9) where 0 < e < / j/ < l-e ad r 3/- II. REMARK.1. It s terestg to ote that the estmate (.9) s sharper tha that obtaed terms of the fmum of the excess of the postve value of the leadg dagoal elemets over the sum of the postve values of other elemets each row. For, adoptg the latter approach we see from (.) that II C-I II 0. whereas (.9) together wth the fact that lj-l_ (l+r) Z r +r (l-r) (+r) shows that the II C- II does ot exceed Proof of Lemma.3. Takg b ad a I/ Q(a,b) observe that the coeffcet matrx C satsfes the followg dfferece equato 4 ICI 44 IQ_ (1/,11/) I- IQ_ 3 (I/,11/) I. (.10) Thus, t follows from Lemma. that (ll+r) q ICI (ll+r/) r (llr + /) (.11) - I order to estmate C (0j) we obta the elemets 0j from the cofactors of the traspose matrx. Thus, for 0 < j - or j 0 (cofer Ahlberg, et al. [6], pp ) ICI O (qr) j- j Q(I/,11/) Q j(i/,11/) ad for 0 < j < -, ICI Ooj (qr)j Q--j (I/,11/). Thus usg the result of Lemma. ad (.10), we observe that for 0 < j -, (ll+r) (l-r)sj r j- (l-r+) (l-r-j-), -- + (ll+r/) (l-r)s_ r (l-r for 0 -< - rj --j (ll+r/) (l-r)8oj (l-r for 0 < j < - - ad (ll+r/) (l-r)8o_ r (ll+r)

4 66 H. P. DIKSHIT ad S. S. RANA From the above expressos for 8.. the result of Lemma.3 follows drectly. Sce C s vertble, t follows from the proof of Theorem or more precsely from (.3), that there exsts a uque sple s S*(3,P) satsfyg the terpolatory codto (1.3). THEOREM.1. Let s S* (3,P) be the sple terpolat of a perodc fucto f satsfyg (1.3). Let f(4) exst ad be a oegatve mootoc cotuous fucto, the for ay fxed pot x such that 0 < x < as +=. s (x) f (x) f(4) (x) [((t+ x) 4 (t -x)4)/p p((x+l-x) (x_x + 1.9(x_x_l) ) 13.9p /4] /4 + o(p 3) (.1) Proof of Theorem. I. We frst proceed to obta the dervatve s" of the md pot sple terpolat s e S*(3,P) of f. Cosderg the terval [X_l,X], we observe that, sce s" s quadratc the terval [X_l,X ] p s (x) =-M_l(X-X) + Mx-x_ I) + pc (.13) where the costats C s are to be determed by the requremet that s e C [0,I]. Thu s, ad we have M p C+ C (.14) p s(x) M_l(X-X) + M(x-x_l) + 3pC(x-x-x_l) + 6pb. (.15) Aga usg the cotuty requremet, we get p (C + C+ I) (b+ -b ) (.16) Usg (.14) (.16) ad the terpolatory codto (1.3), we have 48ps" (x) M_l[p-4(x-x) 3 + 4M [(x-x_l) p ] _pm+l + 48 [f(t+ I) f(t)3 Thus replacg M by e" (x ) (.17), we see that _p 48 p s (x) [p-4(x-x) ] e" (x_ I) + 4 (x-x_ I) ]e"(x) e,, p (x+ I) + R(f) (.17) (.18) where f,, R(f) [p 4(x_x) (x_ I) + 4 [(x-x_l)-p ] f"(x) p f,, (x+ I) + 48 [f(t+ I) f(t)] Fbr coveece, we deote by uj approprate pots of (xj_,xj+ I) ecessarly the same at each occurece. Thus by Taylor s Theorem, we have whch are ot R(f)/48 p f (x) + f(4)(u ) [{(t+l-x)4-(t-x)4} /p- p {(X+l-X) + 4(x-x) (X_l-X) } /4] /4 + o(p 3) (.19)

5 DERIVATIVE OF A MID POINT CUBIC SPLINE INTERPOLATOR 67 r. +j )(-1) ( + j Now wrtg B J E--I j=+l - ), we have +j F. = Z (-) (4p-j Bj Bj_ I) 1 j=+l so that by Ta1or s Theorem, we have +j (4) ). F,, p (_ Z + Y. (-I) f (u.) + o(p j=l j=+l -I Recallg the equato (.3) ad otcg that C (Sj) we have (.0) (e" (x)) l + l (er F R IR- m IR-l m (T I) + (T), say, where m s a suffcetly large postve teger. We shall estmate T ad T separately. Suppose that - x s a fxed gve pot (0, I) ad let x [x] / where x deotes the largest teger ot greater tha x. The t s clear that as =- x ad - (l-x). Now usg the mootocty of f(4) ad applyg Abel s Lemma to the er sums, we have for some postve costat K by vrtue of Lemma.3. (TI)I m I(0-9) m p (.1) Next we see that for the vales of R occurg T x R x 0(p) (.) Thus, usg the hypothess that f(4) s cotuous ad applyg the result of Lemma.3, we have l(t)-(/(+r)) R Z rir-lp (-jl + l )(-I) R-l < m j=r+i (x) o(p j+rf (4) ). Combg the estmates of (TI) ad (T) ad otcg that m s arbtrary, we complete the proof of Theorem.1 vew of (.18). REFERENCES I. MEIR, A. ad SHARMA, A.. Covergece of a Class of Iterpolatory Sples, Theory! (1968), Jm Approx.. DIKSHIT, H.P. O Cubc Iterpolatory Sples, J.. Approx. Theory (1978), BONEVA, L.I., KENDALL, D.G., ad STEFANOV, I. Sple Trasformatos: Three New Dagostc Ads for the Statstcal Data Aalyst, J..Royal Stato Soc. Ser. B 33 (1971), SCHOENBERG, l.j. Sples ad Hstograms Sple Fucto ad Approxmato Theory Proceedg, Edtors Mer, A. ad Sharma, A. Uversty of Alberta, Edmoto, 197, ISNM 1 (1973), Brkhuserverlag. 5. ROSENBLATT, M. The Local Behavor of the Dervatve of Cubc Sple Iterpolator, J. Approx. Theory. (1975), AHLBERG, J.H., NILSON, E.N., ad WALSH, J.L. The Theory of. Sples ad Ther Applcatos, Academc Press, New York, 1967.