En=

INEQUALITYFOREXPONENTIALSUMS

´sErd´PeterBorweinandTamaelyi

DedicatedtoProfessorGeorgeG.Lorentzontheoccasionofhis85-thbirthday.

AsubtleBernstein-typeextremalproblemissolvedbyestablishingthe

sup

0=f∈E2n

equality

|f󰀂(0)|

n−1

e−1

kf󰀋[a,b]

≤

2n−1

1991MathematicsSubjectClassification.Primary:41A17.

Keywordsandphrases.BernsteinInequality,ExponentialSums.

Researchofthefirstauthorsupported,inpart,byNSERCofCanada.Researchofthesecondauthorsupported,inpart,byNSFunderGrantNo.DMS-9024901andconductedwhileanNSERCInternationalFellowatSimonFraserUniversity.

TypesetbyAMS-TEX

1

2

´ERDELYI´PETERBORWEINANDTAMAS

1.Introduction

In“NonlinearApproximationTheory”,Braess[3]writes“Therationalfunctions

andexponentialsumsbelongtothoseconcretefamiliesoffunctionswhicharethemostfrequentlyusedinnonlinearapproximationtheory.Thestartingpointofconsiderationofexponentialsumsisanapproximationproblemoftenencounteredfortheanalysisofdecayprocessesinnaturalsciences.Agivenempiricalfunctiononarealintervalistobeapproximatedbysumsoftheform

n󰀒j=1

ajeλjt,

wheretheparametersajandλjaretobedetermined,whilenisfixed.”

Theaimofthispaperistoprovethe“right”Bernstein-typeinequalityforex-ponentialsums.Thisinequalityisthekeytoprovinginversetheoremsforapprox-imationbyexponentialsums,aswewillelaboratelater.Let

En:=

󰀎

f:f(t)=a0+

n󰀒j=1

󰀏

ajeλjt,

aj,λj∈R.

SoEnisthecollectionofalln+1termexponentialsumswithconstantfirstterm.

Schmidt[10]provedthatthereisaconstantc(n)dependingonlyonnsothat󰀍f󰀅󰀍[a+δ,b−δ]≤c(n)δ−1󰀍f󰀍[a,b]󰀁

foreveryp∈Enandδ∈0,1

2,thereisaconstantc(α)dependingonlyonαso

thatc(n)intheaboveinequalitycanbereplacedbyc(α)nαlogn(Xuimprovedthistoallowα=1

kf󰀍[a,b]

≤

2n−1

n−1

e−1

kf󰀍[a,b]

,y∈(a,b)

isestablishedinTheorem3.3.Theorem3.2followseasilyfromourothercentralresult,Theorem3.1.Thisstatesthattheequality

sup

0=f∈E2n

|f󰀅(0)|

ASHARPBERNSTEIN-TYPEINEQUALITY3

holds,where

󰀎

󰀔2n:=E

f:f(t)=a0+

n󰀒󰀁j=1

ajeλjt+bje

󰀂−λjt

󰀏

,

aj,bj,λj∈R.

TheseresultscomplementNewman’sbeautifulMarkov-typeinequality[9],see

also[2],thatstates

2

󰀍f󰀍[0,∞)

≤9

n󰀒j=0

λj,

whereEn(Λ):=span{e−λ0t,e−λ1t,...,e−λnt}foranysequenceΛofdistinctnon-negativenumbersλj.

DenotebyPnthesetofallpolynomialsofdegreeatmostnwithrealcoefficients.

Bernstein’sclassicalinequalitystatesthefollowing.

Proposition1.1(Bernstein’sInequality).Theinequality

|p󰀅(x)|≤

holdsforeveryp∈Pn.

Thisimpliesbysubstitutionandscaling(thoughnotentirelyobviously)that

|f󰀅(y)|≤

2n

n

󰀍p󰀍[−1,1],

1−x2

−14

´ERDELYI´PETERBORWEINANDTAMAS

Soonededucesinastandardfashion[4,6],forexample,thatifthereisasequence

(fn)∞fonanintervaln=1ofexponentialsumswithfn∈Enthat󰀁−mapproximates󰀂[a,b]uniformlywitherrors󰀍f−fn󰀍[a,b]=on,m∈N,thenfismtimescontinuouslydifferentiableon(a,b).

ThefollowingslightimprovementofBernstein’sinequalitymaybefoundinNatanson[8].

Proposition1.2.Theinequality

|p󰀅(0)|≤(2n−1)󰀍p󰀍[−1,1]

holdsforeveryp∈P2n.

NotethatProposition1.1impliesProposition1.2onlywith(2n−1)replacedby2n.ThefollowingcorollaryofProposition1.2canbeobtainedbyalineartransformation.ItplaysanimportantroleintheproofofTheorem3.1.Proposition1.3.Theinequality

|p󰀅(x)|≤

2n−1

󰀋1/p

󰀁

foreveryf∈En,p∈(0,2],andδ∈0,1

δ

󰀍f󰀍Lp[a,b]

ASHARPBERNSTEIN-TYPEINEQUALITY5

areusedthroughoutthispaperformeasurablefunctionsfdefinedonameasurablesetA⊂Randforp∈(0,∞).IfA:=[a,b]isaninterval,thenthenotationLp[a,b]:=Lp(A)isused.

󰀔2naredefinedintheIntroduction.TheclassesE∗andE∗cTheclassesEnandEnn

areintroducedinSection8.

Thespaceofallreal-valuedcontinuousfunctionsdefinedonA⊂RequippedwiththeuniformnormisdenotedbyC(A).

3.NewResults

Theorem3.1.Wehave

sup

0=f∈E2n

|f󰀅(0)|

󰀍f󰀍[a,b]

≤

2n−1

n−1

e−1

holdsforeveryn∈Nandy∈(a,b).Theorem3.4.Theinequality

󰀍f󰀍[a+δ,b−δ]≤22/p

2

kf󰀍[a,b]

󰀊

n+1

󰀂(b−a).2

4.ChebyshevandDescartesSystems

Theproofofourmainresultreliesheavilyontheobservationthatforevery

0<λ0<λ1<···,

(sinhλ0t,sinhλ1t,...)isaDescartessystemon(0,∞).InthissectionwegivethedefinitionsofChebyshevandDescartessystems.TheonlyresultofthissectionthatisnottobefoundinstandardsourcesisthecriticalLemma4.5.Theremainingtheorycanbefoundin,forexample,[5]or[4].

6

´ERDELYI´PETERBORWEINANDTAMAS

Definition4.1(ChebyshevSystem).LetI⊂Rbeaninterval.The

sequence(f0,f1,...,fn)iscalleda(real)Chebyshevsystemofdimensionn+1onIiff0,f1,...,fnarereal-valuedcontinuousfunctionsonI,span{f0,f1,...,fn}overRisann+1dimensionalsubspaceofC(A),andanyf∈span{f0,f1,...,fn}thathasn+1distinctzerosonIisidenticallyzero.

If(f0,f1,...,fn)isaChebyshevsystemonI,thenspan{f0,f1,...,fn}iscalledaChebyshevspaceonI.

Thefollowingsimpleequivalencesarewellknownfactsoflinearalgebra.Proposition4.2.Letf0,f1,...,fnbereal-valuedcontinuousfunctionsonanin-tervalI⊂R.Thenthefollowingareequivalent.

a]Every0=p∈span{f0,f1,...,fn}hasatmostndistinctzerosonI.

b]Ifx0,x1,...,xnaredistinctelementsofIandy0,y1,...,ynarerealnumbers,thenthereexistsauniquep∈span{f0,f1,...,fn}sothat

p(xi)=yi,

i=1,2,...,n.

c]Ifx0,x1,...,xnaredistinctpointsofI,then

󰀇

󰀇f0(x0)...󰀋󰀊󰀇

f0f1...fn󰀇...:=󰀇.D..x0x1...xn󰀇

󰀇f0(xn)...

󰀇

fn(x0)󰀇󰀇

󰀇..=0..󰀇󰀇

fn(xn)󰀇

Definition4.3(DescartesSystem).Thesystem(f0,f1,...,fn)issaidtobeaDescartessystem(orordercompleteChebyshevsystem)onanintervalIifeachfi∈C(I)and󰀋󰀊

fi0fi1...fim

>0D

x0x1...xmforany0≤i0Thisisapropertyofthebasis.ItimpliesthatanyfinitedimensionalsubspacegeneratedbysomebasiselementsisaChebyshevspaceonI.WeremarkthetrivialfactthataDescartessystemonIisaDescartessystemonanysubintervalofI.Lemma4.4.Thesystem

(eλ0t,eλ1t,...),

λ0<λ1<···

isaDescartessystemon(−∞,∞).Inparticular,itisalsoaChebyshevsystemon(−∞,∞).

Proof.ThedeterminantinDefinition4.3isageneralizedVandermonde.See,forexample,[5,p.9].󰀁

ThefollowinglemmaplaysacrucialroleintheproofofTheorem3.1.

ASHARPBERNSTEIN-TYPEINEQUALITY7

Lemma4.5.Suppose0<λ0<λ1<···.Then

(sinhλ0t,sinhλ1t,...)

isaDescartessystemon(0,∞).

Proof.Let0≤i0(sinhλi0t,sinhλi1t,...,sinhλimt)

isaChebyshevsystemon(0,∞).Indeed,let

0=f∈span{sinhλi0t,sinhλi1t,...,sinhλimt}.

Thenandsince

0=f∈span{e±λi0t,e±λi1t,...,e±λimt}

span{e±λi0t,e±λi1t,...,e±λimt}

isaChebyshevsystem,fhasatmost2m+1zerosin(−∞,∞).Sincefisodd,ithasatmostmzerosin(0,∞).

Sinceforevery0≤i0󰀋󰀊

sinhλi0tsinhλi1t...sinhλimtD

x0x1...xmisnon-zeroforany0remainstoprovethatitispositivewhenever0sinhλi0tsinhλi1t...sinhλimt

D(α):=D

αx0αx1...αxmand

D(α):=D

∗

󰀊1

2e

λi1t

...

1

D∗(α)

Since

(eλi0t,eλi1t,...,eλimt)

=1.

8

´ERDELYI´PETERBORWEINANDTAMAS

isaDescartessystemon(−∞,∞),D∗(α)>0foreveryα>0.Sotheabovelimit

relationsimplythatD(α)>0foreverylargeenoughα,henceforeveryα>0.Inparticular,

󰀊

D(1)=D

sinhλi0tsinhλi1t...x0x1...

sinhλimt

xm

󰀋>0,

whichfinishestheproof.󰀁

ThefollowingpropositiononpolynomialsfromthespanofaDescartessystemwithamaximalnumberofzerosisalsoneededinthenextsection.Itsproofisstandard,andcanbepiecedtogetherfrom[5,pp.25-36]orfoundin[2,p.108].Lemma4.6.Suppose(f0,f1,...,fn)isaDescartessystemon[a,b].Suppose

aThenthereexistsaunique

p=fn+

sothat(1)p(ti)=0,

i=1,2,...,n.

n−1󰀒i=0

aifi,ai∈R

Further,thispsatisfies(2)p(t)=0,

t∈/{t1,t2,...,tn},

(3)p(t)changessignateacht1,t2,...,tn,(4)aiai+1<0,

i=0,1,...,n−1,an:=1,

(5)p(t)>0fort∈(tn,b],and(−1)np(t)>0fort∈[a,t1),(6)(−1)n−ip(t)>0,

t∈(ti,ti+1),i=1,2,...,n−1.

5.ChebyshevPolynomialsforspan{sinhλ0t,sinhλ1t,...,sinhλnt}Westudythespace

Hn:=span{sinhλ0t,sinhλ1t,...,sinhλnt},

where

0<λ0<λ1<···<λn.

WecandefinethegeneralizedChebyshevpolynomialTnforHnon[0,1]bythe

followingthreeproperties:(5.1)

Tn∈span{sinhλ0t,sinhλ1t,...,sinhλnt},

ASHARPBERNSTEIN-TYPEINEQUALITY9

thereexistsanalternationsequence(x0󰀍Tn󰀍[0,1]=1.

(−1)iTn(xi)=󰀍Tn󰀍[0,1],

i=0,1,...,n,

TheexistenceanduniquenessofsuchaTnfollowsfromthepropertiesofthebestuniformapproximationtosinhλ0ton[󰀄,1]fromann-dimensionalChebyshevspaceon[󰀄,1](󰀄>0issufficientlysmall).See[5,p.35],forexample.

ThefollowingextremalpropertyoftheChebyshevpolynomialTnwillbeneededinthenextsection.

Theorem5.1.Usingthenotationabove,wehave

sup

|p󰀅(0)|

󰀍Tn󰀍[0,1]

󰀅=Tn(0).

0=p∈Hn

Proof.Supposep∈Hnwith󰀍p󰀍[0,1]<1andp󰀅(0)>0.ObservethatTn−p

hasatleastonezeroineachoftheintervals(x0,x1),(x1,x2),...(xn−1,xn),where(x0󰀅

(0)wouldimplythatTn−phasatleastonezeroin(0,x0),thereforep󰀅(0)>Tn

0=Tn−p∈Hnhasatleastn+1zerosin(0,1),whichisimpossible.󰀁

6.AComparisonTheorem

TheheartoftheproofofTheorem3.1isthefollowingcomparisontheorem,whichcanbeprovedbyazerocountingargument.ThemethodofourproofisverysimilartothatofacomparisontheremofPinkusandSmith[11]forDescartessystems.Infact,thesimpleproofofTheorem6.1wassuggestedbyAllanPinkus.Theorem6.1.Let

0<λ0<λ1<···<λn

Supposeγi≤λiforeachi.Let

Hn:=span{sinhλ0t,sinhλ1t,...,sinhλnt}

and

Gn:=span{sinhγ0t,sinhγ1t,...,sinhγnt}.

Then

0=p∈Hn

and

0<γ0<γ1<···<γn.

max

|p󰀅(0)|

󰀍p󰀍[0,1]

.

10

´ERDELYI´PETERBORWEINANDTAMAS

Proof.Wehave

0=p∈Hn

sup

|p󰀅(0)|

󰀍Tn󰀍[0,1]

,

whereTnistheChebyshevpolynomialforHnon[0,1].Inparticular,Tnhasn

distinctzerosin(0,1).Let

Tn(t)=:

n󰀒j=0

cjsinhλjt,cj∈R.

ByLemma4.6,(−1)jcj>0.Letk∈{1,2,...n}befixed.Let(γj)nj=0besuchthat

γ0<γ1<···<γn,

γj=λjforj=k,

λk−1<γk<λk

(weletγ−1:=0).Toprovethistheorem,itissufficienttostudytheabovecase

sincethegeneralcasefollowsfromthisbyafinitenumberofpairwisecomparisons.Lett1Qn(x)=

sothat

Qn(ti)=Tn(ti),

i=0,1,...,n.

BytheuniqueinterpolationpropertyofChebyshevspaces,Qnisuniquelydeter-mined,hasnzeros(thepointst1,t2,...,tn),andispositiveatt0.ByLemma4,6,(−1)jdj>0foreachj=0,1,...,n.Wehave

(Tn−Qn)(t)=cksinhλkt−dksinhγkt+

n󰀒j=0,j=k

n󰀒j=0

djsinhγjt,dj∈R

(cj−dj)sinhλjt.

ThefunctionTn−Qnchangessignon(0,∞)strictlyatthepointsti,i=0,1,...,n,

andhasnootherzeros.Also,byLemma4.5,

(sinhλ0t,sinhλ1t,...,sinhλk−1t,sinhγkt,sinhλkt,sinhλk+1t,...,sinhλnt)isaDescartessystemon(0,∞).Hence,byLemma4.6,thesequence

(c0−d0,c1−d1,...,ck−1−dk−1,−dk,ck,ck+1−dk+1,...,cn−dn)strictlyalternatesinsign.Since(−1)kck>0,thisimpliesthat

(−1)n(Tn−Qn)(t)>0,

Thusfort∈(tj,tj+1)wehave

(−1)jTn(t)>(−1)jQn(t)>0,

j=−1,0,1,...,n,t>tn.

ASHARPBERNSTEIN-TYPEINEQUALITY11

wheret−1:=0andtn+1:=∞.Inaddition,werecallthatQn(0)=Tn(0)=0andQn(t0)=Tn(t0)>0.

Theobservationsaboveimplythatift0∈(0,x0)issufficientlycloseto0,then

󰀍Qn󰀍[0,1]≤󰀍Tn󰀍[0,1]=1

Thus

and

󰀅

Q󰀅n(0)≥Tn(0)>0.

|Q󰀅n(0)|

󰀍Tn󰀍[0,1]

=

0=p∈Hn

sup

|p󰀅(0)|

∈(f(t)

−f(−t)).

Observethat

g∈span{sinhλ1t,sinhλ2t,...,sinhλnt}.

Itisalsostraightforwardthat

g󰀅(0)=f󰀅(0)

Foragiven󰀄>0,let

Gn,󰀃:=span{sinh󰀄t,sinh2󰀄t,...,sinhn󰀄t}

and

󰀅󰀆

Kn,󰀃:=sup|h󰀅(0)|:h∈Gn,󰀃,󰀍h󰀍[0,1]=1.

and

󰀍g󰀍[0,1]≤󰀍f󰀍[−1,1].

ByTheorem6.1,itissufficienttoprovethatinf{Kn,󰀃:󰀄>0}≤2n−1.Observethateveryh∈Gn,󰀃isoftheform

h(t)=e−n󰀃tP(e󰀃t),

P∈P2n.

12

´ERDELYI´PETERBORWEINANDTAMAS

Therefore,usingProposition1.3combinedwithalineartransformationfrom[−1,1]

to[e−󰀃,e󰀃],weobtainforeveryh∈Gn,󰀃that

|h󰀅(0)|=|󰀄P󰀅(1)−n󰀄P(1)|

󰀄(2n−1)≤

󰀋

+n󰀄en󰀃󰀍h󰀍[−1,1]−󰀃1−e

󰀊

󰀄(2n−1)=

󰀋

1−e−󰀃

+n󰀄en󰀃.

Soinf{Kn,󰀃:󰀄>0}≤2n−1,andtheresultfollows.Nowweprovethat

sup

0=f∈E2n

|f󰀅(0)|

e󰀃−1

−

1

󰀊

e−1

and

Rn(t):=Tn

󰀊

e

exp

t−b

e−1󰀋−

1

󰀋

a−y

ASHARPBERNSTEIN-TYPEINEQUALITY13

ObviouslyQn,Rn∈Enand

|Q󰀅n(y)|

e−11

y−an

kRn󰀍[a,b]

=

foreveryy∈(a,b).Theproofisnowcomplete.󰀁

ProofofTheorem3.4.WithoutlossofgeneralitywemayassumethatΛ:=(λj)nj=1

isasequenceofdistinctnon-zerorealnumbers.Forthesakeofbrevity,letEn(Λ):=span{1,eλ1t,eλ2t,...,eλnt}.

󰀃1󰀄on−2satisfyingTakeanorthonormalsequence(Lk)nk=0(1)Lk∈span{1,eλ1t,eλ2t,...,eλkt},and(2)

󰀑1/2

−1/2

k=0,1,...,n

LiLj=δi,j,

0≤i≤j≤n,

whereδi,jistheKroneckersymbol.Onwritingf∈En(Λ)asalinearcombination

ofL0,L1,...,L󰀃n,1and󰀄usingtheCauchy-Schwarzinequalityandtheorthonormalityn

of(Lk)k=0on−2,weobtaininastandardfashionthat

max

|f(t0)|

0=f∈En(Λ)

∈,

1

󰀌

kf󰀍L2[−1/2,1/2]

=

n󰀒k=0

󰀍1/2

L2k(t0)

≤

√

󰀍f󰀍L2[−1,1]

≤

√

󰀍f󰀍Lp[−2,2]

Then

max

|f(y)|

∈−|y|

󰀋1/p

.

0=f∈En(Λ)

≤21/pC,

y∈[−1,1].

14

´ERDELYI´PETERBORWEINANDTAMAS

Therefore,foreveryf∈En(Λ),

|f(0)|≤

√n+1≤√

󰀁󰀂1−p/2

n+121/pC󰀍f󰀍Lp[−2,2]

√1/p−1/2

√

󰀈

2−p󰀍f󰀍pLp[−1,1]󰀍f󰀍[−1,1]

󰀉1/2

=2

󰀍f󰀍Lp[−2,2]

≤2

1/p−1/2

󰀂

δt+t0∈En(Λ),∈

weobtain

󰀍f󰀍[a+δ,b−δ]≤22/p

2

−1/p

(n+1)1/p

󰀊

2

2(b

−a)).

Proof.Notethatf∈En(Λ)impliesf󰀅∈En(Λ).ApplyingTheorem3.4tof󰀅with

p=1,weobtain

|f󰀅(0)|≤2(n+1)󰀍f󰀅󰀍L1[−2,2]=2(n+1)Var[−2,2](f)≤4(n+1)2󰀍f󰀍[−2,2]

foreveryf∈En(Λ).Nowiff∈En(Λ)andt0∈[a+δ,b−δ],thenonapplyingtheaboveinequalityto

g(t):=f

󰀁1

ASHARPBERNSTEIN-TYPEINEQUALITY15

Remark8.2.Theorems3.2and3.4triviallyextendtotheclasses

󰀎

∗En:=

f:f(t)=

l󰀒j=1

Pkj(t)eλjt,

λj∈R,Pkj∈Pkj,

l󰀒j=1

󰀏

(kj+1)=n.

Remark8.3.Theorem3.4extendstotheclasses

󰀎

∗c

:=En

f:f(t)=

l󰀒j=1

Pkj(t)eλjt,

λj∈C,

c

Pkj∈Pk,j

l󰀒j=1

󰀏

(kj+1)=n,

c

denotesthefamilyofallpolynomialsofdegreeatmostkjwithcomplexwherePkj

coefficients.Thisfollowsbytrivialmodificationsoftheproof.

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4.DeVore,R.A.&G.G.Lorentz,ConstructiveApproximation,Springer-Verlag,Berlin,1993.5.Karlin,S.&W.J.Studden,TchebycheffSystemswithApplicationsinAnalysisandStatistics,Wiley,NewYork,N.Y.,1966.6.Lorentz,G.G.,ApproximationofFunctions,2nded.,Chelsea,NewYork,N.Y.,1986.7.Lorentz,G.G.,Notesonapproximation,J.Approx.Theory56(19),360–365.8.Natanson,I.P.,ConstructiveFunctionTheory,Vol.1,,Ungar,NewYork,N.Y.,19.9.Newman,D.J.,DerivativeboundsforM¨untzpolynomials,J.Approx.Theory18(1976),360–362.10.Schmidt,E.,ZurKompaktheitderExponentialsummen,J.Approx.Theory3(1970),

445–459.11.Smith,P.W.,AnimprovementtheoremforDescartessystems,Proc.Amer.Math.Soc.

70(1978),26–30.