For a carefully chose system of odes := f ; ; :::; g; ; our results imply i particular, that the Lebesgue costat k (W k; ; )k L (R) satises uiformly f

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1 The Lebesgue fuctio ad Lebesgue costat of Lagrage Iterpolatio for Erd}os Weights S. B. Dameli Jauary, 5 Jauary 997 bstract We establish poitwise as well as uiform estimates for Lebesgue fuctios associated with a large class of Erd}os weights o the real lie. Erd}os weight is of the form: W := exp( Q) where Q : R R is eve ad is of faster tha polyomial growth at iity. The archetypal examples are (i) (ii) where W k; (x) := exp ( Q k; (x)) Q k; (x) := exp k (jxj ) ; > ; k : Here exp k := exp (exp (exp(:::))) deotes the kth iterated expoetial. where Q ;B (x) := exp W ;B (x) := exp ( Q ;B (x)) B log + x ; B > ad > :

2 For a carefully chose system of odes := f ; ; :::; g; ; our results imply i particular, that the Lebesgue costat k (W k; ; )k L (R) satises uiformly for N ; := sup xr j (W k; ; )j (x) k (W k; ; )k L log : (R) Moreover, we show that this choice of odes is optimal with respect to the zeros of the orthoormal polyomials geerated by W : Ideed, let U := fx j; : j g; ; where the x k; are the zeros of the orthogoal polyomials p W ; : geerated by W : The i particular, we have uiformly for N; k 6 k (W k; ; U )k L (R) 6 j= log j Here, log j := log (log (log (:::))) deotes the jth iterated logarithm. We deduce sharp theorems of uiform covergece of weighted Lagrage iterpolatio together with rates of covergece. I particular, these results apply to W k; ad W ;B : : Itroductio ad Statemet of Results I this paper, we ivestigate Lebesgue bouds ad uiform covergece of Lagrage iterpolatio for Erd}os weights. We recall that a Erd}os weight has the form: W := exp ( Q) where Q : R R is eve ad is of faster tha polyomial growth at iity. The archetypal examples are (i) W k; (x) := exp ( Q k; (x)) ; (.)

3 where Q k; (x) := exp k (jxj ) ; k ; > : Here exp k := exp (exp (exp(:::))) deotes the kth iterated expoetial. (ii) W ; (x) := exp ( Q ;B (x)) (.) where Q ;B (x) := exp log + x B ; B ad is large eough but xed. Throughout, let f : R R be cotiuous ad satisfy the decay coditio, We set lim jfw j (x) = : (.3) jxj E [f] W; := if P P k(f P ) (x) W (x)k L (R) (.) to be the error of best weighted polyomial approximatio to f from P ; : Here, P deotes the class of polyomials of degree : It is well kow ([9]) that E [f] W; as : Now let := f ; ; :::; g; ; be a arbitrary set of odes. The Lagrage iterpolatio polyomial to f with respect to is deoted by L [f; W; ]: Thus, if l j; ( ) P ; j ; are the fudametal polyomials of Lagrage iterpolatio at j ; j ; satisfyig, l j; ( ) ( j; ) = j;k ; k ; the, L [f; W; ] (x) = j= f ( j; ) l j; ( ) (x) P : (.5) 3

4 Now write, kw (f L [f; W; ])k L (R) E [f] W; + W (x) j= = E [f] W; + k (W; )k L (R) jl j; ( )(x)j W ( j ) L (R) C (.6) where k (W; )k L is called the Lebesgue costat with respect to the (R) weight W ad the set of odes ; ad (W; ) is the correspodig Lebesgue fuctio. Usig (:6), we see that estimates of the size of the Lebesgue costat eable oe to deduce theorems o uiform covergece of Lagrage iterpolatio. s the subject of weighted Lagrage iterpolatio is a extesively researched ad widely studied subject, we refer the reader to [; 5; 6; 7; ; ;, 3; ; 5]: Now give a weight W : R (; ] as above, we may dee orthoormal polyomials satisfyig p (x) := p (W ; x) = x + :::; with = (W ) > ; Z R p (W ; x)p m (W ; x)w (x)dx = m : We deote the zeros of p by Put < x ; < x ; < ::: < x ; < x ; < : U := fx j; : j g; : (.7) To formulate our results, we eed a suitable class of Erd}os weights from [8] : Deitio.. Let W := exp ( Q) ; where Q : R R is eve, cotiuous, Q exists i (; ) ; Q (j) i (; ) ; j = ; ; Q () > i (; ) ad the fuctio T (x) := + xq (x) Q (.8) (x)

5 is icreasig i (; ) with lim x T (x) = ; T + := Moreover, we assume that for some C j > ; j 3; ad for every " > ; The, we write W E: C T (x) xq (x) Q(x) lim T (x) > : (.9) x + C ; x C 3 (.) T (x) = O ((Q(x)) " ) ; x : (.) The priciple examples of W E are W k; ad W ;B give by (:) ad (:) respectively. For more o this subject we refer the reader to [; 3; ; 8]: To state our results, we eed some more otatio: We eed the Mhaskar-Rakhmaov-Sumber a u deed as the positive root of the equatio u = Z a u tq (a u t) dt p ; u > : (.) t Here, a u exists ad is a strictly icreasig fuctio of u [8; 9] : mogst its uses is the iite-ite rage iequality kp W k L (R) = kp W k L[ ;] ; P P ; (.3) Note that depeds oly o the degree of the polyomial P ad ot o P itself. Now choose y [ ; ] so that jp W (y )j = kp W k L (R) : (.) s W is eve, we may assume that y : We will show later that i fact y > ad is very \close" to : Fix y as above. Fially set := (T ( )) 3 ; ; (.5) 5

6 ad (x) := 8 >< >: max 8 < q : jxj ; q T () jxj ; ; jxj a +L : (.6) ( ) ; jxj Here, L > is xed, but large eough throughout. For more o these special sequeces of fuctios, we refer the reader to [5; 8] : Here ad throughout, 9 = = O (b ) ; b ad = o (b ) will mea respectively that there exist costats C j > ; j = ; ; 3; idepedet of ; such that b C ; C b C 3 ad lim a = : b Similar otatio will be used for fuctios ad sequeces of fuctios. Bouds for Lebesgue costats ad uiform covergece of Lagrage Iterpolatio for U ; : We begi our ivestigatio with the sequece of odes, U ; ; deed by (:7) : We prove: Theorem.. Let W E : The, uiformly for N ; k (W; U )k L (R) 6 T (a ) 6 : (.7) I particular, give " > ; there exists C > idepedet of such that k (W; U )k L (R) C 6 +" : We deduce: 6

7 Corollary.3. Let W E ad r : The there exists C j > j = ; idepedet of ad f so that for N, (a) k(f L [f; W; U ]) W k L (R) Here, C E [f] W; 6 T (a ) 6 C r; (f; W; ) 6 T (a ) 6 : (.8) r; (f; W; t) := sup <ht W r h (f) t (x) L (jxj(t)) + if k(f P ) W k P P L 5 (jxj(t)) ; t > r is the weighted modulus of smoothess of f, (t) := if a u : a u u t ; (.9) ad for a iterval J ad h > ; r h(f; x; J) := t (x) := jxj (t) + T ( (t)) ; x R; (.) 8 >< >: P ri= ( ) i f x + rh ih r ; x rh i J ; otherwise (b) Moreover, if f satises f (r) W L (R); the give " > ; 3 9 >= >; : Here C 3 > is idepedet of : k(f L [f; W; U ]) W k L (R) C 3 a r 6 T (a ) 6 (.) C 3 6 +" r : (.) Thus we ca esure uiform covergece for every r : 7

8 Remark. It is istructive at this poit to recall that for Q = Q k; of (:) ; T ( ) = ky j= log j : Moreover, i geeral, give " > ad ; T ( ) = O ( " ) : (See also (.7)). We thus observe that we may dispese with the T ( ) 6 o the right had side of (:7) by isertig a extra weightig factor ito the left had side of (:7) i the followig sese: Uder the hypotheses of Theorem :; we have uiformly for N ; (W; U ) jxj + T ( ) 6 L (R) This follows easily usig the proof of (:7) ad (:) : 6 : (.3) better behavig Lebesgue fuctio. We observe that although (:) yields uiform covergece for every r ; we ca substatially improve our results, by choosig our iterpolatio poits more carefully. For weights o the real lie, J. Szabados was the rst to exploit this idea ad may of the proofs i this sectio rely heavily o his ideas [] : Motivated by (:3) ad recallig the deitio of y i (:) ad U i (:7) ; we set: ad prove: V + := f y ; y g [ U ; ; Theorem.. Let W E: The uiformly for N ; k + (W; V + )k L log : (.) (R) 8

9 Thus, by addig two completely ew poits of iterpolatio, we ca achieve the much better order log i compariso to the order (T ( )) 6 that we obtaied merely usig the zeros of p : We deduce, Corollary.5. Let W E ad r : The there exists C j > j = ; idepedet of f ad so that for N ; (a) k(f L + [f; W; V + ]) W k L (R) C E [f] W; log C r; (f; W; ) log : (.5) (b) Moreover, if f satises f (r) W L (R) the, give " > ; Here C 3 > is idepedet of : k(f L [f; W; U ]) W k L (R) a r C 3 log (.6) C 3 r+" log : (.7) Remark. atural questio arises as to whether (:) holds (i a lower boud sese) for ay system of odes, at least for some Erd}os weight. This ad related questios will be cosidered i a future paper. Poitwise estimates for (W; U ): We preset poitwise estimates for (W; U ): We emphasize our results ad briey sketch their proofs i Sectio 5 as the argumets are straightforward, but rather legthy. Theorem.6. Let W E: (a) The for N ;there exists C > such that for jxj + L ; (W; U )(x) C [ + p jp W j (x) (.8) 33 jxj jxj + 55 : (x) 9

10 Moreover, we have uiformly for jxj x ; ad ; (W; U )(x) + p jp W j (x) (.9) 33 jxj jxj + 55 : (x) (b) Uiformly for N ad ( + L ) jxj ; (W; U )(x) p jp W j (x)[ + ]: (.3) (c) Uiformly for N ad jxj ; (W; U )(x) a 3 jp W j (x) [ + ]: (.3) jxj Structure of this paper. We close this sectio with some otatio ad remarks cocerig the structure of this paper. Throughout, C; C ; C ::: > will deote costats idepedet of ; x ad P P : The same symbol does ot ecessarily deote the same costat i dieret occurreces. We write C 6= C (L) to idicate that C is idepedet of L: This paper is orgaized as follows: I Sectio, we preset our techical lemmas. I Sectio 3, we preset the proofs of our upper bouds for (.7) ad (.). I Sectio, we prove Theorems. ad. ad Corollaries.3 ad.5. Fially i Sectio 5, we sketch briey the mai ideas i the proof of Theorem.6. Techical Lemmas Lemma.. Let W E ad set x ; := x ; ( ) ad x ;+ := x ; : (a) There exists > idepedet of ad L such that for ; x ; : (.)

11 (b) Uiformly for ad j ; x j; x j+; (x j; ) : (.) (c) Uiformly for ad < j ; ad ad (d) For ; jx j;j jx j+;j (.3) (x j; ) (x j+; ) : (.) sup jp W j (x) xr jxj a sup jp W j (x) 6 T (a ) xr 6 a (.5) : (.6) Proof: This is part of Lemma. of [5] : Now x i (:) : Lemma.. Let W E: (a) Give " > ad ; there exists C > idepedet of such that, C " ; T ( ) C " ad CT ( ) " : (.7) (b) Give < < ; we have uiformly for C; T ( ) T ( ) : (.8) (c) Uiformly for u (C; ); v [ u ; u] ; we have a u a v u v T ( ) : (.9) (d) Give m N ad N ; we have for every fp k g m k= P m W k= jp k j L (R) = m W k= jp k j L [ ;] : (.)

12 Moreover, give r > ; there exists C = C (r) > idepedet of ; m ad P k such that W jxj + T ( ) 6 m k= C W jp k j jxj + T ( ) L (R) 6 m k= jp k j L [ a r(+) ;a r(+) ] : (.) Proof. (a){(c) are part of Lemma.3 of [5] ; (:) follows as i Lemma of [] ad the (:) follows usig (:) ad the method of Lemma 3.3 i [3] : Our ext lemma establishes how \close" y is to : Lemma.3: Let W E; N ad y as i (:) : The, we have for some B > idepedet of ad L: ( B ) y (.) Proof. By (:5) ; (:6) ad the deitio of [see (:5)] ; there exist C j > ; j = ; such that The, this gives C a (T ( )) 6 jp (y )j W (y ) C a mi ( y ; ) : (.3) max y ; C 3 : (.) Now by the deitio of y ; we have clearly that y : Moreover, if y ( ) the (:) is satised with B = : Suppose the, that y < ( ): The (:) becomes y C

13 which agai implies (:) with B = C : Now, x B i (:) : Lemma.. Let W E: (a) Uiformly for ; j ad x R; jl j; (U )(x)j a 3 W (x j; ) jx j;j p (x) : (.5) x x j; (b) There exists C > such that uiformly for ; j ad x R; jl j; (U ) (x)w (x)j W (x j; ) C: (.6) (c) Uiformly for ad j ; a 3 (x j; ) jx j;j jp W j (x j; ) a jp W j (x j; ) jx j;j : (.7) (d) For ; j ad jxj ; there exists C > such that jp (x)j W (x) C a 3 (x) (x j; ) jx j;j jx x j; j : (.8) 3 5 Proof. (a), (b) ad (c) are (:3) ; (:) ad (:) resp i [5] : (d) is (:8) i [8] : Lemma.5. Let W E ad let l +;+ (V + ) ad l +;+ (V + ) be respectively the fudametal polyomials of degree + at the poits y ad -y : The there exists C > such for all x R; jl +;+ (V + )j (x) W (x) W (y ) C (.9) 3

14 ad jl +;+ (V + )j (x) W (x) W ( y ) C: (.) Proof. We prove (:9) : (:) is similar. First observe that l +;+ (V + ) (x) = p (x) (y + x) y p (y ) P + (.) ad satises l +;+ (V + ) (y ) = ; (.) l +;+ (V + ) (x j; ) = ; j (.3) ad l +;+ (V + ) ( y ) = : Observe that by (:) ; we may assume that jxj + : The by (:6) ; (:9) ; the deitio of y ; (:) ad (:) ; l +;+ (V + )W (x) W (y ) C W (x) jp (x)j jy + xj y jp (y )j W (y ) a 6 T ( ) 6 a C ( B ) a 6 T ( ) 6 C : We ext eed a lemma which gives a estimate of the distace betwee y ad jx j; j ; j : Lemma.6. Let W E: The for N ad uiformly for j ; we have jy jx j; jj jx j;j : (.)

15 Proof. We begi with our lower boud. We cosider two cases: Case : jx j; j ( L ) : Note that here, Moreover (:) implies jx j;j 3L : jx j;j 3L (.5) if L is large eough. Next observe that by (:) ad the deitio of [see (:6)] ; we have that (y ) T ( ) (B + L) : (.6) Now as Q ad jp j are both eve fuctios, the deitio of (:6) ; (:6) ; (:8) ; (:5) ad (:6) yield uiformly for j : jy jx j; jj C C Case : jx j; j ( L ) : Observe that if L is large eough, jy jx j; jj jx j;j jx j;j ( ( ) y ) : (.7) Now by (:) ; ( ( ) y ) " jx j;j # (.8) if jx j;j B + L : (.9) 5

16 But the it is easy to see that jx j; j ( L ) implies (:9) if L is large eough ad so we have (:8) : (:7) the becomes jy jx j; jj jx j;j ad we have our lower boud for this case as well. The upper boud is easier. We agai distiguish two cases: Case : jx j; j : Here, if L is large eough, we have by (:) ; jy jx j; jj L + jx j;j = jx j;j : Case : jx j; j ( + ) : Here if L is large eough, we have by (:) ad (:) ; jy jx j; jj B + x ; (B + ) " jx j;j # : The lemma is proved. Let us put x j; := x j; x j+;, j : We prove: Lemma.7: Let W E; N ; r > ad jxj a r : The there exists C j > j = ; such that for j ; (a) W (x) l j; (U ) (x) W (x j; ) C jx j;j jxj x j; jx x j; j : (.3) 6

17 (b) W (x) l j;+ (V + ) (x) W (x j; ) C jx j;j 3 jxj 3 x j; jx x j; j : (.3) Proof. We begi rst with (:3) : First ote that (:5) ad (:6) show that uiformly for ad x; The by (:3) ; jp (x)j W (x) C a W (x) l j; (U ) (x) W (x j; ) = W (x) jp (x)j W (x j; ) jp (x j; )j jx x j; j a C C jxj jxj + L W (x j; ) jp (x j; )j jx x j; j jx j;j by (:) ad (:7) : So we have (:3) : We ow proceed with (:3) : First observe that for j ; jxj l j;+ (V + ) (x) = y x y x j; x j; : (.3) jx x j; j : l j; (U ) (x) : (.33) Next, we claim that jy xj C 3 jxj : (.3) We cosider two cases: 7

18 Case : jxj : Here much as i the proof of Lemma.6, if L is large eough. Case : < jxj a r : Here, usig (:9) ; jy jxjj B + jxj C 3 jxj so that jxj a r C T ( ) C 5 jxj jy jxjj j y j + j jxjj C 6 jxj so (:3) is established. The (:) ; (:3), (:33) ad (:3) yield (:3) : 3 The Proofs of our Upper Bouds I this sectio we establish our upper bouds for (:7) ad (:) : Throughout we assume that W E; x R is xed ad x k(x); is that zero of p closest to x: We eed two lemmas. Lemma 3.. There exist M ad > with the followig properties: (a) If jxj h ; + L i the: 8

19 (i) fj : jj k (x)j g j : jx x j; j M (ii) x xk(x)k; (x) ; k = ; : (iii) (b) If jxj [ ( + L ); ); for all j : x x k(x)3; > M jx x j; j > M h i Proof. Suppose rst that x ; + L j ; we have jtj jx j;j x j; x j+; jx j;j by (:6) ; (:) ad (:) if L is large eough. We coclude usig (:6) ad (3:) that (x) : (3.) (x) : (3.) (x) (3.3) : Observe that if t [x j+; ; x j; ] ; x j; t jx j;j C (x j; ) (L ) (3.) (t) (x j; ) uiformly for j; ad t [x j+; ; x j; ] : (3.5) h i Now by deitio of x k(x); ; we must have x x k(x)+; ; x k(x); or x h i x k(x); ; x k(x) ; at least whe x x; : Usig (:3) h ad (:) if ecessary, i we may assume without loss of geerality that x x k(x)+; ; x k(x); : The by (:) ad (3:5) ; x x k(x); xk(x) ; x k(x)+; x k(x); C 9 (x) : (3.6)

20 Usig (3:6) ad (:) we see that it is possible to choose M such that (3:) holds at least whe x x ; : Suppose x x ; : We may the suppose that L is chose large eough such that x 3; L ad the jx x 3; j + L L (x) usig (:5) ad (:6) : Thus also i this case, it is possible to choose M such that (3:) holds. Parts (ii) ad (iii) of the lemma the follow similarly. Now x M ad i Lemma 3. ad put J := [x ; ; x ; ] h x k(x)+; x k(x) i (3.7) if jxj h ; + L i ad J := [x ; ; x ; ] (3.8) if jxj [ ( + L ; ): We modify the deitio i (3:7) accordigly if jxj x ; : We have the followig estimate. Lemma 3.. Uiformly for j ad N ; j= j =[k(x)+;k(x) ] x j; jx x j; j = 8 >< >: O (a ) ; < < O (log ) ; = O (x) ; > : 9 >= >; (3.9) Proof. First ote that if jxj + L ;we have uiformly for N ad j ; jx tj jx x j; j ; t [x j+; ; x j; ] ; j = [k (x) + ; k (x) ] : (3.)

21 This follows much as i [3] usig Lemma 3: (a) ad (:) sice, x t x x j; = t x j; x x j; x j; x j+; x x j; C ad similarly we ca boud x x j; x t : The, from (:) ad the deitio of J i (3:7) ; we obtai j= j =[k(x)+;k(x) ] = 8 >< >: x j; jx x j; j = O O (a ) ; < < O (log ) ; = O (x) > : Z jtja(+) tj 9 >= >; : dt jx tj The case for jxj + L is similar but easier. We may ow proceed with the proofs of our upper bouds. We begi with: The Proof of the Upper boud i (.7). From (:3) we have for j ; W (x) l j; (U ) (x) W (x j; ) jx j;j Thus, by (:6) ad usig the above, we have (W; U ) (x) = j= j[k(x)+;k(x) ] + C j =[k(x)+;k(x) ] + L jxj + L W (x) jl j; (U ) (x)j W (x j; ) W (x) jl j; (U ) (x)j W (x j; jx j;j + L jxj + L x j; jx x j; j : x j; jx x j; j : (3.)

22 First observe that we may write jx j;j jxj + L + L + jx x j; j jxj + L : (3.) Next we observe that usig (:) ; we may assume without loss of geerality that jxj : The (3:) becomes usig the deitio of ; [see (:5)] jx j;j jxj + L + L = O () + 6 T ( ) 6 jx xj; j Thus usig (:6) ; (3:9) ad (3:3) ; we ow rewrite (3:) as, (W; U ) (x) ad so we have takig sups, as required. We ow preset, j[k(x)+;k(x) ] + a j =[k(x)+;k(x) ] x j; + O jx x j =[k(x)+;k(x) ] j; j = O () + O (log ) + O 6 T (a ) 6 = O 6 T (a ) 6 k (W; U )k L (R) = O 6 T (a ) 6 The Proof of our Upper boud i (.). Firstly, from (:3) we have for j ; W (x) l j;+ (V + ) (x) W (x j; ) jx j;j 3 + L jxj + L : (3.3) 6 T ( ) 6 xj; a jx x j; j 3 (3.) (3.5) x j; jx x j; j : (3.6)

23 Thus by (:6), (:9), (:) ad (3:6) ; we have + (W; V + ) (x) O () + + C = O () + j= j[k(x)+;k(x) ] j= j =[k(x)+;k(x) ] (x) W (x) jl j;+ (V + ) (x)j W (x j; ) jx j;j + L jxj + L 3 x j; jx x j; j (3.7) (x) (3.8) where ad (x) := (x) := C j= j[k(x)+;k(x) ] j= j =[k(x)+;k(x) ] W (x) jl j;+ (V + ) (x)j W (x j; jx j;j + L jxj + L 3 x j; jx x j; j : We observe that usig (:) ; we may assume without loss of geerality that jxj + : We begi with the estimatio of P (x) : Note, that by (:) ; (:33) ad (:3) ; (x) = = O j= j[k(x)+;k(x) ] j= j[k(x)+;k(x) ] y x y x W (x) jl j;(u ) (x)j W (x j; ) jxj jx j;j + L + L W (x) jl j; (U ) (x)j W (x j; ) : 3 (3.9)

24 Next, usig (:) ; (:) ad (:) ; it is easy to see that if L is large eough, we have uiformly for x ad j [k (x) + ; k (x) ] ; so that by (:6) : (x) = jxj jx j;j j= j[k(x)+;k(x) ] + L + L W (x) jl j; (U ) (x)j W (x j; ) = O () (3.) We ow tur to the delicate estimatio of P (x) : Much as i (3:) ;we observe that for j we jxj jx j;j + L + L 3 + The, usig (3:) ; we may write where, (x) = O js x j; jx x j; j + js a 3 jx x j; j jx x j; j 3 a 3 jx j; j 3 + L x j; S = fj : j ; j = [k (x) + ; k (x) ]g ; = O (log ) + O by (3.9) = O (log ) + js x j; C : (3.) jx j; j 3 + L jx x j; j (ja jx j; jj + L ) 3 js jx j; j( ) x j; jx x j; j (a jx j; j) 3 C C

25 + O js jx j; j>( ) x j; T ( ) jx x j; j a 3 C : (3.) Next, usig the Geometric ad rithmetic mea iequality ad (3:9) agai, we may cotiue (3:) as (x) = O (log ) + O B js jx j; j( ) + O T ( ) a 3 = O (log ) + O x j; C jx x j; j + O js jx j; j>( ) js jx j; j>( ) C x j; jx x j; j js jx j; j( ) C x j; jx j; j C (3.3) where i the last lie we used (:5) ; (:6) ; (:) ad (:) : Now it remais to observe that the spacig (:) ad (:6) ; imply that there exist at most ite umber of j such that jx j; j > ( ) : The (3:) yields, (x) = O (log ) + O () = O (log ) : (3.) Combiig (3:3) with (3:9) ad takig sups yields as required. k + (W; V + )k L = O (log ) (3.5) (R) The Proofs of Theorems. ad. ad Corollaries.3 ad.5. I this sectio we preset the proofs of our lower bouds i (:7) ad (:) : We deduce Theorems. ad. ad Corollaries.3 ad.5. 5

26 We begi with, The Proof of our lower boud i (.7). Write (W; U ) (x) = W (x) jp (x)j j= p (x j; ) W (x j; ) jx x j; j : (.) I particular, (:) becomes usig (:6) ; (:9) ; (:) ad (:7) ; (W; U ) (y ) C x j; a C 5 6 T (a ) 6 a a 6 T ( ) 6 jx j; j x j; a : (.) Now it remais to observe that the spacig (:) ad (:6) imply that there exist C 3 j such that x j; [; a ]: The (:) becomes so that as required. (W; U ) (y ) C 6 T (a ) 6 k (W; U )k L (R) (W; U ) (y ) C 5 6 T (a ) 6 ; (.3) We ow tur to the proof of our lower boud (:) : Here a choice of x = y is ot suciet to achieve our lower boud ad we eed to proceed more carefully. Ideed, we will show that the poit we eed sits \far" away from : The Proof of our lower boud for (.). First we claim that there exists y R satisfyig jyj ; for some < < ad uiformly for ; a p W (y) : (.) 6

27 To see this, observe rst that if < < is give, the by (:6) ; (:) ad (:9) ; there exists > C j, j + such that jx j;+ j [; ]. Now choose y = y = x k;+ for some k + such that jy j [; ]. The (:9) ad (:7) give a jp W j (y ) ad (:) is established. Fix y as above. We ow proceed as follows. Sice y < cy ; for some < c < ; we have by (:9) ; (:) ; (:33) ad (:) ; + (W; V + ) (y ) Thus, j= C W (y ) W (x j; ) y y y j= W (y ) W (x j; ) l j; (U ) (y ) l j; (U ) (y ) C (W; U ) (y ) C 3 a jp W j (y ) log C log : ad we have proved our lower boud. We may ow preset: k + (W; V + )k L (R) C log (.5) The Proof of Theorem.. This follows immediately from (3:5) ad (:3) The Proof of Corollary.3: (:8) follows from the represetatio (:6) ; (:7) ad Theorem : of [] : (:) ad (:) follow from (:8) ;Corollary :7 of [3] ad (:7) : The Proof of Theorem.. This follows immediately from (3:) ad (:5) : The Proof of Corollary.5. (:5) follows from the represetatio (:6) ; (:) ad Theorem : of [] : (:6) ad (:7) follow from (:5) ;Corollary :7 of [3] ad (:7) : 7

28 5 Poitwise estimates of (W;U ) I this sectio, we sketch briey the proof of Theorem :6: Fix x; x k(x); ; M; ad J as i Sectio 3: Step : ad Set S := Now write: ( S := ( j : j ; jx x j; j j : j ; S 3 := ) (x) ; (x) jx x j; j M j : j ; jx x j; j > M (U ; W ) (x) := js (x) + js (x) + (x) : js 3 (x) : ) (x) Step : Estimatio of P js (x) ad P js (x). First observe that it suces to estimate the above sums for x h ; + L for they are idetically zero outside this rage of x: Moreover, recall that we may assume by symmetry that x > : The the followig holds: Lemma 5.. Let W E. (a) There exists C such that uiformly for ad x h ; + L i, js (x) C : (5.) Moreover, uiformly for ad x [; x ; ] ; js (x) : (5.) 8 i

29 (b) Uiformly for x h ; + L i ad N ; js (x) p jp W j (x) jxj : (5.3) Proof. First ote that (:6) gives js (x) = W (x) jl j; (U ) (x)j W (x j; ) js C C js for some C > idepedet of x ad as the above sum is ite. For the lower sum, we use the weighted Erd}os-Tura iequality (see for example [5]) ; l j; (U ) (x) W (x) W (x j; ) + l j+; (U ) (x) W (x) W (x j+; ) (5.) valid for ; j ad x [x j+; ; x j; ] : If x x ; ;we may assume without loss of geerality that x h x k(x)+; ; x k(x); i : The (5:) gives js (x) W (x) Thus (5:) ad (5:) follow. k(x)+ j=k(x) l j; (U ) (x) W (x j; ) C : It remais to show (5:3) : Here we rst observe that by (:) we have uiformly for j S ; (x j; ) jx j; x j; j jx x j; j : (5.5) The (:5) ad (5:5) easily yield js (x) p jp W j (x) jxj 9

30 as required. Prelimiary estimatio of P js 3 (x): Lemma 5.. Let W E. (a) If jxj ;we have uiformly for x ad N ; js 3 (x) p p W (x) Z jtja(+) tj jtj jx tj (b) If jxj ;we have uiformly for x ad N ; js 3 (x) p a p W (x) jxj Z jtja(+) tj dt: (5.6) jtj dt: (5.7) h Proof. We cosider the case x ; + L i ad x x; : The other cases are similar. By (:) ad (:5) ; Z xj; jx j; j js 3 (x) p p W (x) j[;][k(x)+;k(x) ] x j+; jx x j; j The much as i (3:) ; (5:8) readily yields (5:6) for this case. dt: (5.8) Step : Estimatio of J := Z jtja(+) tj jtj dt: We ow record the followig techical estimate for J : h i Lemma 5.3. Let W E ad suppose that x ; + L uiformly for x ad N ; J jxj jxj 3 : The + : (5.9) (x)

31 Step 5: The Proof of Theorem.6. Observe that for jxj + L jxj (x) ; > if L is large eough. The (5:) ; (5:) ; (5:3) ad (5:9) yield the result for this case. Theorem.6 (b) ad (c) are similar but easier. 6 ckowledgemet The author would like to thak J. Szabados for allowig him the use of the preprit [] : The author also thaks the referees for their useful commets with regard to the presetatio of this paper. Refereces [] S. S. Boa, Weighted Mea Covergece of Lagrage Iterpolatio, Ph.D. Thesis, Ohio State Uiversity, Columbus, Ohio, 98. [] S. B. Dameli, Smoothess theorems for Erd}os weights II, submitted to Joural of pproximatio Theory. [3] S. B. Dameli, Coverse ad smoothess theorems for Erd}os weights i L p ( < p ); submitted to Joural of pproximatio Theory. [] S. B. Dameli ad D. S. Lubisky, Jackso theorems for Erd}os weights i L p ( < p ) ; submitted to Joural of pproximatio Theory. [5] S. B. Dameli ad D. S. Lubisky, Necessary ad suciet coditios for mea covergece of Lagrage iterpolatio for Erd}os weights, Caad. Math. J., (996) ;

32 [6] G. Freud, Lagragesche iterpolatio uber die ullstelle der Hermitesche orthogoalpolyome, Studia Sci. Math, Hugar. (969), [7]. Kopmacher, \Liear operators ad Christoel fuctios associated with Orthogoal polyomials", Ph.D. thesis, Uiversity of the Witwatersrad, Johaesburg, 985. [8]. L. Levi, D. S. Lubisky ad T. Z. Mthembu, Christoel fuctios ad orthogoal polyomials for Erd}os weights o ( ; ), Redicoti di Matematica e delle sue pplicazioi (di Roma), (99), 99{89. [9] D. S. Lubisky, Ideas of Weighted polyomial approximatio o ( ; ) ; Proc. Eighth Texas Symposium o pproximatio Theory, eds.(chui et al), World Scietic, Sigapore, (995) : [] D. Matjila, Lagrage Iterpolatio for Freud weights, Ph.D. Thesis, Uiversity of the Witwatersrad, 993. [] D. M. Matjila, Bouds for Lebesgue fuctios for Freud weights, Joural of pproximatio Theory, 79 (99), [] P. Nevai, Lagrage iterpolatio at the zeros of Laugerre polyomials, Mat. Lapok., (97) ; 9 6: [3] P. Nevai, Fourier Series ad iterpolatio II, Mat. Lapok (973), 9-6. [] J. Szabados, \Weighted Lagrage iterpolatio ad Hermite-Fejer iterpolatio o the real lie", Preprit. [5] J. Szabados ad P. Vertesi, Iterpolatio of Fuctios, World Scietic, Sigapore, 99. S. B. Dameli, Departmet of Mathematics, 3

33 Uiversity of the Witwatersrad, PO Wits 5, South frica. 33