复平面上扩充Hermite-Fejer插值多项式的收敛性

复平面上扩充Hermite-Fejer插值多项式的收敛性 复平面上扩充Hermite-Fejer插值多项式的 收敛性 总29卷第2期 1996年6月 数学研究 Journalof~,lathemagcalStudy V0J29blo.2 Jun.1996 THECONVERGENCEPR0BLEMoFM0D??ED HERM/TE-FF_JER?rERP0LAT10NPOINo??AL INTHECOMPLEXPLANE国 . 踟?阳咖 (2~aangzlaouInstituteofsurveyingendMapping,450052) AbstractInthispaper’modifiedHetmit+-t~j+r啦onpoIyl?1ialinthe?plm-mw酗 sn曲edThe?r-cI心mthatmp.帅m砒do晦notconvergem~tm’mlyf(z) ?A(1zl?1)w酗曲taiI’ed AndtheorderofapproximationthatthepolynomialConvergestof(z)?A(IzI?1)in也eSenseofrr舢was given. KeywoodsInterpolationt3olyPer.westudymodifiedHermide-Fejerinterpo lationpolyaomi~thatomitsthederivateat apointwithorderu(n)=2n一2suchthat @Seeeived30Oct.1995 第2期zhua删1gqiIl吕:Theo叽v盯目帅?ProblelrtofModified?l3? ―】(,,)一,(),—J,2,…,n.(7) (.1一(,,)一0一12,…,一1,+1…,.(8) Frornt1],thepolynommlsaceiIlg(7)and(8)withorderp(再)一2再一2is )(,,z)一—l(,,) +(2州))2(cd(z~娄(9) Itisknown(see[2,31)thatLalgra~geinterpolationpoIialandHmnjte—FejerinterI~lation polynomialdoesnotoonvergetof(z)?A(1zl?1)inthesenseofc0nver直enoeuniformlyonIzI?1? From[],wehavefor,(z)?(Iz1?1) I1(,,z)一,(z)ll,h一1(,,z)一,(z)dz1),?%(,,l/再),0<<+.. IH~.-t(,,)一,()I?%’,,/, (10) (11) where(,,1,再)denote~themodulusofcontkntous,0nIzI1andDis妇 luteconstant— InchIspaper,wearegoingtogivethec0nve理_e)一+.._(19) Lemma2[5] O(1nn) . , Le岫a]ForanypolynomialP.一l(z),0<P<+o.,wehave . 圳詈骞 Lenmm4 flfI(z)I,Idzl?詈,l<p<+o.(22) ProofFrom(2)and(3),weknowthat(z)isapolynomialwithordern一1and )一f.’峥 Hence,u血gLemma3,,?ehave . ?1,l如l?导(一詈.m—i— Itiseasytocalculate 【enm5 骞=:=黑, 3ProofofTheorelns Proofoftheorem1.From(3).Wehave)一一,()墨n(曩一1)_..Using(1),(2), ()and(9),wehave )一窨一引() +(z一(北耋) 一 ?c耋-cz--z~, 第2期zhu(~angqtng:TheConvta-ge~hobletnModified……’15’ +(z-z,若c耋(27 Becallseof 1()1?1,.()1?3抽,lz一I?百I, wehave l一’兰箍而l—i2?南?导, Hcnc~wehave l}?三二()砉)l?!.}?2???譬?i8?(2s) Consuque.ntlycombming(25),(27),(28)with(24),we0btain02)? This0rovestheThc~orem1completely. ?fofthcoren~s2.FromJacksontheoremU3,廿Ieexig恒apoiynom~口一一20)withordert一2such that n1a)(1,(;)一口.一2()l?踟(,,l,-)(29) Supposing0.-2(z)一q一2一.+q3一.+…%n0,then;Lemma6wehave ?口.一2()=?(2一’+s一+…+a0z.)一0?(30) I/sins(30),(29),(24),wehave 日?(,,z)一,():H—l(,,z)一,(z) 一,)0~(z0一..(31) Suppling,>l,wehave ? 16?数学研究996妊 IIz‖)(,,z)一,()II,?I1Ha一(,,z)一,(:)I1, +It?,(:)(,()一一 :())II,(32).. Byusing(10),wehave lIH一(,,z)一,(=)?mU,1/再),(33) Busing(29)andLemma4,wehave II---1?(:)奎t-i (,()一.卜z())II,?mu,l/再)一C#-u/r~o(f,1,霄), (34) Bycombining(33),(34)with(32),weobtain(14). Suppose0<p?1.r=suchm砒?+{=lIthenusingHoldexineqality胁dthe resultforp>1,wehave 【In.(.)(,,z)一,(z)Id4?(1In?(,,z)一,(z)I,I出1)?(Il出1) I一I=I矗一 ?(目一(?(,,i/目)))’–0(m(,,i1,0)’, HenOe,vehave H?(,,z)一,ithlzl<l,wehave (36) 日.(,一f(z)一』堡一0,一..).(37)I刳一i torah’ruing(36),(37)with(35),forzwithIzI<1weobtain 日?(,,z)一,(z),(再一?) Heno~weobtaintheresuRthat日-](,,z)eonverge~uniformlyto,(z)?A?1)in?<1. 第2期uChangqing:TlaeC?v日ProblemModified?17? Thispa’ovestheTheorem3completety References 1TuranRAremarkonHermite-Fejertntel’polation.A/’lri.urIiv..Buddpeat. EOtVOlllSeeuMatl~1960.3‖),60, 66 2CmapHOBB.M.HJlo6e/leBIOH.A.,KoHc’rpykTnBHanTc~op/laTeotm 中印a~3Mmllzk-CHOro .一几1964 3WalshJ.L?Interpolationandapproximationbyrationfunction-mtbeeomI~ explane.Amer.Math.S0c.Pr.vn啦 &L1969 5 6 7 st盥?xC.TheorderofapproxSmation时(0,1,….q)Hecmite-F~.m‖int=a’polati~po】yn0mi8l血1too协ofunity (1aine~AnnalsMathemati~.1992,13A(2)i146,测绘学院.邦卅I市450052) 摘要得到了复平面上忽略—点导散要求的扩充Z-Imna.e-Fejcr插值多项式在?1上不一致收敛于, (EA(?1)的结论,并得到了其平均收敛阶和内闭一致收敛性. 关键调插值多项式,一致收敛,平均收敛,逼近阶