Research Article Gauss-Lobatto Formulae and Extremal Problems

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1 Hdaw Publshg Corporato Joural of Iequaltes ad Applcatos Volume 2008 Artcle ID pages do:055/2008/ Research Artcle Gauss-Lobatto Formulae ad Extremal Problems wth Polyomals Aa Mara Acu ad Mugur Acu Departmet of Mathematcs Uversty Luca Blaga of Sbu 5 7 Io Ratu Street Sbu Romaa Correspodece should be addressed to Aa Mara Acu acuaa77@yahoocom Receved Jauary 2007; Accepted 5 December 2007 Recommeded by Jozsef Szabados Usg quadrature formulae of the Gauss-Lobatto type we gve some ew results for extremal problems wth polyomals Copyrght q 2008 A M Acu ad M Acu Ths s a ope access artcle dstrbuted uder the Creatve Commos Attrbuto Lcese whch permts urestrcted use dstrbuto ad reproducto ay medum provded the orgal work s properly cted Itroducto By we deote the space of polyomals of degree ot greater tha To obta our results we eed the followg results of Duff ad Schaeffer ad of Gautsch ad Notars 2 Lemma Duff ad Schaeffer If qx c x x s a polyomal of degree wth dstct real zeroes ad f p s such that p x q x the for k wheever q k x 0 p k x q k x 2 Lemma 2 Gautsch ad Notars A real polyomal r of exact degree 2 satsfes rx > 0 for x fadolyf rx bb 2ax 2 2cb ax a 2 c 2 3 wth 0 <a<b c <b a b / 2a

2 2 Joural of Iequaltes ad Applcatos By P αβ x where s a oegatve whole umber ad α β > we deote the th Jacob polyomal It s kow that Jacob polyomals wth the same parameters α ad β are orthogoal o wth respect to the weght fucto ρx x α x β We wll eed the followg propertes of Jacob polyomals 3: d { dx P αβ P αβ P αβ β } x α α β P αβ x 6 Let P αβ x be the Jacob polyomal of degree ormalzed to have the leadg coeffcet equal to The P αβ x 2 Γ α β! αβ Γ2 α β P x 7 From the relatos 6 ad 7weobta d { } P αβ x P αβ dx x 8 The Jacob polyomals orthogoal o wth respect to the weght fucto ρx / x 2 are the so-called Chebyshev polyomals of frst kd These polyomals are gve by T x cos arccos x x ad T /2 T are the Chebyshev polyomals of frst kd of degree wth the leadg coeffcet equal to The Jacob polyomals orthogoal o wth respect to the weght fucto ρx x2 are the so-called Chebyshev polyomals of secod kd These polyomals are gve by U x s arccos x x x 2 ad Ũ /2 U are the Chebyshev polyomals of secod kd of degree wth the leadg coeffcet equal to Letusdeotebyx cos2 π/2 the zeroes of T the Chebyshev polyomal of the frst kd The followg problem was rased by Turá Problem Let φx 0for x ad cosder the class P ϕ of all polyomals of degree such that p x ϕx for x How large ca max x p k x be f p s a arbtrary polyomal P ϕ?

3 A M Acu ad M Acu 3 He poted out two cases: ϕx x 2 ad ϕx x 2 I papers 4 5 the author cosders the soluto the weghted L 2 -orm for the maorat ϕx / x 2 Let H be the class of real polyomals p such that p x x 2 where the x are the zeroes of the Chebyshev polyomal of frst kd Note that U H From paper 5 was obtaed the followg result Theorem 3 see 5 If p H the oe has x 2 k /2 [ p k ] 2dx x k! k 2 2 3k 2π k 2! 2k 32k 2k 5 2 k 0 2 wth equalty for p U We deote by H the class of all real polyomals p such that p x 2 x 2 3 where the x are the zeroes of the Chebyshev polyomal of frst kd Note that P /2/2 H The ext theorem ca be obtaed the same way of Theorem 3 Theorem 4 If p H the oe has x 2 [ ] k /2 2dx p k x π k! 2 2 k 2! k 2 2 3k 2k 32k 2k 5 4 k 0 2 wth equalty for p P /2/2 Let H αβ be the class of real polyomals p such that p x P αβ x 5 where the x are the zeroes of P αβ Remark 5 For α β /2 the class H /2 /2 cocdes wth the class H I ths paper we wat to gve a geeralzato of these results

4 4 Joural of Iequaltes ad Applcatos 2 Quadrature formulae of the Gauss-Lobatto type I ths secto we recall some geeral cocepts about quadrature formulae ad we prove some lemmas whch help us provg our result Let b a ρxfxdx A f p a B f b Rf 2 be a quadrature formula where ρ s a oegatve weght fucto b / a b parefxed ad dstct odes The odes a a b wll be determed from the codto that the quadrature formula 2 has maxmal degree of exactess These quadrature formulae are the so-called Gauss quadrature formulae wth fxed odes The ext theorem gves the ecessary ad suffcet codto such that the quadrature formula 2 has maxmal degree of exactess Theorem 2 see 6 The maxmal degree of exactess r 2 p of quadrature formula 2 s obtaed f ad oly f the odes a are the zeroes of a orthogoal polyomal of degree wth respect to the weght fucto wx ρx p x b x a b Let b a ρxfxdx à f p a B f p b C f b Rf 22 be a quadrature formula Smlarly the ext theorem gves the ecessary ad suffcet codto such that the quadrature formula 22 has maxmal degree of exactess Theorem 22 see 6 The maxmal degree of exactess r 2 2p of quadrature formula 22 s obtaed f ad oly f the odes a are the zeroes of a orthogoal polyomal of degree wth respect to the weght fucto wx ρx p x b 2 x a b Remark 23 The coeffcets A are postve à from Gauss quadrature formulae 2 ad 22 The Gauss-Lobatto quadrature formulae are the Gauss quadrature formulae wth two fxed odes amely b a b 2 b I ths paper we wll cosder the case a b ad the weght fucto s ρx x α x β These formulae of umercal tegrato are called the Gauss-Jacob-Lobatto quadrature formulae Lemma 24 For ay gve ad k 0 k let k be the zeroes of P αkβk The the quadrature formulae k x kα x kβ k fxdx B f B 2 f A f Rf 23

5 A M Acu ad M Acu 5 where B 2 2kαβ Γk β Γ α Γ kγk β 2 24 Γ α β k 2Γ β B 2 2 2kαβ Γk α Γ β Γ kγk α 2 Γ α β k 2Γ α 25 A > 0 26 x kα x kβ k 2 fxdx B f B 2 f C f C 2 f à f Rf 27 where { k 2 k α β 3 B C 2β k 3 { k 2 k α β 3 B 2 C 2 2α k 3 } k k α β k β } k k α β k α C 2 2kαβ2 Γk β 2Γ α Γ k Γβ k 3 Γ α β k 3Γ β 20 C 2 2 2kαβ2 Γk α 2Γ β Γ k Γα k 3 Γ α β k 3Γ α 2 à > 0 22 have the degree of exactess equal to 2 2k Proof If the quadrature formula of the Gauss-type 2 we cosder a b ρx x kα x kβ k p 2 b b 2 the by Theorem 2 the quadrature formula 23 has the maxmal degree of exactess r 2 2k I order to compute the coeffcets B ad B 2 we eed the followg formulae: x α x λ P αβ m x λ x β P αβ m xdx m 2 αλ Γλ Γm α Γβ λ m λ < β Γm Γβ λγm α λ 2 23 xdx 2βλ Γλ Γm β Γα λ m λ < α 24 Γm Γα λγm β λ 2 If the quadrature formula 23 we cosder fx xp αkβk x theby k usg the relato 24 we obta 25 whle by usg fx xp αkβk x ad the k relato 23 we obta 24 If the quadrature formula of the Gauss-type 22 we cosder a b ρx x kα x kβ k 2 p 2 b b 2 the by Theorem 22 the quadrature formula 27 has maxmal degree of exactess r 2 2k

6 6 Joural of Iequaltes ad Applcatos If the quadrature formula 27 we cosder fx x x 2 P αk2βk2 x k 2 respectvely fx x 2 xp αk2βk2 x the by usg the formulae 23 ad 24 k 2 we obta the coeffcets 2 ad 20 If the quadrature formula 27 we choose fx x 2 P αk2βk2 x respectvely k 2 fx x 2 P αk2βk2 x the by usg the formulae 23 ad 24 we obta the k 2 coeffcets 29 ad 28 Lemma 25 Let rx bb 2ax 2 2cb ax a 2 c 2 be a real polyomal For ay gve ad k 0 k let k be the zeroes of P αkβk The the quadrature formulae k rx x kα x kβ k fxdx D f D 2 f A r f Rf where D 2 2kαβ Γk β Γ α Γ kγk β 2 a b c 2 Γ α β k 2Γ β D 2 2 2kαβ Γk α Γ β Γ kγk α 2 a b c 2 Γ α β k 2Γ α A > 0 rx x kα x kβ fxdx k 2 D f D 2 f G f G 2 f à r where { D C 2 b 2 2abbc ac [ k 2kαβ3 2βk3 { D 2 C 2 2 b 2 2abbc ac [ k 2kαβ3 2α k 3 f Rf 27 k kαβ2 ] a bc } 2 2kβ k kαβ2 ] a b c } 2 2k α G C a b c 2 G 2 C 2 a b c 2 à > 0 28 wth C C 2 defed 20 ad 2 have degree of exactess 2 2k Proof The proof follows drectly by replacg f wth rf Lemma 24 3 Extremal problems wth polyomals I ths secto we wat to gve exact estmatos of certa weghted L 2 -orms of the kth dervatve of polyomals whch are the class H αβ

7 A M Acu ad M Acu 7 Remark 3 Sce P αβ c P αβ for k 0 the polyomals P αkβk k zeroes k c R ad P αβ k P αkβk t follows that k P αkβk ad P αβ k have the same k Lemma 32 If p H αβ thefork 0 oe has P αβ k p k 3 wheever P αβ k 0 for k 32 p k p k k P αβ 33 k P αβ 34 Proof By the Lagrage terpolato formula based o the zeroes of P αβ we ca represet ay algebrac polyomal p by p x P αβ x x x P αβ p x x P αβ x x x p x P αβ x 35 We also have P αβ x P αβ x x x P αβ x P αβ x P αβ x 36 x x Dfferetatg wth respect to xweobta p x P αβ x x P αβ x P αβ x x x 2 x x x P αβ p x P αβ x x P αβ x x x 2 37 Sce y 0 are the zeroes of P αβ xwehave P αβ y 0 0ad p y 0 P αβ y 0 p x y 0 2 x P αβ x P αβ y 0 P αβ y 0 y 0 x 2 38

8 8 Joural of Iequaltes ad Applcatos We fd P αβ p y 0 y 0 αβ P αβ P αβ P y 0 y 0 y 0 y 0 2 x p x y 0 2 αβ x P x y 0 2 P αβ x y 0 39 Now applyg the Duff-Schaeffer lemma we have p k P αβ k k 30 By the Lagrage terpolato formula based o the zeroes k of P αβ k we ca represet the polyomals p k p k k x x ad P αβ k by kx P αβ P αβ k p k kx kx P αβ k P αβ x 3 Sce P αβ k P αβ k k 32 usg relato 3wehave p k P αβ k k P αβ k k p k P αβ k P αβ k 33 We recall that the zeroes of the orthogoal polyomal o a terval a b are real dstct ad are located the terval a b I our case we have The relato 34 ca be obtaed a smlar way so the proof s completed

9 A M Acu ad M Acu 9 Lemma 33 The followg formulae hold: k P αβ 2 k 2!Γ α β k 3Γ α Γ2 α β Γ k Γk α 3 P αβ k k 2 2 k 2!Γ α β k 3Γ β Γ2 α β Γ k Γk β 3 34 Proof Relato 8 yelds kx P αβ! P k! αkβk x 35 k The proof s completed by usg relatos 4 5 ad7 Theorem 34 If p H αβ the x kα x kβ[ ] 2dx p k x [ 2 2αβ 2! Γ2 α β [ k β 2 α β k 3 k 2 2k β 2k β 3 α β k 3 k 2 k α 2 2k α 2k α 3 holds for all k 0 2 wth equalty for p P αβ ] 2 Γ α β k 3Γ α Γ β Γ k k k α β 2 2k β k β 2 k k α β 2 2k α k α 2 ] 36 Proof Accordg to Lemma 24 ad postvty of the coeffcets the quadrature formulae we have x kα x kβ[ ] 2dx p k x B p k 2 B2 p k 2 k A p k 2 [ k ] 2 [ k ] 2 B P αβ B 2 P αβ x kα x kβ [ P αβ x k ] 2 dx B [ P αβ 2 C P αβ k 2 k ] 2 [ k ] 2 B 2 P αβ Ã [ P αβ k P αβ k ] 2 k k2 2 C 2 P αβ A [ P αβ k k P αβ ] 2 k2 37

10 0 Joural of Iequaltes ad Applcatos Sce P αβ k 0 k 2 ad by usg Lemma 33 we obta the equalty 36 Remark 35 If we choose α β /2 the above theorem we obta Theorem 4 Theorem 36 If p H αβ adfrx bb 2ax 2 2cb ax a 2 c 2 s a real polyomal wth 0 <a<b c <b a b / 2a the rx x kα x kβ[ ] 2dx p k x ] 2 Γ α β k 3Γ α Γ β [ 2 2αβ 2! Γ2 α β Γ k { 2 b 2 2abbc ac [ αβk3 k!2 k β 2 2k β 3 b c2 a k β 2 2 b 2 2ab bc ac k α 2 [ α β k 3 k 2 2k α 3 k k α β 2 2k α k k α β 2 2k β } ] a b c2 k α 2 ] 38 holds for all k 0 2 wth equalty for p P αβ Refereces R J Duff ad A C Schaeffer A refemet of a equalty of the brothers Markoff Trasactos of the Amerca Mathematcal Socety vol 50 pp W Gautsch ad S E Notars Gauss-Krorod quadrature formulae for weght fuctos of Berste- Szegö type Joural of Computatoal ad Appled Mathematcs vol 25 o 2 pp G Szegö Orthogoal Polyomals vol23ofcolloquum Publcatos Amerca Mathematcal Socety Colloquum Publcatos Provdece RI USA 4th edto I Popa Markoff-type equaltes weghted L 2 -orms Joural of Iequaltes Pure ad Appled Mathematcs vol 5 o 4 artcle 09 pp I Popa Quadratures formulae ad weghted L 2 -equaltes for curved maorat Mathematcal Aalyss ad Approxmato Theory Proceedgs of the 6th Romaa-Germa Semar o Approxmato Theory ad Its Applcatos pp Medamra Scece Basoara Romaa Jue D D Stacu G Coma ad P Blaga Aalză Numercăş Teora Aproxmăr Vol II Presa Uverstară Clueaă Clu-Napoca Romaa 2002