Structural theorems for quasiasymptotics of Schwartz distributions

unwieldycodpieceΗλεκτρονική - Συσκευές

8 Οκτ 2013 (πριν από 3 χρόνια και 10 μήνες)

128 εμφανίσεις

StructuraltheoremsforquasiasymptoticsofSchwartz
distributions
JassonVindas
jvindas@math.lsu.edu
LouisianaStateUniversity
InternationalConferenceonGeneralizedFunctionsGF07
MathematicalResearchInstituteandConferenceCenter
Bedewo,Poland,September7,2007
StructuraltheoremsforquasiasymptoticsofSchwartzdistributionsp.1/19
Summary
Theaimofthistalkistocommunicatenewstructuraltheorems
forquasiasymptoticsofSchwartzdistributions.

Reviewofthedenitionofquasiasymptoticsatinnityand
attheoriginandtheknownproperties.

Integrationofthequasiasymptoticandrelationshipwith
asymptoticallyandassociateasymptoticallyhomogeneous
functions.

StructuralTheoremsforquasiasymptoticsatinnityandat
theorigin.

Particularcase:thequasiasymptoticoforder-1atinnity.

Consequence:Characterizationofjumpbehaviorof
FourierseriesintermsofCesarosummability.

Evaluationofdistributionsinthee.v.Cesarosense.

Consequence:PointwiseFourierinversionformula.
StructuraltheoremsforquasiasymptoticsofSchwartzdistributionsp.2/19
Notation

DandD

denotetheSchwartzspacesoftestfunctionsand
distributions.

SandS

arethespacesofrapidlydecreasingfunctionsand
thespaceoftempereddistributions.

Allofourfunctionsanddistributionsareovertherealline.

TheFouriertransforminSisdenedas
ˆ
φ(x)=
￿

−∞
φ(t)e
ixt
dt.
StructuraltheoremsforquasiasymptoticsofSchwartzdistributionsp.3/19
SlowlyVaryingFunctions
Recallthatreal-valuedmeasurablefunctiondenedinsome
intervaloftheform[A,∞),A>0,iscalledslowlyvarying
functionatinnityifLispositiveforlargeargumentsand
lim
x→∞
L(ax)
L(x)
=1,
foreacha>0.
Similarlyonedenesslowlyvaryingfunctionsattheorigin.
StructuraltheoremsforquasiasymptoticsofSchwartzdistributionsp.4/19
Quasiasymptoticatinnity
LetLbeslowlyvarying.Wesaythatf∈D

hasquasiasymptotic
behavioratinnityinD

withrespecttoλ
α
L(λ),α∈R,iffor
someg∈D

andeveryφ∈D,
lim
λ→∞
￿
f(λx)
λ
α
L(λ)
,φ(x)
￿
=hg(x),φ(x)i.
Wealsosaythatfhasquasiasymptoticoforderαatinnitywith
respecttoL.
Wealsoexpressthisby
f(λx)=λ
α
L(λ)g(x)+o(λ
α
L(λ)),λ→∞inD

.
Wemayalsohave
f(λx)=λ
α
L(λ)g(x)+o(λ
α
L(λ)),λ→∞inS

.
StructuraltheoremsforquasiasymptoticsofSchwartzdistributionsp.5/19
QuasiasymptoticattheOrigin
Similarly,onedenesthequasiasymptoticinD

andS

atthe
origin.

Byshifting,onecandenethequasiasymptoticof
distributionsatanypoint.

Forexample,ojasiewiczdenedthevalueofadistribution
f∈D

atthepointx
0
asthelimit
f(x
0
)=lim
ε→0
f(x
0
+εx),
ifthelimitexistsinD

.

Notation:Iff∈D

hasavalueγatx
0
,wesaythat
f(x
0
)=γinD

.Themeaningoff(x
0
)=γinS

,...,mustbe
clear.
StructuraltheoremsforquasiasymptoticsofSchwartzdistributionsp.6/19
Previousknownpropertiesatinnity

Iff∈D

hasquasiasymptoticatinnityinD

.Then,f∈S

.

Structuraltheoremswhentheorderofthequasiasymptotic
α/∈−N.

IffhasquasiasymptoticinD

whoseorderisnotanegative
integer,thenfhasthesamequasiasymptoticinS

.For
α∈−NtheresultwasknownonlyundertheassumptionL
bounded.
StructuraltheoremsforquasiasymptoticsofSchwartzdistributionsp.7/19
Previousknownpropertiesattheorigin

StructuralTheoremforα>0.

StructuralTheoremforα∈(−1,0]undertheassumptionL
bounded.

Iff∈S

hasquasiasymptoticattheorigininD

,thenithas
thesamequasiasymptoticinS

inthefollowingtwocases,

α≤0andα/∈−N.

α>0andLbounded.
StructuraltheoremsforquasiasymptoticsofSchwartzdistributionsp.8/19
IntegrationoftheQuasiasymptotic
Suppose
f(λx)=L(λ)g(λx)+o(λ
α
L(λ)),inD

,
(hereλ→∞or0).SupposethatgadmitsaprimitiveG
k
oforder
kwhichishomogeneousofdegreek+α.Then,foranygiven
k-primitiveF
k
off,thereexistfunctionsb
0
,...,b
k−1
,suchthat
F
k
(λx)=L(λ)G
k
(λx)+
k−1
￿
j=0
λ
α+k
b
j
(λ)
x
k−1−j
(k−1−j)!
+o
￿
λ
α+k
L(λ)
￿
,inD

,
StructuraltheoremsforquasiasymptoticsofSchwartzdistributionsp.9/19
IntegrationoftheQuasiasymptotic
Suppose
f(λx)=L(λ)g(λx)+o(λ
α
L(λ)),inD

,
(hereλ→∞or0).SupposethatgadmitsaprimitiveG
k
oforder
kwhichishomogeneousofdegreek+α.Then,foranygiven
k-primitiveF
k
off,thereexistfunctionsb
0
,...,b
k−1
,suchthat
F
k
(λx)=L(λ)G
k
(λx)+
k−1
￿
j=0
λ
α+k
b
j
(λ)
x
k−1−j
(k−1−j)!
+o
￿
λ
α+k
L(λ)
￿
,inD

,
where
b
j
(aλ)=a
−α−j−1
b
j
(λ)+o(L(λ)).
StructuraltheoremsforquasiasymptoticsofSchwartzdistributionsp.10/19
AsymptoticallyHomogeneousFunctions
DenitionAfunctionbiscalledasymptoticallyhomogeneous
ofdegreeαatinnity(resp.at0)if
b(ax)=a
α
b(x)+o(L(x)).
Properties

Inthecaseatinnitywhenα<0,orat0whenα>0,
b(x)=o(L(x)).

Inthecaseatinnitywhenα>0,orat0whenα<0,
b(x)=βx
α
+o(L(x)).
StructuraltheoremsforquasiasymptoticsofSchwartzdistributionsp.11/19
StructuralTheoremforSomeCases
Theorem1Letf∈D

havequasiasymptoticbehavioratinnity
(resp.attheorigin)inD

,
f(λx)=C

L(λ)
(λx)
α

Γ(α+1)
+C
+
L(λ)
(λx)
α
+
Γ(α+1)
+o(λ
α
L(λ)).
(1)
Ifα/∈{−1,−2,...},thenthereexistapositiveintegerm,a
m-primitiveFoffsuchthatFiscontinuous(resp.continuousin
[-1,1])and
lim
Γ(α+m+1)F(x)
|x|
α+m
L(|x|)
=C±.
(2)
Conversely,iftheseconditionshold,then(bydifferentiation)(1)
follows.
StructuraltheoremsforquasiasymptoticsofSchwartzdistributionsp.12/19
AssociateAsymptoticallyHomogeneousFunctions
Inthecaseofnegativeintegerorder,themaincoefcientof
integrationofthequasiasymptoticsatisesthefollowing
denition.
DenitionAfunctionbiscalledassociateasymptotically
homogeneousofdegree0atinnity(resp.at0)withrespect
toLif
b(ax)=b(x)+βL(x)loga+o(L(x)).
StructuraltheoremsforquasiasymptoticsofSchwartzdistributionsp.13/19
StructuralTheoremfortheOtherCases
Theorem2fhasthequasiasymptoticbehaviorinD

atinnity
(resp.attheorigin),
f(λx)=γλ
−k
L(λ)δ
(k)
(x)+
(−1)
k−1
β
(k−1)!
λ
−k
L(λ)Pf
￿
1
x
k
￿
+o
￿
λ
−k
L(λ)
￿
,
ifandonlyifthereexistm∈N,m≥k,afunctionbsatisfying
b(aλ)=b(λ)+βlogaL(λ)+o(L(λ))andam-primitiveF,which
iscontinuous(resp.continuousin[−1,1])suchthat
F(x)=b(|x|)
x
m−k
(m−k)!
+γL(|x|)
x
m−k
2(m−k)!
sgnx
−βL(|x|)
x
m−k
(m−k)!
m−k
￿
j=1
1
j
+o
￿
|x|
m−k
L(|x|)
￿
StructuraltheoremsforquasiasymptoticsofSchwartzdistributionsp.14/19
SecondversionoftheStructuralTheorem
Thisisaversionfreeofb
Theorem3Letf∈D

.Thenfhasquasiasymptoticatinnity
(resp.attheorigin)oforder−k,k∈{1,2,...}ifandonlyifthere
existsacontinuousm-primitiveFoff(resp.continuousin
[-1,1]),m>k,suchthatforeacha>0,
lim
x→∞
(m−k)!
￿
a
k−m
F(ax)−(−1)
m−k
F(−x)
￿
x
m−k
L(x)
=I(a).
(3)
InsuchcaseIhastheformI(a)=γ+βloga.
StructuraltheoremsforquasiasymptoticsofSchwartzdistributionsp.15/19
AParticularCase
RecallthedenitionofCesarolimitsofdistributions.Letg∈D

,
lim
x→∞
g(x)=η(C,k),
ifthereexistsak-primitiveGofg,beingaregulardistribution,
suchthatG(x)=ηx
k
/k!+o(x
k
),asx→∞.Thenthestructural
theoremforthequasiasymptoticoforder-1isthefollowing:
f(λx)=γδ(λx)+βPf
￿
1
λx
￿
+o
￿
1
λ
￿
asλ→∞
ifandonlythereisk∈Nsuchthatforall1-primitiveFoffand
a>0
lim
x→∞
F(ax)−F(−x)=γ+βloga(C,k).
StructuraltheoremsforquasiasymptoticsofSchwartzdistributionsp.16/19
FirstConsequence:LocalbehaviorofFourierSeries
fissaidtohaveajumpbehavioratx
0
if
f(x
0
+ǫx)=γ

H(−x)+γ
+
H(x)+o(1)inD

asǫ→0
+
.
Supposethatf(x)=
￿

n=−∞
a
n
e
inx
,thenithasthisjump
behavioratx
0
ifandonlyifthereexistsk∈Nsuchthatforeach
a>0
lim
N→∞
￿
−N≤n≤aN
a
n
e
inx
0
=
γ
+


2
+
i


+
−γ

)loga(C,k).
StructuraltheoremsforquasiasymptoticsofSchwartzdistributionsp.17/19
e.vCesaroevaluations
Denition1Letg∈D

,andk∈N.Wesaythattheevaluation
hg(x),φ(x)iexistsinthee.v.Cesàrosense,andwrite
e.v.hg(x),φ(x)i=γ(C,k),
(4)
ifforsomeprimitiveGofgφand∀a>0wehave
lim
x→∞
(G(ax)−G(−x))=γ(C,k).
Ifgislocallyintegrablethenwewrite(4)as
e.v.
￿

−∞
g(x)φ(x)dx=γ(C,k).
Remark:Inthisdenitiontheevaluationofgatφdoesnothave
tobedened,weonlyrequirethatgφiswelldened.
StructuraltheoremsforquasiasymptoticsofSchwartzdistributionsp.18/19
PointwiseFourierInversionFormula
Now,wecharacterizethepointvaluesofadistributioninS

by
usingFouriertransforms.
Theorem4Letf∈S

.Wehavef(x
0
)=γinS

ifandonlyif
thereexistsak∈Nsuchthat
1

e.v.
￿
ˆ
f(t),e
−ix
0
t
￿
=γ(C,k),
whichincase
ˆ
fislocallyintegrablemeansthat
1

e.v.
￿

−∞
ˆ
f(t)e
−ix
0
t
dt=γ(C,k).
StructuraltheoremsforquasiasymptoticsofSchwartzdistributionsp.19/19