HnUf> xk) = S0Jf(xk) (k = 1,..., «; j = 0,..., m - 1).

Transcription

1 PROCEEDINGS of the AMERICAN MATHEMATICAL SOCIETY Volume 09, Number 4, August 990 ON (0,,2) INTERPOLATION IN UNIFORM METRIC J. SZABADOS AND A. K. VARMA (Communcated by R. Danel Mauldn) Abstract. From the well known theorem of G. Faber t follows that for any gven matrx of nodes there exsts a contnuous functon for whch the Lagrange nterpolaton polynomal [/, x], generated by the n th row of the matrx, does not tend unformly to f(x). In ths paper we shall provde analogous results for the related operator Hn }[f, x] as defned below. Let () (-!<)*, <x2<-<xn(< ) be an arbtrary system of nodes of nterpolaton (xk = xkn,k=l,...,n;n =,2,...), and for an arbtrary contnuous functon f(x) n [-, ] (.e., / G C[l, ]) and nteger m >, consder the (0,,..., m) Hermte-Fejér nterpolaton of order m defned by HnUf> xk) = S0Jf(xk) (k =,..., «; j = 0,..., m - ). Evdently, Hn m(f, x) s a unquely determned polynomal of degree at most mn. Hn,(/, x) s the Lagrange nterpolaton polynomal of f(x) ; the classcal result of G. Faber [4] shows that ths cannot be unformly convergent for all / G C[-l, ] for any system of nodes (); whle another classcal result of P. Erdös and P. Turan [2] asserts that f () are the roots of the «th orthogonal polynomal wth respect to an arbtrary L -ntegrable weght functon w(x) > 0,then Hn,(/, x) converges n weghted L metrc for any /gc[-, ]. For m 2, the stuaton s dfferent. There exst systems of nodes ( ) such that Hn 2(f, x) unformly converges for all f C[-l, ] (e.g., for the roots of the Jacob polynomals 7^Q \x) wth - < a, ß < 0; see G. Szegö [6], Theorem 4.6). Hence n ths case the L and s of no nterest. convergence follows automatcally Receved by the edtors December, 988; ths paper was presented by the second author at the Sxth Texas Internatonal Symposum on Approxmaton Theory, January Mathematcs Subject Classfcaton (985 Revson). Prmary 4A05. Ths paper was completed whle the frst author vsted the Unversty of Florda n Ganesvlle. Research partally supported by Hungaran Natonal Foundaton for Scentfc Research Grant No Amercan Mathematcal Socety /90 $.00+$.25 per page

2 976 J. SZABADOS AND A. K. VARMA For «7 = 3 the results are less complete. R. Saka [5] proved that for the Chebyshev roots Hn 3(f,x) cannot converge for all /gc[-, ] (actually he proved ths for all odd m 's), and later P. Vértes [7, Theorem 2.7] generalzed ths result for arbtrary Jacob nodes (under some addtonal condton whose valdty s checked only for m < 5 ). Our result here s that for any system of nodes (), 77n 3(f, x) cannot converge unformly for all f C[-l, ]. Ths follows from the followng more quanttatve result on the norm of Hn 3. Theorem. For any system of nodes () we have (2) 77 3 >Clog«. Proof. An easy calculaton shows that (3) Hn,(f,x) = J2f(xk)Ak(x), k= where (4) Ak(x) = {l- 3l'k(xk)(x - xk) + [6l'k(xk)2 - lk(xk)](x - xk)2}lk(xf (k=l,...,n) wth the usual notaton lk(x) for the k th fundamental polynomal of Lagrange nterpolaton. Hence ntf,3n= Í2W* k= where s the supremum norm of the correspondng functon. Snce 3x + 4.5x > 0 for any real x, we get from (4) \-Vk(xk)(x-xk) + Here «<**)-#<**: (*-**) > \ mk^k> - <**: (x-xk) fe)2-fc)= A*k X. - X;.e., y _!_ t% (Xk-X^X.-Xj) *j y 6í (**-*/)' (5) \Ak(x)\ > - 3 (x-xk 2( x, Z2\lk (x)\ (2<k<n). " k "k-> In the rest of the proof of Theorem we use a modfcaton of the dea of proof of Theorem n P. Erdös and P. Turan [3]. We dstngush two cases. Case. There s a k0, < kq < «such that lk(«o)l = >n (6, G [-,]).

3 ON (0,, 2) INTERPOLATION IN UNIFORM METRIC 977 Then by Markov's theorem (6) y*) > y *»? ( x~^< Thus choosng a, G [-, from (5) and (6) whch s stronger than (2). ] such that \ZX- xk\ > Aj > \ x- 0\ we obtan 3 ( \2 ( V 32 Wrrte) \2n) =?"' Case 2. \\lk\\ < n (k =, 2,..., «). In ths case, accordng to a result of P. Erdös [], we have (wth xk = cos6k ) Thus > - -n <l0g«(7ç[0,7t]). 9t l Y! ^ tw " f l7l ^ 4^-^, 7 C [0, t] and n > «0. Therefore usng the harmonc-geometrc-arthmetc (?) eke (xk Now let xk_l, n (8) w (x)=n(*-**m = = > > *e/! (Ee,e/; 5I7 «) -4, 3. log «> 0 7 «f 7 >4 7Ç[0,7t],«>«c /T log«' log«mean nequaltes we get 7 = 2log«' 2log«(9) Mn = mx\œn(x)\ = \œn(ç)\,mn = max\a)n(x)\ = \a>h{z0)\ (x = cos0)..v < xt/n Accordng to Lemma n [3], we have (0) max «' (x) = 0(tj nm), where nn = max, - -f-. ee' \log «Mnj Subcase 2a. Mn < Mn/log2 «. Then by (0) max <u>) ee' = 0(^) \log nj

4 978 J. SZABADOS AND A. K.. VARMA Choosng 7 = 7^ n (7), we obtan from (5), (8), (9), and (7) ^(«* 8k l'n 8*e/:»««) c^y "3 g^-^t-) ">«(**) t-** xfe-l' Subcase 2b. A/n > Afn /log «. Then smlarly as above, ^ t 5 j > C2log «. and max eo (jc) = 0(nM ),.ee (] Y \Ak(í0)\ > 2 0k&'n oke'k *>,«<>) œ'n(xk) Let n = cos 0n and 0 < ön < f. Defne l^o ^rlv** x/t-p > y"_ ""3 À lío-**!(**-**-)2 0-2 '«,A u «u «C7 Then by (7) (wth I = Ink) A = 0,,..., 0 log3«m l -.-_ -^ :- 0k l'n Ko xk\(xk xk-\> x=\ eke x \"o "k\(xk xk-> m.e., by () -C4Wntx^e^Jxk-xk_x)2 n 2 2 > C,-t log «log ««log «= c5«log«, XI I4t(f0)l >c3c5log«. The theorem s completely proved. Remark. We conjecture that the statement of our theorem remans true for any odd m.

5 ON (0,,2) INTERPOLATION IN UNIFORM METRIC 979 References. P. Erdös, On the unform dstrbuton of the roots of certan polynomals, Ann. of Math. 43 (942), P. Erdös and P. Turan, On nterpolaton. I, Ann. of Math. 38 (937), _, An extremal problem n the theory of nterpolaton, Acta Math. Acad. Se. Hung. 2 (96), G. Faber, Über de nterpolatorsche Darstellung stetger Funktonen, Jahresber. der Deutschen Math. Ver. 23 (94), R. Saka, Hermte-Fejér nterpolaton prescrbng hgher order dervatves, J. Approx. Theory (to appear). 6. G. Szegö, Orthogonal Polynomals, vol. 23, Amer. Math. Soc. Colloq. Publ., Provdence, RI, P. Vértes, Hermte-Fejér nterpolatons of hgher order. I, Acta Math. Hungaran Academy, (-2) 59 (989), Mathematcal Insttute of the Hungaran Academy of Scences, Budapest, Reáltanodau.3-5 H-053 Unversty of Florda, Department of Mathematcs, Ganesvlle, Florda 326