Marcinkiwiecz-Zygmund Type Inequalities for all Arcs of the Circle

Transcription

1 Marcikiwiecz-ygmud Type Iequalities for all Arcs of the Circle C.K. Kobidarajah ad D. S. Lubisky Mathematics Departmet, Easter Uiversity, Chekalady, Sri Laka; Mathematics Departmet, Georgia Istitute of Techology, Atlata, GA July 8, 003 Abstract We establish Marcikiewicz-ygmud Iequalities of the form X jp ( j )j p ( j j ) C jp ()j p d; j= valid for all trigoometric polyomials P of degree m, ad for = 0 < < ::: < =, uder appropriate spacig coditios. The emphasis is o uiformity i the legth of the iterval, irrespective of whether it is close to 0 or. We also establish weighted versios ivolvig doublig weights. Itroductio The classical Marcikiewicz-ygmud iequality has the form C jp ()j p d X k p P + C jp ()j p d; () 0 k=0 0

2 valid for all trigoometric polyomials P of degree. Here < p < ad C ad C are idepedet of P ad. This iequality is useful i studyig covergece of Lagrage iterpolatio, orthogoal expasios ad discretizatio of itegrals. It has also bee exteded i may directios. For example, Mastroiai ad Totik [6, p. 46] established a versio of the right-had iequality ivolvig doublig weights: X k=0 k k p W + P + C 0 jp ()j p W () d: () Here W is a doublig weight. That is, there is a costat L > 0 such that if I is ay iterval, ad I is the cocetric iterval with double the legth, the W L W: I The smallest such L, idepedet of I, is called the doublig costat. Moreover, + W () = W: (3) Geeralized Jacobi weights are doublig weights, ad so are may others. Thus Mastroiai ad Totik greatly exteded the scope of earlier iequalities. They also allowed o-equally spaced poits ad trigoometric polyomials of degree C. See [7] ad [4] for surveys of Marcikiewicz-ygmud Iequalities. The large sieve of umber theory is closely related to (). Oe formulatio of it is [, p. 08] jp ( j )j p " ( j ) C k= I jp ()j p d; (4) with C idepedet of m; ; P; p; ; ; f j g. Here P is a trigoometric polyomial of degree ; " " () = # = p + si si + p + while 0 < < ::: < m ;

3 0 < p < ad m. The parameter is a measure of the umber of j i small itervals, give by = max jfj : j [ " () ; + " ()]gj : [;] We ote that i [] P could be a geeralized trigoometric polyomial, ot just a ordiary trigoometric polyomial. The key achievemet there was idepedece of the size of [; ] as shriks to 0. The oe drawback of the result is that as [; ] approaches [0; ], we do ot recover the usual Marcikiewicz iequality for [0; ], for the case of equally spaced poits. For the full iterval [0; ], this shortcomig ca be repaired by two applicatio of (4), but for itervals [; ] close to [0; ], it is ot clear how to derive a uiform result. I this paper, we preset a versio of (4), which will have the correct form for all choices of [; ] - whether is very small or close to. We ca do this usig a Berstei iequality of the authors, which is sharp i order for all arcs o the circle, or equivaletly, all subitervals of [0; ] [3]. The drawback, however, is that we obtai iequalities oly for p >, ad for trigoometric polyomials, ot geeralized trigoometric polyomials. We prove: Theorem Let 0 < ad for, de e " () = 6 si 4 cos si + + Let K, m ; p < ad satisfy = ; [; ] : (5) = 0 < < < ::: < m+ = (6) j+ j K" ( j ) ; j = 0; ; ; :::; m: (7) The for all trigoometric polyomials P of degree K; jp ( j )j p ( j+ 3 j ) C jp j p ; (8)

4 where C is idepedet of ; m; ; ; f j g ad P. The essetial feature is the uiformity i [; ], irrespective of whether is small or close to. Thus as [; ] approaches [0; ], we see that " " ()! si + # si + ad the right-had side lies betwee ad (), so we recover the form of the classical Marcikiewicz-ygmud iequality. O the other had for ay ;, we have " " () # = si si + : (9) ad so the large sieve iequality (4) is implied by Theorem, with appropriate chage of otatio. We shall also prove a result ivolvig doublig weights, by usig a iequality of Erdelyi []. For simplicity we formulate it oly o itervals of the form [!;!], where! <, to coform with Erdelyi. Thus its chief use is o small itervals. We also itroduce the otatio ad " () =! +! si si + W () = () +() () # =! (0) W (y) dy: () This is a extesio of the otatio W used by Mastroiai, Totik, Erdelyi ad others, from costat icremet to a variable icremet () : Theorem Let 0! <. Let W : [!;!]! R be such that W (! cos t) is a doublig weight o [0; ] : Let K, m ; p < ad! = 0 < < < ::: < m =! () satisfy j+ j K ( j ) ; j = 0; ; ; :::; m: (3) 4

5 The for all trigoometric polyomials P of degree K; W ( j ) jp ( j )j p ( j+ j ) C jp j p W; where C is idepedet of ; m; ; ; f j g ad P. The proofs are preseted i the ext two sectios. Ackowledgemet C K Kobidarajah ackowledges the support of the Coferece Orgaisers that made attedace of the coferece possible. Both authors thak the orgaisers for the excellet coferece. The Proof of Theorem We use Nevai s method [8] for establishig such iequalities together with the Berstei iequality jp 0 " j p C jp j p ; (4) valid for all trigoometric polyomials P of degree [3, p. 345]. We also ote the iequality [3, p.355, Lemma 3.(c)] that e i e i 8 " () ) " () " () 3 : (The otatio there is a little di eret). A little re ectio the shows that, give K ; there exists L > such that o j j mi ; K" () ) L " () " () Here L > depeds o o K (ot o ; ; ). L. (5) Proof of Theorem Let us assume (6) ad (7). Fix 0 j m ad choose s [ j ; j+ ] such that jp (s)j = mi jp j : [ j ; j+ ] 5

6 The so jp ( j )j p = jp (s)j p + j s d d jp ()jp d; (6) jp ( j )j p ( j+ j ) mi jp j p ( j+ j ) + K" ( j ) [ j ; j+ ] j+ j jp j p + C j+ j jp j p jp 0 j " ; j+ j p jp j p jp 0 j by rst (7) ad the (5), with C idepedet of P; ; j; :::. Addig over j, followed by Hölder s iequality ad our Berstei iequality (4) give as desired. jp ( j )j p ( j+ j ) C jp j p + C jp j p + C jp j p ; jp j p jp 0 j " p jp j p p jp 0 j p " p p 3 Proof of Theorem We begi by presetig some backgroud: (I) A trasformatio of [ ; ] oto [!;!] : Let us de e, as did Erdelyi, a trasformatio L (t) = arcsi [(si!) (cos t)] ; t [ ; ] : It maps [0; ] (ad [ ; 0]) oto [!;!]. Observe that si L (t) = (si!) (cos t) 6

7 ad sice L (t) [!;!] ; ; L 0 (t) = (si!) (si t) cos L (t) (si!) (si t) ; uiformly i! 0; ad t [ ; ]. The otatio is i the sese stadard i orthogoal polyomials: the ratio of the two sides is bouded above ad below by positive costats idepedet of! ad t. (There are trivial modi catios if both sides vaish). Similar otatio will be used for sequeces ad sequeces of fuctios. We also the have s si L (t) jl 0 (t)j (si!) si! recall (0). Moreover, we have uiformly i! ad t such that = p jsi (L (t)!) si (L (t) +!)j s L (t)! L (t) +! si si C (L (t)) ; (7) jl 0 (t)j (L (t)) (8)! jl (t)j! : (9) (II) Trasform the f j g ito ft j g. Sice L is strictly icreasig o [ ; 0], it has a strictly icreasig iverse L [ ] that maps [!;!] oto [0; ]. So give f j g as i (), we ca de e t j = L [ ] ( j ), j = L (t j ) : We shall frequetly use the fact that give C, there exists C > 0 such that j j C () ) () C () C : (0) For a proof of this, see [5, p. ], ad apply the iequality there several times. This ad (8) also show that L 0 does ot grow by faster tha a 7

8 costat multiple i correspodigly small itervals. Usig the mea value theorem, we see that for some betwee t j ad t j+, ad moreover, j+ j = L 0 () (t j+ t j ) C ( j ) (t j+ t j ) ; () j+ j L 0 (t j ) (t j+ t j ) ( j ) (t j+ t j ) () uiformly i j (ad m; ; ; ) such that (0) holds for t = t j. From our spacig restrictio (3) o the f j g, we deduce that uiformly i j (ad m; ; :::), t j+ t j C : (3) (Oe eeds a mior modi catio to this argumet for t j close to!, violatig (9)). (III) The relatio betwee W ad W ;! Let us de e, as did Erdelyi, ad W! (t) = W (L (t)) t+ W!; (t) = t W! : Erdelyi otes that W ;! is a doublig weight with costat idepedet of, depedig oly o the doublig costat of W (! cos t). Moreover, because of the spacig (3) o the ft j g, we have uiformly i j Next, from (), Here W ( j ) = () W!; (t) W!; (t j ) ; t [t j ; t j+ ] : L [ ] ( j + ( j )) L [ ] ( j ( j )) W (L (t)) L 0 (t) dt: L [ ] ( j ( j )) = L [ ] ] dl[ ( j ) ( j ) ( j ) () = t j d L 0 (L [ ] ()) ; 8

9 where is betwee j ad j ( j ). Usig (8), (0) ad (), we see that L [ ] ( j ( j )) = t j + O ; uiformly i j. From (8) ad (0) ad the doublig properties, we obtai tj + W ( j ) W! = W ;! (t j ) W ;! (t) ; t [t j ; t j+ ] : (4) t j We are ow ready for the Proof of Theorem As i the proof of Theorem, we obtai jp ( j )j p = jp (L (t j ))j p mi jp Lj p + [t j ;t j+ ] tj+ t j p jp Lj p jp 0 Lj L 0 : Now we use the spacig (3), (0), (), ad the fact that W ;!,, ad L 0 do ot chage much i small itervals (i the form (8), (0), (4)) to deduce that W ( j ) jp ( j )j p ( j+ j ) C tj+ t j jp (L (t))j p W ;! (t) L 0 (t) dt + C tj+ t j jp Lj p j(p 0 ) Lj W ;! L 0 ; with C idepedet of P; j; ; m; :::. Add over j : W ( j ) jp ( j )j p ( j+ j ) C C jp (L (t))j p W ;! (t) L 0 (t) dt + C jp (L (t))j p W ;! (t) L 0 (t) dt +C jp Lj p W ;! L 0 p jp Lj p j(p 0 ) Lj W ;! L 0 j(p 0 ) Lj p W ;! L 0 =p ; 9

10 by Hölder s iequality. Usig Erdelyi s Berstei iequality i the form (.9) i [, p. 334], with appropriate chages of otatio, we have j(p 0 ) Lj p W ;! L 0 C jp Lj p W ;! L 0 ad hece we have show that W ( j ) jp ( j )j p ( j+ j ) C jp (L (t))j p W ;! (t) L 0 (t) dt: Usig Theorem. i [, p. 33], we ca replace W ;! i the last right-had side by W!, so we ca cotiue this as W ( j ) jp ( j )j p ( j+ j ) C = C!! jp (L (t))j p W! (t) L 0 (t) dt jp j p W: Refereces [] T. Erdelyi, Markov-Berstei-Type Iequality for Tigoometric Polyomials with Respect to Doublig Weights, Costr. Approx., 9(003), [] L. Goliskii, D.S. Lubisky ad P Nevai, Large Sieve Estimates o Arcs of a Circle, J. Number Theory, 9(00), [3] C.K. Kobidarajah ad D.S. Lubisky, L p Markov-Berstei Iequalities o All Arcs of the Circle, J. Approx. Theory, 6(00), [4] D. S. Lubisky, Marcikiewicz-ygmud Iequalities: methods ad results, (i) Recet Progress i Iequalities (ed. G. Milovaovic et al.), Kluwer Academic Publishers, Dordrecht, 998, pp [5] D.S. Lubisky, L p Markov-Berstei Iequalities o Arcs of the Circle, J. Approx. Theory, 08(00), -7. 0

11 [6] G. Mastroiai, V. Totik, Weighted Polyomial Iequalities with Doublig ad A Weights, Costr. Approx., 6(000), [7] G. Mastroiai, M.G. Russo, Weighted Marcikiewicz Iequalities ad Boudedess of the Lagrage Operator, (i) Mathematical Aalysis ad Applicatios, (ed. T.M. Rassias), Hadroic Press, Palm Harbor, Fl, 999, pp [8] P. Nevai, Orthogoal Polyomials, Memoirs of the America Mathematical Society, Providece, Rhode Islad, 979.