PROBLEMS AND RESULTS ON THE THEORY OF INTERPOLATION. I

PROBLEMS AND RESULTS ON THE THEORY OF INTERPOLATION. I By P. ERDŐS (Budapest), correspodig member of the Academy X Let xp X ) be a triagular matrix of real umbers i (-,-}-), - X00 < X00 <... < x,, ). It is well kow that the uique polyomial of degree at most -, which assumes the values yk at x 0l)( k ), is give by (the upper idex will be omitted if there is o dager of cofusio) w (X) yk lk (x)7 Ik (x) (xk) (x - CU xk) (X) (X - xi). Clearly Ik (xk) =, Ik (xá)=0 for i, i + k. Let f(x) be a cotiuous fuctio i (-,-{-). Put (3) L,, (f(x)) _ yk Ik (x), i. e. L >t (f(x)) assumes the values f(xk) at xk ( k ). Let f(x) be cotiuous i (-, + ). A well kow theorem of HAHN states that the sequece L (f (xo)) coverges to f(xo) if ad oly if () k- Ik(x-)I < c where c is idepedet of. FABER2 first proved that for every poit group there exists a fuctio f(x) cotiuous i (-,+ ) for which L.(f(x)) does ot coverge uiformly to f(x) i (-,-{-), FABER i fact proved that for every poit group (2) lim max 2:k(#==00 -. =ao - :5x:5 Actually FABER proved slightly more, he shoved that max I I ;Ik (x) I > clog. - :5x:5~ H. HAHN, Über das Iterpolatiosproblem, Math. Zeitschrift, (98), pp. 5-42. 2 G. FASER, Über die iterpolatorische Darstellug stetiger Fuktioe, Jahresb. der Deutsche Math. Ver., 23 (94), pp. 90-20.

382 P. ERDŐS From (2) ad (3) it does ot yet follow that to every poit group there exists a poit xo ad a cotiuous fuctio f(x) so that the sequece L"(f(xj) does ot coverge to f (xo), or what is the same thig (by HAHN's theorem) that there exists a xo (- x, ) so that (4) lim l l ik (x) l = ~. =co S. BERNSTEIN3 proved (4), ad i fact he proved that for a certai xo ad for ifiitely may 7L (5) 2 Ik(xo)l > -x-0() log. : (_2 I the case of the Chebyshev abscissas (6) maxi k(x) I < 2 log + O(). Thus i some sese (5) is best-possible. (5) holds i fact for the Chebyshev abscissas for every xo ad ifiitely may. I this paper we shall prove that for every poit group (4) holds for almost all x. I fact, we shall prove the followig stroger THEOREM. Let s ad A be ay give umbers, ad let > o= o(s, A). Further let - x, < x2 <... < x. be ay poits. The the measure of the set i x (- oo < x < c-) for which,z (7) lk(x)i -- A holds, is less tha s. Theorem I clearly implies that (4) holds for almost all x i (-,+ ). It is well kow that (4) does ot have to hold for all x i (-,+ ), i fact, it is easy to see that there exists a Cator set S so that for every x i S 9L (8) Ik (x) < c /L= where c does ot deped o or x. To see this it suffices.áo start with the roots of T.(x) ad push two cosecutive roots close together ; we suppress the details which ca easily be supplied by the reader. By slightly more complicated argumets oe ca show that the set S ca have Hausdorff dimesio. I S. BERNSTEIN, Sur la limitatio des valeurs etc., Bull. Acad. Sci. de l'urss, (93), No. 8, pp. 025-050.

PROBLEMS AND RESULTS ON THE THEORY OF INTERPOLATION. I 383 By more complicated argumets we could prove the followig stroger THEOREM 2. Let > o(a, c, #, 8) be sufficietly large, - - x, < x2 < < x,,. The the measure of the set i x for which I I Ik (x) - A holds, is less tha c/log (c = c(a)). Further if _ ~(F) is sufficietly small, the the measure of the set i x for which I I Ik (x) :5 log holds, is less tha -. The case of the Chebyshev abscissas shows that Theorem 2 is bestpossible. We do ot give the proof of Theorem 2 sice it is similar but more complicated tha that of Theorem. I a subsequet paper I shall prove that for every poit group there exists a poit xo (- xo -) so that for ifiitely may t (9) 'y ~lk(xa)~ >? log -c. fu I view of (6) (9) is best-possible. Here the followig questio could be asked : For which distributio of poits - < xi <... < x,, is t max J j lk (x) -I=5;.xz~l a miimum? It has bee cojectured that this poit group X, X2, - X is uiquely determied by the fact that all the + maxima of Jlk(x)J i (xi, xi+) (0 i +, xo = -, x,,+, + ) are equal. Deote the value of this miimum by U. We would further cojecture that for every other poits i (-,+ ) (- x; < x2 <... : x z -_ ) at least oe of the -}- maxima of lk (x) I i (xí, xí+,) (0 i + ) is less tha U,,. The oly result I ca prove i this directio is that at least oe of these maxima is less tha í2.4 It might be further worth-while to formulate the followig cojecture : Let ~ zi I=: ( i ). The t max I I lk (z}l zj :5 is a miimum if ad oly if the zi's are the roots of z" = a, I a =. 4 P. ERDŐS, Some remarks o polyomials, Bull. Amer. Math. Soc., 53 (947), pp, 69-76, Theorem 2 (p. 7).

384 P. ERDŐS G. GRÜNWALD5 ad J. MARCINKIEWIC6 proved simultaeously ad idepedetly that there exists a cotiuous fuctio f(x) so that the sequece of Lagrage iterpolatory polyomials L (f(x)) take at the Chebyshev abscissas diverge for every x (- x ). I fact, they prove that for all x (--x ) lim L,,(f(x))= m. I a subsequet paper I hope to prove the followig result : X() Let xz> x22 be ay poit group. The there exists a cotiuous fuctio f (x) so that for almost all x, lim L(f(x))= oo. =w O the other had, it is ot true that there always exists a cotiuous f (x) so that L (f (x)) diverges at all those x for which (4) holds. I fact, I ca costruct a poit group so that for every cotiuous fuctio f (x) there are cotiuum may poits xo for which (4) holds ad evertheless L(f(xa))-, f(xo) as teds to ifiity. If (4) holds for every xo (- _ x,, :5 ), I ca ot decide whether there exists a f (x) so that L (f (x)) diverges for every x (- _ x-:5 ). Oe fial remark. It is ot difficult to costruct a poit group so that for every cotiuous fuctio f(x) ad every xo there should exist a sequece k (depedig o f(x) ad xo) so that (0) lim Lk (AX 0)), A X0)* k=m We ca, i fact, costruct a poit group so that for every x. () lim lk (xo) = which by HAHN's theorem implies (0). The proof of () is easy, we just outlie it. Let xl,..., xi, xi+2,..., x,t. The a simple compu- be - roots of T,,(x) ad xi+-xi = o ( 2 (log ) tatio shows that for xi x xi+ V. I Jlk(x)J - +0()- Now I/2 thus it is easy to see that we ca costruct a poit = (log ) group whose th row is the above-modified Chebyshev poit group ad so G. GRÜNWALD, Über die Divergezerscheiuge etc., Aals of Math., 37 (936), pp. 908-98. 6 J. MARCINKIEWICZ, Sur la divergece des polyómes d'iterpolatio, Acta Sci. Math. Szeged, 8 (937), pp. 3-35.

PROBLEMS AND RESULTS ON THE THEORY OF INTERPOLATION. I. that every x o (- xp ) should be cotaied ifiitely ofte i a short" iterval (xi, x ;+,) ad thus () follows. Now we prove Theorem. We put xo = -, x,,+, = + l. Heceforth c l, c, f... will deote positive absolute costats idepedet of E, A ad. The poits x (- - < x < oc) which satisfy (7) we shall call bad poits. Thús we have to prove that the measure of the bad poits is less tha ~.. First we prove the followig LEMMA. The measure of the bad poits x which satisfy ay of the coditios a) are ot i (-, + I), b) for which mi j x-xi j <,02, I:-k- c) which are i itervals (xi, xi+) satisfyig xi+i -xi > c l Ale, is less tha E/2. Let xo be a bad poit ot i (-,+ ). By iterpolatig r - o the poits x i ( - i ) we obtai for all x X"- - ~ xk _l lk(x) l~l or (2) xó - < ~ Ilk(X-) k- Now if Ix,, j - -}- R, the x - > A, or we have from (2) that for j x I ~ -{- A/a I Ilk(x) >A, k- or if x(, is bad we have j x(, I < -}- A/. Thus the measure of the bad poits : satisfyig a) is less tha 2A/ < c/6 for > o. By similar argumets we ca easily show that the measure of the bad poits satisfyig a) is less tha c2/2. I fact, it is easy to show that the measure of the poits x outside (-l, -J-) for which max IIk (x)i < A, I=k~_,,. is less tha c3/'. Sice the umber of the poits x i is -{- 2, it is clear that the measure of the poits i x satisfyig b) is less tha -/6. Now we have to deal with the bad poits satisfyig c). Let - :!!E~; 9 :!~. It is well kow' that there exists a polyomial p _ (x) of degree at most 7 See e. g. P. ERDÖS ad P. TURÁN, O iterpolatio. II, Aals of Math., 39 (938), p. 72.

3 86 P. ERDŐS. - for which (3) p _ ( ) =, l p(x)) < mi c4' ( zj ~) Assume ow that (4) mi ~4-xk > c5a/. -k- The we obtai from (3) ad (4) for - :5 x :s., ( 5) j p-(xk) ~ < Á ( k ). _ Thus from (5) ad the idetity p,, (')= 2: p-(xk)ik we (9),or a poit 9 ca be bad oly if Ik (f) l > A, A-- (6) mi -xk c5a/. :~5k= obtai The umber of itervals (x i, xi+) satisfyig x2+- xi > c A/E is clearly less tha E/c,A ad thus from (6) the measure of the bad poits satisfyig -c) is less tha 2c,A E _ 2c 5 E E < 6 ' c A c l if cl > 2c5. Thus the proof of Lemma is complete. To complete the proof of our theorem we oly have to prove that the measure of the bad poits ot satisfyig ay of the coditios a), b) ad c) is less tha E/2. I other words, we have to prove that the measure of the poits x (satisfyig - x ) which are i itervals (x i, x,+) satisfyig 6 < xi+-xi < c, ad for which (7) xi+ 2 <x<xi+- 2 ad which satisfy (7), is less tha E/2. If the above statemet is false there clearly must exist at least E2 /2c A itervals (x i, xi+), xi+-xi < c A/E, which cotai bad poits satisfyig (7). But the there must exist at least vll4c A such itervals (8) (Xi, xi'+) ' l which do ot have a. edpoit i commo ad poits satisfyig (7). Now we prove É~ r 4 cl A each of which cotais bad

PROBLEMS AND RESULTS ON THE THEORY OF INTERPOLATION. I 387 LEMMA 2. Assume without loss of geerality that I w' (xi) I I w' (xü-). The for ay x satisfyig (7) ad xi+-xi : c,a/e (i. e. ot satisfyig c)) we have li+ (x) I > có e2 /A. It is kow (ad easy to see) that for x i < x < xi+ (9) li (x) + li+(x) > I, or by I w' (xi) I I w' (xi) I we have for xi < x < xi+ (20) w (x) -f- 0(X) >. W (xi+) (X- xi) CU (Xi.+) (X- xi+) Now sice x satisfies (7) ad (x i, xi+) does ot satisfy c) we have (2) j x - xil > 2cÁ e2 x-xi+- Thus from (20) ad (2) which proves the lemma. LEMMA 3. Let yl, y2,..., yt be ay t (t > t o distict umbers i (-, + ) ot ecessarily i icreasig order. The for at least oe u ( u t) If the lemma would be false we would have (22) < t, log t l-igj~t Yi- Yj, 8 where the y"s are ordered by size, i. e. - y' < y2 <... < yt. But it is easy to see that (22) is false. To see this observe that t-~ (Yí+k-Yz) - 2k, i= ad usig the iequality betwee the arithmetic ad harmoic mea, we obtai t-k- (t- k)2 0,, ~ (t-k) t-k = i= Yi+k - yi 2k 2k or t- t-k > t- (t-k)2 > r- log t l~i<j t Y - k-- yi+k - yi 2k 8. fort > to which proves Lemma 3. 8 See P. ERDŐS ad P. TURáN, O iterpolatio. III, Aals of Math., 4 (940), Lemma IV, p. 529. 0 Acta Mathematics IX/3-4 i+ (x) I + 2ec,A > 2 ) U- _ > t log t Y2í- Yi 8.

W P. ERDŐS : PROBLEMS AND RESULTS ON THE THEORY OF INTERPOLATION. I Cosider ow the itervals (8). By assumptio each of them cotais bad poits satisfyig (7). Defie yir = xa, -if I (.Q I j I ' (xir+i) I, otherwise yi, = xi,,+. Reorder the yi, as follows (23) I w'(y) I I (0'(Y2) I... :: l w'(yt) I, t = F2 -< By Lemma 3 there is a yu. so that (24) > t log t it l yu-yi I 8 We have (say) Y" = xa,.. Now we show that the iterval (25) x',, + 2 c x < xi+-2 does ot cotai ay bad poits, ad this cotradictio will complete the proof of Theorem. To see this let x satisfy (25). We have form (24) ad (23) by a simple computatio that for the x's satisfyig (25) we have for > o(f) (26) l > t log t> log i= I x-yi j 6 20 Further clearly t (27) k(# > I' k(# k.- where the dash idicates that the summatio is exteded oly over those xk's which are equal to oe of the yi ( - i u). By (23) we have (xi,, = y.) (28) I lh (x) I = ~, w (x) cv' (xk)(x-xk) w (x) w' (Yu) (x -Yu) U~.Y á= ] 4c A ' x-yu x-ya Now, by Lemma 2, (23) ad (26) we have I x--yu I > 2 by (25) (29) 03(x) U- i= x-yu. x-y ;, which completes the proof of Theorem. c,-0 F C6F3 log > A 2 i- x-y4 > 2 20 > A (Received August 958)