On polynomial and barycentric interpolations

Transcription

1 Special Issue for the 10 years of the Padua poits, Volume Pages O polyomial ad barycetric iterpolatios László Szili a Péter Vértesi b Abstract The preset survey collects some recet results o barycetric iterpolatio showig the similarity to the correspodig Lagrage (or polyomial) theorems Namely we state that the order of the Lebesgue costat for barycetric iterpolatio is at least log ; we state a Grüwald Marcikiewicz type theorem for the barycetric case; moreover we defie a Berstei type process for the barycetric iterpolatio which is coverget for ay cotiuous fuctio As far as we ow this is the first process of this type The aalogue results for the polyomial Lagrage iterpolatio are well ow 1 Itroductio I the last 30 years the barycetric iterpolatio has bee ivestigated first of all cosiderig its practical importace The aim of this paper is to state some recet results showig the similarity betwee the barycetric ad polyomial Lagrage iterpolatio Amog others we see that the order of the correspodig Lebesgue costat is at least log We state a Grüwald Marcikiewicz type theorem for the barycetric case Moreover we defie the aalogue of the polyomial Berstei process for the barycetric iterpolatio As far as we ow it is the first barycetric iterpolatio procedure which is coverget for every cotiuous fuctio The aalogo for the polyomial case is well ow 2 Lagrage ad barycetric iterpolatio 21 Let C = C(I) deote the space of cotiuous fuctios o the iterval I := [ 1, 1], ad let P deote the set of algebraic polyomials of degree at most stads for the usual maximum orm o C Let X be a iterpolatory matrix (array), ie, X = x = cos ϑ ; k = 1,, ; = 1, 2,, with 1 = x +1, x < x 1, < < x 2 < x 1 x 0 = 1 (21) ad 0 ϑ π Cosider the correspodig Lagrage iterpolatio polyomial L (f, X, x) := f (x )l (X, x), (22) Here, for, with l (X, x) := ω (X, x) ω (X, x )(x x ), 1 k, ω (X, x) := (x x ), are polyomials of exact degree 1 They are the fudametal polyomials, obeyig the relatios l (X, x j ) = δ k j, 1 k, j By the classical Lebesgue estimate, usig the otatios λ (X, x) := l (X, x),, (23) Λ (X ) := λ (X, x),, (24) (Lebesgue fuctio ad Lebesgue costat (of Lagrage iterpolatio), respectively,) we obtai if L (f, X, x) f (x) λ (X, x) + 1 E 1 (f ) (25) a Lorád Eötvös Uiversity, Departmet of Numerical Aalysis, Pázmáy P sétáy I/C, H-1117 BUDAPEST, Hugary szili@caesareltehu b Alfréd Réyi Mathematical Istitute of the Hugaria Academy of Scieces, Reáltaoda u 13 15, H-1053 BUDAPEST, Hugary vertesi@reyihu

2 Szili Vértesi 18 ad L (f, X ) f Λ (X ) + 1 E 1(f ), (26) where 22 E 1 (f ) := mi P P 1 f P The above estimates clearly show the importace of λ (X, x) ad Λ (X ) Usig the obvious idetity we ca write L (f, X, x) as where l (X, x) 1, (27) w ω (X, x) f (x ) x x L (f, X, x) =, (28) w ω (X, x) x x 1 w = ω (X, x ) = 1, 1 k (29) (x x j ) j k A simple cosideratio shows that the right-had expressio of (28) at x j takes the value f (x j ), 1 j, usig arbitrary w 0, 1 k Thus, choosig w = ( 1) k+1, we get the classical barycetric iterpolatio formula for f C: where B (f, X, x) := f (x )b (X, x),, (210) ω (X, x) ( 1)k x x b (X, x) := = ( 1) j ω (X, x) j=1 x x j j=1 ( 1) k x x (211) ( 1) j x x j The first equatio of (211) shows that b is a ratioal fuctio of the form P /Q, where P (X, x) = ω (X, x ) l (X, x), 1 k, (212) Q (X, x) = Above, P P 1 \ P 2 ad Q P 1 As we remarked, the process B has the iterpolatory property, ie ω (X, x j) lj (X, x), (213) j=1 B (f, X, x ) = f (x ), b (X, x j ) = δ k j, 1 k, j ; (214) (cf (210) ad (211)) Moreover, it is ot so difficult to prove the ext fudametal relatio valid for arbitrary matrix X : Q (X, x) 0 if x,, (215) where = (, ) (see J-P Berrut [3, Lemma 21]) By the above defiitios we ca prove that {b (x), 1 k } is a Haar (Tchebycheff) system (or briefly, T-system) for ay fixed (see [6] ad [17]) Actually, T = T(x ) where x = (x 1, x 2,, x ) Modifyig our previous otatio (210), we defie for f C, x ad L (f, x, x) := f (x )b (x, x), (216) λ (x, x) := b (x, x), (217) Λ (x ) := λ (x, x), (218) (cf (22) (24)) By defiitio, they are the Lagrage iterpolatory T-polyomials, T-Lebesgue fuctios ad T-Lebesgue costats, respectively, cocerig the above defied T-system Dolomites Research Notes o Approximatio ISSN

3 Szili Vértesi 19 3 Divergece-type results 31 As we metioed the estimatio (26) shows the importace of λ (X, x) ad Λ (X ) So it was fudametal the result of G Faber [7] from 1914 which says that Λ (X ) 1 log, 1 (31) 12 for ay iterpolatory X However we ca prove much more Namely improvig some results of P Erdős ad P Vértesi (cf [16, 22]) P Vértesi [15] proved Theorem 31 There exists a positive costat c such that if ɛ = {ɛ } is ay sequece of positive umbers the for arbitrary matrix X there exist sets H = H (ɛ, X ), H ɛ for which if x [ 1, 1] \ H ad = 1, 2, 32 λ (X, x) > cɛ log Usig (31) oe ca see that for ay fixed matrix X there exists a f C such that lim L (f, X, x) = May papers deal with the barycetric iterpolatio (cf J-P Berrut ad G Klei [4] ad its refereces) ad defie poitsystem havig T-Lebesgue costat of order log We metio two recet papers Firstly the work L Bos, S De Marchi, K Horma ad J Sido [5], where the so called well spaced odes are defied; secodly the paper of B A Ibrahimoglu ad A Cuyt [10] statig that for the odes e = e = 1 + 2k 1 ; k = 1, 2,, Λ (e ) = 2 log + O(1) π However, the ext fudametal Faber-type statemet, as far as we ow, is ew (cf G Halász [8]) Theorem 32 For arbitrary system x Λ(x) > log, 3 (32) 8 More detailed cosideratios show that a statemet aalogous to Theorem 31 ca be proved Namely we have Theorem 33 There exists a positive costat c such that if ɛ = {ɛ } is ay sequece of positive umbers, the for arbitrary matrix X there exist sets H = H (ɛ, X ), H ɛ, for which if x [ 1, 1] \ H ad = 1, 2, 33 The proofs of these are i [17] λ (x, x) > cɛ log (33) A fudametal egative result i Lagrage iterpolatio is due to G Grüwald ad J Marcikiewicz from 1936 Theorem 34 There exists a fuctio f C for which lim L (f, T, x) = for every x [ 1, 1], where T = t = cos 2k 1 π ; k = 1, 2,, ; 2 The reader ca cosult for the iterestig history of Theorem 34 i [16, p 76] Now we quote the correspodig result for the barycetric case proved by Ágota P Horváth ad P Vértesi, i 2015 (see [9]) Theorem 35 Oe ca defie a g C such that for every x [ 1, 1] Let us remark that i both cases so they have the best possible order! lim L (g, e, x) = Λ (T) = 2 π log + O(1) ad Λ (e ) = 2 log + O(1), π Dolomites Research Notes o Approximatio ISSN

4 Szili Vértesi 20 4 O the covergece of a Berstei type process 41 Let be the polyomials Q 1 (f, T, x) defied accordig to S Berstei [1] ad [2] /2 Q 1 (f, T, x) = f (t 2k 1, ) l 2k 1, (T, x) + l 2k, (T, x), f C (41) (for simplicity let be eve) Actually, [1] ad [2] took the ext more geeral process Let l, q be atural umbers; for simplicity we suppose that = 2lq We divide the odes ito q rows as follows x 1 x 2 x 2l, x 2l+1, x 2l+2, x 4l, x 2l(q 1)+1, x 2l(q 1)+2, x 2lq, Accordig to the above scheme let Q l (f, T, x) Q l (f ) = (42) = f 1 (l 1 + l 2l ) + f 2 (l 2 l 2l ) + f 3 (l 3 + l 2l ) + + f 2l 1 (l 2l 1 + l 2l ) + f 2l+1 (l 2l+1 + l 4l ) + f 2l+2 (l 2l+2 l 4l ) + f 2l+3 (l 2l+3 + l 4l ) + + f 4l 1 (l 4l 1 + l 4l ) + f (2l 1) (l (2l 1) + l ) + + f 1 (l 1 + l ) q You may cosult with [1] or [2] (above f k = f (x ) ad l k l (T, x)) If N = + r, = 2lq, 0 < r < 2l, the defiitio of Q N l is as follows By the above defiitios we have with e 0 (x) 1 Q N l (f ) := Q l (f ) + Q l (e 0, x) N k=+1 f k l k l (T, x) 1, (43) Q l (f, x ) = f (x ) if k 2l, 4l,, 2lq, (44) ie Q l iterpolates at q = 2lq q odes This umber is "very close" to if the (fixed) l is big eough while q (ad, too) teds to ifiity, ie for large l our Q l is "very close" to the Lagrage iterpolatio L However, Q l coverges for every f C, whe (cf Theorem 41 ad Theorem 42), which geerally does ot hold for L 42 Actually, (43) ad (44) hold true for arbitrary poit system I [1] S Berstei proved Theorem 41 Let l be a fixed positive iteger ad f C The lim f (x) Q l(f, x) = 0 Actually, he proved for N = + r, too; the case whe N = + r demads oly small techical chages i the proof Remark 1 The Berstei process ad its geeralizatios were exhaustively ivestigated by O Kis (sometimes with coauthors) For more details we suggest the refereces i [12] 43 I [12] we geeralized Theorem 41 usig the roots of the Jacobi polyomials P (α,β) For the roots x (α,β) (1 x) α (1 + x) β P (α,β) (x) = ( 1) 2! = cos ϑ (α,β), 0 < ϑ (α,β) d < π of P (α,β) (x) we have 1 < x (α,β) d x (1 x) α+ (1 + x) β+ < x (α,β) 1, < < x (α,β) 1 < 1 (x) So let P (α,β) (x) be defied by (α, β > 1) Dolomites Research Notes o Approximatio ISSN

5 Szili Vértesi 21 Let l (α,β) P (α,β) (x) (x) = P (α,β) (x )(x x ) (k = 1, 2,, ) For a fixed positive iteger l, we defie Q (α,β) (f, x) accordig to (41) ad (42), where l l k ad f k stad for l (α,β) (x) ad f (x (α,β) ), respectively As we oticed we have the properties aalogous to (43) ad (44) for Q (α,β) (f, x), too l We proved i [12] (cf [14]) Theorem 42 Let l be a fixed positive iteger, = 2lq (q = 1, 2, ) ad f C The lim f (x) Q(α,β) (f, x) = 0 l for ay processes Q (α,β) l supposig 1 < α, β < 05 Our statemet follows from the ext more iformative poitwise estimatios Theorem 43 Let l be fixed atural umber The for arbitrary fixed α, β > 1 ad f C Q (α,β) 1 x 2 (f, x) f (x) l = O(1) ω f ; i + i2 1 2 i γ uiformly i ad x [ 1, 1], where γ = mi(2; 15 α; 15 β) ω(f ; t) is the modulus of cotiuity of f (x) i=1 Remark 2 As it is well ow, Q ( 1/2, 1/2) (f, x) = Q l l (f, T, x) For this case some improvemets of Theorem 42, icludig saturatio, were proved recetly by J Szabados [11] 44 We defie the barycetric Berstei-type operators by /2 B (f, e, x) = f e 2k 1, b2k 1, (e, x) + b 2k, (e, x) (agai, let be eve) I our forthcomig paper [13] amog others we prove as follows Theorem 44 We have lim B (f, e, x) f (x) = 0 for ay f C Moreover the sequece {B } is saturated with the order {1/}; the trivial class is the set of costat fuctios Remark 3 As far as we ow the {B } process is the first barycetric iterpolatio operator sequece which is coverget for arbitrary f C; actually Theorem 44 ca be proved for the sequeces aalogue to {Q l } eve if l > 1 5 Some ope problems ad remarks 1 Statemets aalogous to Theorems 32, 33, 35 ad 44 ca be proved for the so called Floater Horma iterpolats (see [4, 4]) 2 We ited to prove the Berstei Erdős cojecture i detail for the barycetric case The iterested reader may cosult with [17, 33] for further orietatio 3 Oe ca try to prove the almost everywhere divergece theorem of P Erdős ad P Vértesi for the barycetric case (cf [16, 25]) 4 Theorems for other "well-spaced" ode systems (see [4, 6]) will be proved icludig the {x (α,β) } systems 5 Probably it is quite difficult to get the saturatio ad saturatio class for the operator sequeces defied o other "well-spaced" odes Acowledgemets We thak the uow referee for the remarks May of them were icorporated i the paper Dolomites Research Notes o Approximatio ISSN

6 Szili Vértesi 22 Refereces [1] S Berstei Sur ue modificatio de la formule d iterpolatio de Lagrage (Frech) Commuicatios Kharkow, 5:49 57, 1932 [Collected works Vol II The costructive theory of fuctios [ ] (Russia), Izdat Akad Nauk SSSR, Moscow, 1954, ] [2] S Berstei Sur ue formule d iterpolatio (Frech) C R, 191: , 1930 [Collected works Vol I The costructive theory of fuctios [ ] (Russia), Izdat Akad Nauk SSSR, Moscow, 1952, ] [3] J-P Berrut Ratioal fuctios for guarateed ad experimetally well-coditioed global iterpolatio Comput Math Applic, 15:1 16, 1988 [4] J-P Berrut ad G Klei Recet advaces i liear barycetric ratioal iterpolatio J Comput Appl Math, 259:95 107, 2014 [5] L Bos, S De Marchi, K Horma ad J Sido Boudig the Lebesgue costat for Berrut s ratioal iterpolat at geeral odes J of Approx Theory, 169:7 22, 2013 [6] VK Dzjadik Itroductio to the Theory of Uiform Approximatio of Fuctios by Polyomials "Nauka" Moscow (1977) (Russia) [7] G Faber Über die iterpolatorische Darstellug steiger Fuktioe Jahresber der Deutsche Math Verei, 23: , 1914 [8] G Halász Oral commuicatio, October 2014 [9] Ágota P Horváth ad P Vértesi O barycetric iterpolatio II (Grüwald Marcikiewicz type theorems) Acta Math Acad Sci Hug, (to appear) [10] BA Ibrahimoglu ad A Cuyt Sharp bouds for Lebesgue costats of barycetric ratioal iterpolatio at equidistat poits Experimetal Mathematics (to appear) [11] J Szabados O a quasi-iterpolatig Berstei operator J of Approx Theory, 196:1 12, 2015 [12] L Szili ad P Vértesi O a coverget process of S Berstei Publicatios de l Istitut Mathématique, Nouvelle Série, 96(110): , 2014 [13] L Szili ad P Vértesi O barycetric iterpolatio III (i preparatio) [14] P Vértesi Lagrage iterpolatio for cotiuous fuctios of bouded variatio Acta Math Acad Sci Hug, 35:23 31, 1980 [15] P Vértesi New estimatio for the Lebesgue fuctio of Lagrage iterpolatio Acta Math Acad Sci Hug, 40:21 27, 1982 [16] P Vértesi Classical (uweighted) ad weighted iterpolatio i the book A Paorama of Hugaria Mathematics i the Twetieth Cetury I, Bolyai Society Mathematical Studies, 14:71 117, 2005 [17] P Vértesi O barycetric iterpolatio I (O the T-Lebesgue fuctio ad T-Lebesgue costat) Acta Math Acad Sci Hug, DOI: /s Dolomites Research Notes o Approximatio ISSN