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Copositive approximation by rational functions with prescribed numerator degree

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Copositive approximation by rational functions with prescribed numerator degree

    Copositive approximation by rational

    functions with prescribed numerator degree

    ApplMath.ChineseUniv

    2009,24(4):411416

    Copositiveapproximationbyrationalfunctionswith prescribednumeratordegree

    YUDanshengZHOUSongping.

    Abstract.Thepaperprovesthat,iff(x)?

    thenthereexistsarealrationalfunctionr() thatthefollowingJacksontypeestimate

    l,l,1P<oo,changessign1timesin(1,1),

    ?Rwhichiscopositivewith,(),such

    lp<(,,).

    hods,whereisanaturalnumber+1,andGisapositiveconstantdependingonlyon

    ?1Introduction

    LetL[1,1],1P<..,bethesPaces.fppowerintegrablefuncti.ns.n[1,1],andC【一l,1]

    beo1kctOfontinll0usfuI1cti.ns.n[_1.1].We.usethen0tatiOI1LlI1for

    q111.Andnormisdefinedas

    filLL:=Ilfll:=1P<oo

    P一?.

    ..

    Denotebyx)thesetofpolynomialsofdegreeatmostwithrealcoefficients,andthec1ass of(1,)typerationalfunctionby

    t1

    ,

    l

    :=

<:p舡如))

    Received:20080727

    MRSubjectClassification:41A2O.41A30

    Keywords:copositiveapproximation ,

    rationalfunctions,approximationrate

    DigitalObjectIdentifier(DOI1:10.1O07/sl176600920765 ThefirstauthorissupportedbytheNationalNaturalScienceFoundationofChina(10901044

    1andResearch

    ProjectofHangzhouNormalUniversity(YS05203154)

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    p

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    L.1

    

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    re

    412App1.Math.d.ChineseUnivVol_24,No.4

    Approximationbyreciprocalsofrealpolynomialsisaninterestingsubjectinrationalapprox

    

    imation,andsomeimportantdevelopmentshavebeenmaderecently.Researchofreciprocal

    approximationishard,sincereciprocalsarenotevenclosedunderaddition.Itisclearthatone

    cannotexpecttoapproximatearealfunctionchangingsignin[1,1byreciprocalsofreal polynomials,somostoftheresultsareachievedundertheassumptionthatfkeepsitssign

    inf1,11.Forcontinuousfunctionfon[1,1changingsign1timesin(

    1,1),Leviatanand

    Lubinsky[41haveprovedthatfcanbeapproximatedbyrationalfunctionsinRfn.

    ::l<Y1<y2<<Yz<l:=yz+i),denotebyA.()thesetofall ForYL:f

    functionsf(x)?p_]1suchthat(1),()0forz?fYk,+l,=0,1,',f.Afunction

    gissaidtobecopositivewithfiff(x)g(x)0,forall?【一1,1.

    LeviatanandLubinsky[]establishedthefollowing.

    TheoremA.Given={Yo::1<1<Y2<???<Yl<1:=Yl+lj,forevery,()?

cI1,

    11n?.(),thereexistsar()?copositivewithf(x)suchthatforevery?(1,1),

    ,()r()lC(1+1)(.,)

    Recently,ourwork[generalizesTheoremAtof-1.11spacesfor1P<o.(Meiand Zhou[.]alsogaveaslightlyweakergeneralizationin1

    ,

    l1spacesfor1<P<..)?Infact,we

    obtainedthefollowing

    TheoremB.Iff(x)?lI1],1<P<..,changessignltimesin(1,1),thenthereexists

    r(x)?Rsuchthatforsufficientlylargen,

    ll,rffp,(\,,,

    whereCp,

    6isapositivec.nstant.nlydependingonPandwith

    0

    m

    <

    in

    <f

    {l.j?.j+1l},

    a1<a2<<alaresignchangepointsof,(),ao=1,al+l=1.

    TheoremC.If.()?l'l1changessign1timesin(-1,1),thenthereexistsr()?Rsuch

    thatforsufficientlylargen,

    Cal6wf,

    Inthepresentpaper.andCalwaysdenotesanabsolutepositiveconstant,anddenotes apositiveconstantonlydependingon.Theirvaluesmaybedifferenteveninthesameline. FromtheproofofTheoremBandTheoremC,wenotethattheirconstructionofr(x) stronglydependsonthezerosofSteklovfunction,^).Although^x)hasthesamenumber

    (),thepointsarenotnecessarilythesame.Forthisreason,r(x) signchangepointsas.

    constructedtheredoesnotkeepcopositiveproperlywith,().Furthermore,itisnoteasyto locatethesignchangepointsof^(),andthisfacthassomedifficultiesinapplication.We

alsonotethatTheoremAisprovedforDitzian-Totikmoduluswhichismoreefficientthanthe

    usualmodulusofcontinuity,soitisofinteresttoreplace(.,)pbyDitzianTotikmodulus. ThepresentpaperwillfoCUSonthesetopics.Themainresultisthefollowing

    Theorem.LetZbeapositiveinteger.Given={Y0:=1<yl<2<'<yl<1:=Yl+l,,

    YUDansheng,eta1.Copositiveapproximationbyrationalfunctionswithprescribednumeratordegr

    ee413

    foreverY,(z)?L1,

    11n?.(),1P<..,thereexistsr()?Rn?.()suchthat

    forevery.

    ?刊12tLW~/,

    whereisanaturalnumbern.1essthan;+1,andom<inly1

    Set

    and

    ?2ProofofTheorem

    ,::c0s,0?J,

    1+x/1-x2

    ,

    a,b)=min{A(0),?(6))

    

    WeneedthefollowingLemmas.

    Lemma1.([2],Lemma3.3)Letnbeafixednaturalnumber,andbeapiecewiseconstant

    splinewithknotsatxj,0J.Thenforeveryinterval[a,b]c[1,1],thefollowing inequalityholds:

    )l<c(+)u(<..-

    Lemma2.GivenI()?1,

    l1n?.(),

    piecewiseconstantsplinesnwithknotsatxj satisfies

ll,IJ

    0<P<?.For>N:=Cthereexistsa

    ,0J7/,,suchthatitiscopositivewithfand /,

    Proof.ItcanbederivedfromLemma3.5of[2anditsprocedureofproof. Remark.Sn(x)alsosatisfies()=0,J=1,2,,1.

    Lemma3.([7,Lemma6)Forsufficientlargeandany1,thereexistpolynomials ofdegreenthatarepositivein[--1,1]suchthat where

    Ip(x)l

    p(x)

    <,

    f,I'\\礼鲁

    \21

    ,,\【—

    Yj)Il

    ProofofTheorem.Set

    Qx):=_尸【引()(),

    wherePf~l(x)isthepolynomialofdegreeinLemma3,anduE~l(x)ispositivepolynomial

    ofdegreechosensuchthat

    号】(X)'Sn,,)Sn,)(1)

    n

    /,,?,

    /

    414App1.Math.J.ChineseUnivV_O1.24.No.4 Fromtheresultof[1],such1(z)isobtainable.Withoutlossofgenerality,wecanassume

    thatf(x)0in(bl,1),thenwhatweneedtoveriryisthefollowinginequality:

    lI,_f,)ByLemma2,Wecangetapiecewiseconstantsplinewith itiscopositivewithlandsatisfies

l1,C6w

    Thus(2)holdsifwecanverifythat

    Notethat

    )

    Byusing(1),(3)andthe

    wehave

    Set

    where

    and

    (.,)p

    f,)

    knotsat,0JTt,suchthat

    j.Ip(x)l/

    in

    骞缛

    +

    f(ISn(X)I一丽1)I=I1+ inequality(seeLemma3)

    ll

    <l__1-1] c(s,)cf,)

    (一丽1)

    )

    

    /n,yj+/n)::J=1 7

    :=[1,1/E:=U

    J=0

    =/n]

=]

    =

    //n],.).2,l

    (5)

    圳一?一一

    S

    U

    =

    YUDan

    sheng,eta1.Copositiveapproximationbyrationalfunctionswithprescribednumeratordegr

    ee415

    NotethatSn(yj)=0,J=1,2,,1,byLemma1wehave

    Thus,wehave

    ForanyX?易,J

    /ls()s()Id?E

    <

    (+)

    P

    f,,\

    ?u

    where?dependson5,then 2

    (

    (x,yj:(nnSn,1).dx):(,=) ),nc,dz

    J=1,2,,l1

    ()2(,)I+(,)

    wherethelastinequalityfollowsfromtheinequalities(see(14)(15)of[2)

    (?()).4?()(IxYl+A()),

    1(Ix-

l+?(z))<Ix-l+?()<2(1l+?())

    ByLemma1again,for(一?)P>1,thatis,>?+1,adirectcalculationyieldsthat

    <

    p

    (,)(+

    ×

    (,)

    Xf+(,J) c;l.P

    fP

    c;l21tpP ()

    lXYjl,// n(,J)

    I+(,

    (?()))p

    )P

    ,l,J:1,2,,f1niP

    By(8)and(9),foranyx?Eo,wehave

    (,Y1)

    

    Yll+(,1) ,(一兰)p)

    \()P

    (x,))

    (,)d

    (10)

    p

    \,?/

    ln p

    \,?,/

    1n /,?, \

    p

    < PE pL ,

    Z

    n?

    +

    

    ?

    n?

    e+

    -_ 0

    一一 .

    ,2?? ?

    ?

    L

    

    ,,

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