the correspodig fudametal polyomials, (1. 3) (x) lk(x))1 all = max?1(x) k=1-1 ;x<1 ~ = 1, 2,..) r the Lebesgue fuctios ad the Lebes e costats of the i

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1 ON THE ALMOST EVERYWHERE DIVERGENCE OF LAGRANGE INTERPOLATION > 0 I'. Frdüs ad P. Vértesi Mathematical Istitute ofthellurigaria Academy ql'scieces, Budapest, Hwzga 1. INTRODUCTION I the previous pa per P.Erd8s -,,ated without proot that j Z = 1, 2, (LI) -1 <_x mi 4x -W... <xl,t! ("i, L... ) is a triagular matrix, the zh a re exists a cotiuom-, such that the sequece. of Lagrage '.ergo ario, l mials pc, y,10- L ll (7, Z, x) =L.r, X)= -.' - iu, " x k, _ K-r1' x " 1 diverges almost e i r-1,1 ], ~ j -' d ír, fact. - Z, X) for almost all x ( -see 'j)j. (Here, as usual 0. 2 (X) km, ( k=1, 2,..., r, k=1 ( X-X P. ) ) Th e detailed v e r s i o - ~ v, i 1-1 ap,pear í r, A e t, 11 a th. A e u d. ~e i. T~,r, 270

2 the correspodig fudametal polyomials, (1. 3) (x) lk(x))1 all = max?1(x) k=1-1 ;x<1 ~ = 1, 2,..) r the Lebesgue fuctios ad the Lebes e costats of the iterpolatio Here is the sketch of the proof. The detailed proof (about 30 pages) tured out to be quite complicated ad several ususpected difficulties had to be overcome 2. PRELIMINARY RESULTS I his classical paper 127 G. Faber proved that for ay matrix Z lim ~_ = co from where i ; follows Li e ctly that for every poit group there exists a cotiuous fuctio f 1 (x), -1-<x<1 (shortly f 1 e C ), such that lim IÍ L(f 1, -->o7 x ~ ; = o0 ("fece }I g,.x? ÍÍ _ ;Í g ií max g (x) Í for ge C.) Almost twety 1<x cl years la+.er, i 1931, Berstei showed that for every Z for which ii. _) hcl s there exists a f 2 F (' ad x0-1<x U c 1, such that (see [. lim ~ L r f 2, xu)i = CO 11 -a ")~N

3 E_ ~ -1+2 (k-1) / (- 1) j ad the fuctio I x I lím ' L (j t (, E, x) j = co if x xto x -,- ao The, usig the "good" Chebyshev matrix T = {x kll = cos 2 ) 1 X ; k= t, 2 ; = 1, 2,..., G. Grüwald [4] obtaied that there exists a fuctio f i b C for which (2.2) Um 1 L (f 3' T, x ) = co for almost all x i [-1, 11. Later he ad (idepedetly) J. Marcikiewicz proved that for a suitable f4 6 C, (2.2) is true for every x from E-1,11 (see [5] ad [6]). Quite recetly A. A. Privalov [7] cosidered the Jacobi matt-ices Z(oc,(3) = f xk ' k=1, 2,.... ; =1, 2,...3, (see e. g. [81, Part 2 ), ad showed that for a certai f 5 6 C (2. 3) [ c c f3) lim I L (f 5, Z,X)1= oo ->oo a. e, o (-1, 1], where "a, e. " stads for "almost everywhere ". ( He cosidered some further poit groups, too.) His proof strogly depeds o the properties of the Jacobi roots Fially, he proved (2. 3) for the whole ( -1, 1) (see 3. RESULT As idicated above, we are goig to prove poit group Z, i, e. we state ( 2. 2) for ay fixed THEOREM. For ay matrix Z for which Q.1) holds oe ca fid a fuctio F E C such that 272

4 ,) lim I L (F, Z, x)~ = oo for almost all x i [-I,!] moo O the other had, cosiderig the special matrix x l I ca say that (3, i) geerally is ot true for all xe 1] (see P, urá [9], Problem 111 ; [11, p.384). Fially, let us ote that the "fm" caot be replaced by "lim" lim". Ideed, as P. hrd8s shouted, oe ca costruct a poit rpt,;r so that for every fec ad every x e: [-1, 1] there would exist 0 sequece k (depedig o f ad x0 1, suc?. that ee [i], p. 384 ). lim L (,f, x j =:f (x o ) k k--~.oc ON THE PROOF?s we metioed above, the proof 's rather log ad quite complited aithough it uses oly elemetary techiqueo, Our aim here is sketch it, stressig some chat a cteratíc cosideratios ad lemmas. 4,1 The quoted result (2, 1) of S, herstei ca be obtaied from e fact that for ay matrix Z oe -_a choose the poit x E [-1, 0,r vlach r ~1, (x o(1) I ~ ~' x ' ~} k=1 I 2 r ifir :i_'e.ly _u.1tty 11 k';ue illc, samt papel, U1,1 ).

5 the obviously I, e.. if L x0 }- k=l g (xk ) lk(x0 )= k Í ik (x 0 )l ='1 (x 0 ) O def f (x) - the f(x)ec, moreover k= 1. 2,... L (f, x0 ) _ - F E W 1 k e i i l k gk (x), L (q x i i ' 0 + L k>i from where we obtai (2. 1) with suitably chose {~P W -spa. 4, 2. I 1958 P. Erdős proved, that for ay give A > 0 ad E>0 the measure of the set i x ( - oo<x<oo) for which A (x), A, >, 0 (A, E), holds, is less tha C, whatever is the matrix Z. From this we immediately obtai, that (4.1) I'm -,w k=1 lk (y)%=0q for almost all y i [-1, 1], So, as above, we ca obtai a ucoutable family of fuctios f y (x)gc such that lim 1 L(fy, y) 1 = o0 ->co for almost all y i (-1, i], To prove the origial statemet (3. 1), we shall have to costruct the cotiuous F(x) usig the ucoutable family of fuctios fv. But

6 .As approach does ot seem passable. So we choose aother method, shere we shall have to uite at most coutable family of fuctios. 4. 3, At first let us suppose that (with x O = 1, ad x +1-1 Axk def def xk-xk+1, <- = 1/ I (k=0, l,,,., ; =1, ). Divide the iterval L-1, 13 by equidistat poits as follows, Itm ~,~,~ r2m Imm _1 Q2 X b 2 1 Now, if we codiser, e, g. the expressio k k+1 (4,2) (-1) 1(x) + x (-i) lk(x)i íf xci 2 it is a simple, but a very importat remark that all the terms i (4.2) have the same sig for ay fixed xe12. (Ideed if, e, g. xl <a2, theta Co(X) sig [(-1 ) k cd(xk) ] =S where S = ± 1 for ay k, 15k,, moreover x-xk 7 0, which meas that

7 W e 1a11 l, XI)vk. I %V" 11.1\k, 11+ U111(c 11111N' IA, %\ Itill. 'IloilI Ill oblaill F(x) where the i11(e1 v als I I ai c ul' positive measure. This phcuolcou i exprrssvd by the followig statemet. LEMMA Let A>0 be a arbitrary fixed umber. The cosiderig the arbitrary iteger m,m Q (A), for ay. 0 (m) there exists the set H c 1-1, 1] for which ~t(h ),1/l i m wheever x e [-1, 1] \ H (4.3) 1_ ki 1 1k (x)i>(l m) 1 / 3 >2A íf a 0(m). xk~ 1 j(x), m Here I. is the iterval cotaiig x; 1 (x), m the Lebesgue measure. moreover, stads for This lemma proved to be a very importat part of the proof. It is a rather deep geeralizatio of the statemet by P.Erdds (quoted i 4. 1) because i the sum 7-1 lk(x) I geerally eve the terms for which I x-xk I is "small" are "large". I the proof of (4. 3) we use oly some basic otios of iterpolatio theory ad combiatorial cosideratios Usig Lemma 4.1, we obtai the fiite umber of cotiuous fuctios fi (x) whose Lagrage iterpolatory polyomials are big o the sets B. i. More exactly we get that lim I L ( f., x l { _ 00 o B İ -* >. co (14í ;s ) where 1 I p (Bi ) -2 p, f LIMA 4.2. If ~ > 0 is arbitrary. To combie these f i we use r I (x), r 2(x)EC, moreover lim I L (r l,x)i=co if xeb 1, ~L (B 1}cco, -+co 2 7 6

8 -Slim 0 L(r 2,x)1 oo if xeb2, rt(b 2 KCO, ->c» the ay fixed iterval ( Al' P2 ) ( P1 4 P2) cotais a OC such that zm I L (ar l +r 2' x) 7 = oo a, e, o B 1UB 2 -)-co (a, e. = almost everywhere). Applyig these lemmas ad some other cosideratios we obtai the theorem For the itervals Lxk > ó, istead of Lemma 4. 1, we ca use the followig LEMMA Let Ax > k J ( k is fixed, O,k~ ). The for ay fixed 04gc1f2 we ca defie the idex t = t(k,) ad the set h k C lxk+l, ' xk~ such that Í (hk -< 4gl-x k11' moreover J5 ~lt'x)~>3 íf XE1xk+1 ' \h xk~ k id >, l :q). Fially, by a statemet aalogous to Lemma 4.2 the proof for the case of the log itervals as well. we ca complete Ad at last oe more problem o Lagrage iterpolatio which to be quite a difficult oe ; There is a poitgroup fxkj such that for every cotiuous f ( x ;, L f~ x J1-á f (x~ holds for at least oe x 0 seems for which lim ( x ) =cr (see [13). This is probably true, but at this p ommomet the caot prove it (the origial "proof" was icomplete). REFERENCES i-'. Lrd6s. Problem= a..d results o the theory of iterpolatio. 1.

9 3. S. Berstei, Sur la limitatio des valeurs d' u polyome, Bull. Acad. Sci, de l' URSS 8 (1931), G. Grüwald, Uber 1)ivergezerscaeiuge der Lagragesche Iterpolatios polyo me, Acta Scí. >\,i:ath, Szeged 7 (1935), , Ibber Divergezerscheiuge der Lagragesche Iterpolatiospolyome stetiger Fuktioe, :\als of Math. 37 (1936?, J. Marcikiewícz, Sur la divergece des polyomes d' iterpolatio, Acta Sci. Math. Szeged 8 (1937), A. A. Privalov, Divergece of Lagrage iterpolatio based o the Jacobí abscissas o the set of positive measure ( Russia ), Sibirsk. Mat. Z. 18 (1976), LP. Nataso, Costructive Theory of Fuctios( Russia ) Gos. izdat, tech. -teor. lit., Moscow-Leigrad, P. Turá, Some ope problems of approximatio theory (i Hugaria), Mat.Lapok 25 (1-2), P. Erdős ad T. Gr -Uwald, O polyomials with oly real roots, Aals of Math. 40 (1939), P. Erdős ad P. Turá, O iterpolatio, III, ibid. 41 (1940), P. Erdős ad J. Szabados, O the itegral of the Lebesgue fuctio of iterpolatio, Acta Math. Acad. Sci. Hugar. 32 (1978), , 13. A. A. Privalov, Approximatio of fuctios by iterpolatio polyomials, i "Fourier Aalysis ad Approximatio Theory", I-II, North-Hollad Publ. Co., Amsterdam-Oxford-New York, 1978., pp S. N. Berstei, Quelques remarques sur 1 iterpolatio, Math. A. 79 (1918),