Page 1
EstimatingtheSpectralMeasureofa
Multiv ariate StableDistribution viaSpherical
HarmonicAnalysis
MarcusPiv ato
?
andLuisSeco
y
Octob er ??????
Abstract
A newmethod is develop ed for estimatingthesp ectralmeasure of
amultivariatestableprobabilitymeasure?byrepresentingthemeasure
asasumof sphericalharmonics?
In troduction?
Stable probabilitydistributionsarethenaturalgeneralizationsof
thenormal distribution?andsharewithittwokeyprop erties?
?Stability? Thenormaldistribution isstableinthesensethat?
ifXandYareindependentrandomvariables?withidentical
normal distributions?thenX?Yis alsonormal? and
?
?
???
?X?Y?
?
?
distr
X
?
?
distr
Y
Inasimilarfashion? ifX andYareindependent?identically
distributed?i?i?d?stablerandomvariables?thenX?Yisalso
stable?anditsdistributionisthesameasXandYwhenrenor?
malizedby?
????
?The stabilityexponent?rangesfrom?to
??When????we havethefamiliarnormaldistribution?
?
DepartmentofMathematics?University ofToron to?email?
pivato?math?toronto?edu
y
RiskLabandDepartment ofMathematics?University ofToron to?email?
seco?math?toront o? edu
?
Page 2
?RenormalizationLimit?TheCentralLimit Theorem says
thatthenormaldistribution is thenaturallimitingdistribution
ofasuitablyrenormalizedin?nitesumofindep endentrandom
variableswith?nitev ariance?IfX
?
?X
?
???? isasequenceofsuch
variables?thentherandomvariables
?
N
???
N
X
n??
X
n
converge? indensity? toanormaldistribution?Similarly?iffY
k
g
?
k??
areindependentrandomvariableswhosedistributionsdecay ac?
cordingtoapow erlawwithexponent?????thentherandom
variables
?
N
???
N
X
n??
Y
n
conv erge? indistribution?toan??stabledistribution
Thus?stabledistributions modelrandomaggregations ofmany
small?independentperturbations?F orexample? stabledistributions
mo delthemotionsofMarkovian stochasticprocesseswhose incre?
mentsexhibitpowerlaws?Stabledistributionsarisewithsurprising
frequencyincertainsystems? especiallythose inv olvingmanyinde?
pendentinteractingunitswithsensitive dependenciesb etw eenthem?
Theyhaveappeared inmathematical ?nance????????????? ?????????????????
?????????????????????????????Internettra?c statistics???? ???????? ???????
and ariseinmathematical mo delsofrandomscalar ?elds??????????
radar??? ??andsignalprocessing?????????????telecommunications????
and eventhepowerdistributionofoceanwaves????
Forfurtherexamples?see??????????or?????Thede?nitiverefer?
enceonuniv ariatestabledistributionsis?????thede?nitivereference
onmultivariatedistributionsandstableprocessesis?????Otherre?
cent references are?????? ?????? andaforthcomingbookbyNolan????
sligh tlyolderreferences are????and ????
Although one?dimensionalstabledistributions arewell?understood?
there aremanyopenquestionsinthemultivariateregime?Thesim?
plicityofthemultivariateGaussianuniversedoes notextendto non?
Gaussianmultivariatestable distributions?AnN?dimensionalGaus?
sian distribution iscompletelydeterminedbyits N?Ncovariance
?
Page 3
matrix?whichtransformsnicelyunderlinearchangesofcoordinates?
Inparticular?byorthogonallydiagonalizingthematrix?wecan?nd
anorthonormalbasisforR
N
?relativ etowhichthem ultivariatenor?
malv ariableis rev ealedasa sumof independentunivariatenormal
variables?thisis PrincipleComponentA nalysis?
Fora generalmultivariatestabledistribution?however? thesitu?
ationismuchmorecomplex?Sincethemarginals donot have?nite
variance? it does notmakesensetode?nea?covariancematrix?in
theusualway?noneoftheintegralswouldconverge?V ariousmod?
i?ednotionsof?covariance?havebeenproposed?see?forexample?
??????butthesedonottransformin anysimplewayunderc hanges of
coordinates? In particular?there is nothinganalogous toa ?principle
componen ts analysis??Instead?the correlationstructure ofastable
distributiononR
D
isdeterminedbyanarbitrary measure??? onthe
sphereS
D??
?
?
?x?R
D
?k?xk??
?
?calledthe sp
?
ectralmeasure?
Theorem??Let????????andlet?bean??stableprobability
measureonR
D
?with center???R
D
? Then?hasFourierTransform?
??
?
???exp???
?
???
where? ?the?logF ouriertransform?? isgiven?
??
?
???
D
???
?
?
E
i?
Z
S
D??
?
???
D
??s
E
d??s????
where?
???
??????j?j
?
?B
?
??
h?i
i? ???
with?
h?i
??
?
sign????j?j
?
if ????
??log j?jif???
andB
?
??
?
tan
?
??
?
?
if????
?
?
?
if???
and where?is some nonnegative Borel measure onS
D??
?
Proof?See?????x????p????or?????
??Theorem??
?
Page 4
?iscalledthespectralmeasureofthedistribution
?
? andises?
sentiallyan ?in?nite?dimensional?data?structure?so itisclearthat?
ingeneral? noN?Nmatrixcanpossiblybeadequateforrepresen ting
it?A?principlecomp onents?typedecomp ositionisonlyvalidwhen
thesp ectralmeasureconsistsof?D antipodalypositionedatoms?
Estimating?ismuchdi?cultthanestimatingacovariancematrix?
Whereastheterms ofacovariancematrixcanbedirectlycomputed
byestimating thecorrelationb etweencoordinates?? isonlyindirectly
visible? theimage of?underasort of?sphericalconvolution? appears
inthelo garithmofthecharacteristicfunctionof thedistribution? there
isnomoredirectwaytoobserveit?
Inthispaper?wedevelopanmethod forestimating?fromthelog?
characteristicfunction??Assumeforsimplicitythatthedistribution
iscentered attheorigin?andletthesphericallog?characteristic
functionbethefunctiong?S
D??
??Cdeterminedbyrestricting?
tothesphere?Then?forall
?
??S
D??
?wehave
g?
?
???
Z
S
D??
?
???
D
?
??s
E
d??s? ???
Thecharacteristicfunctionofadistribution iseasy toestimate
fromempiricaldata?andthus?weassumewehaveagood estimate
ofgon somesuitably ?ne meshoverS
D??
?Hence?the problemis to
recov er? fromg?
Abusingnotation?wemightrewriteequation ???as?g??
???
?
???IfD??orD? ??thenS
D??
isatopologicalgroup?andthis
?convolution?canbeinterpretedliterally?via the form ula?
?
???
? ??
?
???
Z
S
D??
?
???
?
?
??s
??
?d??s??
Inotherdimensions?howev er?S
D??
isnota topologicalgroup?
andtherefore?conv olutionper seisnotwell?de?ned?Wemustinstead
thinkofS
D??
asahomogeneousmanifoldundertheactionofSO
D
?R??
andde?neakindof?conv olution?intermsofthisgroupaction?
Theeigenfunctionsof theLaplacianoperator onS
D??
are called
sphericalharmonics?andform an orthonormalbasis forL
?
?S
D??
??
?
Thisterminologyisstandard? butsomewhatunfortunate?since?isunrelatedto
anyoneofhalfadozenother?spectra?and?spectralmeasures?currently existentin
mathematics?P erhapsitwouldbemoreappropriate tocall?aFeldheim measure?
sinceFeldheim????wasthe?rsttode?neit?
?
Page 5
analogoustotheF ourierbasisforL
?
?S
?
?fromclassical harmonicanal?
ysis? TheexpressionofafunctiononS
D??
intermsofthisbasis is
calleditssphericalF ouriertransform?Afunctionf?L
?
?S
D??
?
iscalledzonal ifitis rotationallyinvariantarounda particularco or?
dinate axis?forexample??
???
iszonal?There isawayof convolv?
ingarbitraryfunctionsbyzonalfunctions?and?just asin classical
harmonicanalysis? convolutionofafunctionfby?translates in to
component wisemultiplication oftheirrespectiveF ouriertransforms?
Thus? to deconv olvefand??itsu?cesdividetheF ouriertransform
of??fbythat of??
Theadvantageofthisapproachistw ofold?First?itprovidesa
naturalcontinuousrepresentationofthe sp ectralmeasure?obviating
theneedtoapproximate itwitha sum ofatoms?Second? itiscom?
putationallyfaster? Thecomputationsinvolvedarestillexpensive?
numericallyin tegratingonasphere usingameshofdensity????N
isacomputationof orderO
?
N
?D???
?
? and computingaconv olutionof
twofunctions isth usacomputationoforderO
?
N
??D???
?
?However?
thereis noneedtoexplicitlycompute a matrixin verse?rst? b ecause
aclosed?formexpression existsfor theelements ofthe orthonormal
basis?
Organization ofthisPap er?Inx??wesummarizepreviouswork
on thisproblem? Inx??wedevelopsome backgroundmaterial?treat?
ingS
D??
ashomogeneous manifoldundertheactionofSO
D
?R??and
reviewing zonalfunctions?theeigenfunctions oftheLaplacian? anda
suitablenotionofconvolution?andprovideexplicitformulaeforthe
sphericalharmonics?Inx??wede?nethesphericalFouriertransform
andshow howtocompute?deconv olution?usingthis transform?Inx??
wecharacterizethe rateof conv ergenceofthesphericalFourierseries
asan estimateofthespectralmeasure?andrelate thisto convergence
of theunderlying stabledistribution?
?SummaryofpreviousWork?
Early on?Press????dev elopedan estimationscheme formultivariate
stable distributions?througha straigh tforward generalizationof his
one?dimensionalmethod?Press?s method? however?onlyworksfor
?pseudo?Gaussian?distributions?withlog?characteristicfunctionsof
theform?
?
Page 6
?
X
?
?
???
D
?
????
E
i?
D
?
???
?
?
E
???
where?issomesymmetric?p ositivesemide?nite?covariance ma?
trix??If?has uniteigenv ectors??
?
???????
D
?witheigenvalues ?
?
?????
D
?ie?asa covariancematrix?wehave?principlecomponen ts??
?
??
?
?????
?
?
??
?
??thenthespectralmeasureof thisdistribution issymmetricand
atomic? withatomsateachof???
?
????????
D
?withmasses?
?
?????
D
?inotherw ords?
??
D
X
d??
?
d
?
?
??
d
??
???
d
?
Press prop oses to solve forthe comp onents ofthematrix?byem?
piricallyestimatingthe logc haracteristicfunction atsomecollection
offrequenciesf
?
?
?
?????
?
?
N
g? whereN?D?D? ????? and then solv?
ingasystemofNlinearequations? He claimsthat hismethodwill
generalizetoasumofpseudo?Gaussians?
?
X
?
?
???
D
?
????
E
i?
M
X
m??
D
?
???
m
?
?
E
???
?where?
?
??????
M
arelinearly independent?symmetric?p ositive
semide?nite matrices?? How ever? inthiscase? one no longerendsup
withasystem oflinearequations? so itis notclearthat themethodis
tractable?In anyevent?Press?s methodonlyappliestom ultivariate
distributions withparticularlysimpleatomicspectralmeasures?which
furthermoremustbe symmetricallydistributed?Empiricalevidence
?see? forexample??? ??suggeststhatthestabledistributions foundin
?nancial data aresigni?can tlyskewed? symmetryis notareasonable
assumption?
Cheng?RachevandXin????????developamore sophisticatedmethod?
byexpressingastablerandomv ectorinpolarcoordinates?andthen
examining theorderstatisticsof theradialcomponent? asa function
ofthe angularcomponent?Theyutilizethetheorem ofAraujoand
Gin?e ?Corollary?????b??Chapter??p????of???? statingthatthera?
dialdistributiondecaysmostslowlyinthose angulardirectionswith
the heaviestconcentration ofspectralmass?thesedi?erences indecay
ratearethenusedto estimatethe densitydistribution of thespectral
measure?
?
Page 7
Nolan?Panorska?andMcCullo ch??? ???????developamethodbased
up ona discreteappro ximationofthe spectralmeasure? Ifthespectral
measureistreated asasumofa?niten umberofatoms?
??
X
a?A
?
a
?
a
?
then?forany?xed
?
??S
D??
?the function?
???
?
?
?s???
???
D
?
??s
E
ofTheorem? canberestrictedtoafunction?
???
?
?
?A??C?The
setofall discretemeasuressupportedonAisa?nite?dimensional
vectorspace? whichwecan identifywithC
A
?and?
???
?
?
isjustalinear
functional onthisv ectorspace?If??S
D??
issome?niteset?then
wecande?nea linear map?
F?C
A
??C
?
where? for each
?
?? ??
F ???
?
?
?g?
?
???
Z
S
D??
?
???
?
?
d?
The method of Nolanetal? thencomes do wntoinverting this
lineartransformation to recov er?from anempirical estimateofg?
They explicitlyimplemen tedtheirmethod in thetwo?dimensionalcase
?ie? when thesp ectralmeasure liv esona circle?? andtested itagainst
avarietyof distributions?Theyfoundthatitworked fairlywell fora
variety of measuresonthe circle?andconsistentlyoutperformedthe
methodofChenetal?ThemethodsofChenetal? andNolanetal?
arealsodiscussedinx?of?????
Finally
cussion
?
???
Hurd
?
et al????
??
dev
section
elop???
??
?S
D??
asa Homogeneous Riemannian
Manifold?Zonalfunctions?Laplacians?
andConvolution
?The developmentof backgroundmaterialhere lo oselyfollo ws thedis?
inchapterAmorefriendlyapproach is ?????
?
?
Page 8
S
D??
isacompactRiemannianmanifold?andG?SO
D
?R?isa
?nonab elian?compactLie group?acting transitivelyand isometrically
onS
D??
byrotations?Wewilldevelopaversionof harmonicanalysis
onS
D??
asahomogeneousRiemannianmanifold?thistheoryisactu?
allyapplicable toanyhomogeneous Riemannianmanifold?itmaybe
helpfultokeepthisinmind??
LetL
bsg
be the canonicalv olumemeasure inducedonS
D??
byits
Riemannstructure?Forexample?onS
?
?L
bsg
istheusual ?surfacearea?
measure?S
D??
iscompact? soL
bsg
is?nite?assumeL
bsg
isnormalized
to have totalmass ?? LetL
?
?S
D??
??L
?
?S
D??
?L
bsg
?C?? Theaction
ofGonS
D??
inducesa linearG ?act iononL
?
?S
D??
?inthe obvious
wa y? if??L
?
?S
D??
?andg?G?theng???S
D??
??Cis de?ned?
g ???m????g?m??
LetC
?
?S
D??
?be the spaceofsmo oth?complex?valuedfunctions
onS
D??
?L
bsg
is?nite?soC
?
?S
D??
?isalinear subspaceofL
?
?S
D??
?
?thoughnotaclosed subspace??G actssmoothly onS
D??
?soC
?
?S
D??
?
isG?invarian t?Weconsidertherestrictedaction ofGonC
?
?S
D??
??
Let??C
?
?S
D??
???C
?
?S
D??
? is the Laplacianop erator?
Theorem ?? ?TheL aplacian onS
D
?????
Firstconsider thecaseD???Endow thecircleS
?
withtheangular
coordinatesystem?????????sothatanypointonS
?
?
?S
?
?f??? ??g
has coordinates
? cos???? sin?? ??
Iff?S
?
?
??C?then?relativetothiscoordinatesystem?wehave?
?
S
?
f?
?
?
f
??
?
?
Moregenerally?de?neS
D
?
?S
D
n
?
R
D??
???????f?g
?
?and
then de?nethedi?eomorphism
S
D??
?
????????S
D
?
?s??? ???cos???? sin????s?
Thenwehavethe following inductiveformula?
?
Page 9
?
S
D
f?
?
?
f
??
?
??D???cot???
?f
??
?
?
sin???
?
?
S
D??
f????
?
?comm utes withtheisometricGaction?forallg?G?
??g ????g?????
Let???f??C???isaneigenv alueof?g?and foreach??
?? letV
?
??
?
??C
?
?S
D??
????????
?
be thecorresponding
eigenspace? Th us?V
?
isaG ?inv ariantsubspace?
Theeigenfunctions oftheLaplacianonS
D??
arecalledspherical
harmonics?Further informationon sphericalharmonicscanbe found
inc hapter??section?of?????c hapterIIof ????chapters ?and?of
??? ??c hapters?and? of?????x??andx?? of?????andalsoin?????
??? ??????? ?????? ????? ?? ?????and?????
Lete?????????? ???S
D??
? and de?ne
G
e
?fg?G?g ?e?eg?
thesetof allorthogonaltransformations ofR
D
?xingthee?axis?In
otherwords?G
e
isthesetofall?rotations?oftheremaining?D???
dimensionsaboutthisaxis?hence?thereisanaturalisomorphism
G
e
?
?
SO
D??
?R??G
e
is thusaconnected?compactsubgroupofG?
Theaction ofGuponC
?
?S
D??
?restrictstoanactionofG
e
?andthe
spacesV
?
remain inv ariant under this newaction?
De?nition??Zonalfunction
Afunction??C
?
?S
D??
?iscalledzonal?relativetoGandthe
?xedpointe?S
D??
?ifitisinvariantundertheaction ofG
e
?
F ormally? for anyG
e
?invariantsubspaceV?C
?
?S
D??
??de?ne
Z
e
?V???f??V??g?G
e
?g????g
Thus?thezonalelementsofC
?
?S
D??
?aresmoothfunctionsrota?
tionallyinvariantabouttheeaxis?Clearly?anysuchfunctionmust
be of theform
??x???
?
?x
?
?
where?
?
??????? ??C? andwherex??x
?
?x
?
?????x
D
?isany
elementofS
D??
?
?
Page 10
Proposition??
??IfV?C?S
D??
? isanontrivialG?invariant subspace?thenZ
e
?V?
isnon trivial?
??Ifdim?Z
e
?V?????thenV isanirreducibleG?module?
Proof?
ProofofP art??
Claim??Vcontainsanelement? suchthat??e?????
Pro of?SinceV isnon trivial?thereissome??Vwhichis
nonzerosomewhere?say??x??? ??SinceGactstransitively
onS
D??
? ?ndg?Gso thatg ?e?x? Th us?if??g ???then
??e????g?e????x????? SinceVisG ?inv ariant???V is
theelementwe seek????????????????????????Claim??
Now?G
e
isaclosedsubgroupofthecompactgroupG? thus?G
e
iscompact?soithasa ?nite HaarmeasureH
aar
?De?ne
???
Z
G
e
g??dH
aar
?g?
SinceH
aar
is?nite? thisintegral isw ell?de?ned?SinceVisa
closed?G?invariantsubspace?theelement? is inV?F urthermore?
since??e????e??and??e??? ??weconcludethat?isnontrivial?
Finally?notethat?isG
e
?invariantbyconstruction?inother
words?itiszonal?
ProofofPart??Supp oseV?V
?
?V
?
?whereV
?
?V
?
areG?
invariant?ThenbyPart??wecanconstructlinearlyindependent
zonalfunctions?
?
?Z
e
?V
?
?and?
?
?Z
e
?V
?
??Since?
?
??
?
?
Z
e
?V??thiscontradictsthehypothesisthat dim?Z
e
?V?????
??Proposition??
Theisometricaction ofG
e
on S
D??
inducesalinear?isometric ac?
tion uponthetangent spaceT
e
S
D??
?If?v?T
e
S
D??
isthederivative
ofapath??????????S
D??
with?????e?theng??visthederivative
ofthepath?g????????????S
D??
?TheactionofofG
e
onS
D??
is
rankone?meaningthatG
e
acts transitively onthesetofunittan?
gentvectorsT
e
S
D??
?For anyr???letB?e?r?betheball ofradius
rab oute inS
D??
?relativeto the intrinsicRiemannianmetric?The
following is clear?
??
Page 11
Lemma??Forallr?G
e
actstransitivelyon?B?e?r?inS
D??
?
Proposition??IfS
D??
isofrankone?theneacheigenspace
V
?
of? is anirreducibleG ?module?
Pro of? ByProp osition ??it su?cesto showthat dim?Z
e
?V
?
?????
So?supposethat?
?
??
?
?Z
e
?V
?
?arelinearlyindependent?Since
theyarezonal??
?
?u? and?
?
?u?arefunctionsonlyofthedistance
fromu toe?So?forsomeu?S
D??
with dist?u?e??r?de?ne
z
?
???
?
?u?andz
?
???
?
?u?? andlet? ??z
?
?
?
?z
?
?
?
?Thus?? is
alsozonal?We aimtoshow that? is the zero function?thus??
?
and?
?
arejustscalarmultiplesofoneanother?
Now?byconstruction???u???? and thus???? on?B?e?r??At
thesametime?howev er?? isa linear combination oftwo elements
ofV
?
? hence?it isalso inV
?
?ie?? isa?????eigenfunctionsof
??Fix?? and letrgetsmall?Ifr ismadesmallenough?then
theDirichletboundarycondition?
j?B?e?r?
?? forcesthesmallest
eigenvalue of?tobelargerinabsolutevaluethan??creatinga
contradiction?
??Proposition ??
Oneconsequence of thisirreducibility is
Theorem???Schur?sLemma?????
IfViscomplexBanachspaceandanirreducibleG?module?and
??V??Visacontinuous?complex?linearmapthatcommuteswith
theG?action?then?mustbem ultiplicationbyascalar?
?
No wconsiderthe D ?torusT
D
?equipped with the standardequiv?
ariantmetric?TheeigenfunctionsoftheLaplacianonaretheperiodic
functionsoftheformE
n
?x??exp???i?hn?xi??withn?
c
T
D
?
?
Z
D
?
wherex??????
D
and?????
D
isiden ti?edwithT
D
intheobvious
way?TheseeigenfunctionsformanorthonormalbasisforL
?
?T
D
??
ThesameistrueforarbitraryhomogeneousRiemannian manifolds?
and inparticular?forthesphere?
Theorem??
?If?
?
???
?
?thentheeigenspacesV
?
?
andV
?
?
areorthogonalas
subsetsofL
?
?S
D??
??
??
Page 12
?Theeigenspacesof? spanL
?
?S
D??
??In otherw ords?
L
?
?S
D??
??
M
???
V
?
Proof?Seeforexample????? chapter?? p? ????or???? Theorem?????
p?????Ortreat? asanellipticdi?erentialoperator? anduse
?????x???? Theorem ??p????? Alternately?employ theSpectral
Theoremforunboundedself?adjoint operators?see?????chapter
X?section??p? ?????
??Theorem??
De?nition??EquivariantFunction
If??S
D??
?S
D??
??C?thensaythat? isaG ?equivariantif?
forall m?n?S
D??
andg?G?
??g ?m?g ?n????m?n?
SinceG acts isometricallyandtransitively onS
D??
? thisis equiv?
alentto sayingthat??x?y?isafunction onlyof thedistance
dist?x?y??
Forinstance?ifthefunction?
???
?S
D??
?S
D??
??Cde?nedby
equation???isG?equiv arian t?
G?equivariantfunctions areinterestingbecausewecande?nea
sortofconvolutionwiththem?
De?nition??? Convolution
If? isG?equiv ariant???S
D??
??C?andb othareL
bsg
?integrable?
then de?ne????S
D??
??Cby
??????s??
Z
S
D??
??s???????dL
bsg
???
Forexample?if? isameasure onS
D??
?with Radon?Nikodym
derivative??S
D??
??C?then????S
D??
??Cisde?ned
????s??
Z
S
D??
??s???????dL
bsg
????
Z
S
D??
??s???d????
??
Page 13
Inparticular?if?isaspectral measureand???
???
?thenthis
formulais identicaltoequation???? In otherw ords?
?
???
???g
where
It
gis
out
thespherical
that
log?characteristicfunction?
RecallagainthecaseofT
D
Theeigenfunctionsofthe Laplacian?
?
E
n
?n?Z
D
?
?arew ell?behav edunderconvolution? classicalhar?
monicanalysis tellsusthat
?
?
X
n?Z
D
a
n
E
n
?x?
?
A
?
?
?
X
n?Z
D
b
n
E
n
?x?
?
A
?
X
n?Z
D
?a
n
?b
n
?E
n
?x?
turnsthisphenomenongeneralizes toarbitraryhomo?
geneous Riemannianmanifolds?
Prop osition??? ?Convolutionand Eigenfunctions?
Let??S
D??
?S
D??
??CbeG?equiv arian t?Fix???and
??Z
e
?V
?
?? andde?neA
?
?C
D
b y?
A
?
??
???? ??e?
??e?
?
Then?forany??V
?
?????A
?
???
Proof?LetT
?
?C
?
?S
D??
???C
?
?S
D??
?bede?ned?T
?
????????
Claim??Theop eratorT
?
commuteswiththeG?action?for
allg?G?T
?
?g ????g?T
?
????
Proof?F oranym?S
??
?
T
?
?g ????m??????g?????m?
?
Z
S
D??
??m?n???g ?n?dL
bsg
?n?
?
???
Z
S
D??
??m?g
??
?n
?
???n
?
?dL
bsg
?n
?
?
?
???
Z
S
D??
??g?m?n
?
???n
?
?dL
bsg
?n
?
?
???????g?m?
?g???????m??
??
Page 14
???wheren
?
??g ?n
???Because?isG?equivariant????????????? ?Claim??
Claim ??T
?
comm uteswith??
Proof?Foreachy?S
D??
?de?ne?
y
?S
D??
??Cby?
y
?x??
??y?x????x?y ?? Thus?
??????x??
Z
S
D??
??x?y????y?dL
bsg
?y?
?
Z
S
D??
??y???
y
?x?dL
bsg
?y?
Hence????????x???
Z
S
D??
??y???
y
?x?dL
bsg
?y?
?
Z
S
D??
??y????
y
?x?dL
bsg
?y????
b ecause? isa linearoperator?
Claim??????
y
?x????
x
?y??
Proof?Find someg?G sothatg?x?yandg?y?x?
Thusforanym?S
D??
?
?
x
?m????x?m????g ?x?g?m????y?g ?m?
??
y
?g?m???g??
y
??m?
In otherwords??
x
??g??
y
??
Thus???
x
???g??
y
??g????
y
??
Inparticular???
x
?y??g????
y
??y????
y
?g?y ?
???
y
?x??
??????????????????????????????????????Claim????
Hence?wecanrewriteexpression???as?
Z
S
D??
??y????
x
?y?dL
bsg
?y?
??
Page 15
ButS
D??
isa manifold withoutboundary?so?isself?adjoint
?see?forexample?????chapter???Hence?
Z
S
D??
??y????
x
?y?dL
bsg
?y??
Z
S
D??
???y???
x
?y?dL
bsg
?y?
?
Z
S
D??
??x?y?????y?dL
bsg
?y?
????????x?
????????????????????????????????????????????Claim ??
Itfollowsfrom Claim? thatT
?
must leave inv ariant alleigenspaces
of?? inotherwords?forall????V
?
isinvariant underT
?
?
Butby Claim??therestrictedmap
?T
?
?
j
?V
?
??V
?
isthen anisomorphism of linearG? mo dules? SinceGacts irre?
duciblyonV
?
?byProp osition???it followsfromSchur?s Lemma
thatT
?
mustactonV
?
by scalarmultiplication?thus? thereis
someA
?
?Cso that?forall??V
?
?
T
?
????A
?
??
????inotherw ords?????A
?
???Inparticular?if??Z
e
?V
?
??
then????A
?
??? hencewemusthaveA
?
?
????e?
??e?
?
??Proposition???
Corollary???Let??Z
e
?V
?
?beazonaleigenfunction?nor?
malizedso thatk?k
?
???De?neZ?S
D??
?S
D??
??Cby
Z?x?y????g
??
x
?y?
whereg
x
?Gis anyelementsothatg
x
?e?x?ThenZisw ell?
de?ned?indep endentofthe choiceofg
x
?andisG?e quiv ariant?Ifwe
thende?neP
?
?L
?
?S
D??
???L
?
?S
D??
?by
P
?
??????e???Z???
thenP
?
istheorthogonalprojectionfromL
?
?S
D??
?on to the
eigenspaceV
?
?
??
Page 16
Proof?
Proofof?WellDe?ned?? Ifg
?
?g
?
?Gsothatg
?
?e?g
?
?e?x?
theng
??
?
?g
?
?e?e?thus?g
??
?
?g
?
?G
e
? Thus?since?is zonal about
e?
??g
??
?
?y????g
??
?
?g
?
?g
??
?
?y????g
??
?
?y?
Pro of of ?Equivariant??Letx?y?S
D??
?andh?G?Notethat
wecan pickg
?h?x?
?h?g
x
?Thus?
Z?h?x? h?y????g
??
?h?x?
?h?y??? ??h?g
x
?
??
?h?y????g
??
x
?h
??
?h?y?
???g
??
x
?y??Z?x?y??
Proofof?OrthogonalPro jection?? SinceP
?
isde?nedbya
conv olutionin tegral?it is clearlya linearop erator?Itthensu?ces
to showthatP
?
?xesV
?
?andannihilatesV
?
?
?
If??V
?
?thenbyProposition???
Z???
?Z????e?
??e?
??thus?P
?
?????Z????e????
so it su?ces toshowthat?Z????e????But?
Z???e??
Z
S
D??
Z?e?y???y?dL
bsg
?y?
?
Z
S
D??
??g
??
e
?y????y?dL
bsg
?y?
?
Z
S
D??
??y????y?dL
bsg
?y??sinceg
e
?Id?
?k?k
?
?
???byhypothesis?
Onthe otherhand?if??V
?
?
?thenfor alls?S
D??
?
??
Page 17
Z???s??
Z
S
D??
??g
??
s
?y????y?dL
bsg
?y?
?
Z
S
D??
?
g
??
s
??
?
?y????y?dL
bsg
?y?
?
?
g
??
s
????
?
??
sinceg
??
s
???V
?
???
? ?Corollary???
Prop osition ???? ZonalEigenfunctionsof? onS
D??
?
Theeigenvaluesof?onS
D??
arealloftheform
?
N
?N??N?D? ???
forsomeN?N?Let?
N
bea correspondingeigenfunction? and
assume that?
N
iszonal?relative toSO
D
?R? ande??
CaseD??? Modulomultiplicationby somenormalizingconstant?
?
N
????cos?N???
whereweusethecoordinatesystem??????????? cos????sin?????
S
?
?Ifwe write?
N
intermsofCartesiancoordinatesx??x
?
?x
?
? on
R
?
?wegetthe
?
Ceby?sev polynomials?
?
N
?x???
?N???
x
N
?
b
N
?
c
X
n??
????
n
?
?N????n?
N
n
?
N?n??
n??
?
x
?N??n?
?
?
???
CaseD???Modulom ultiplicationbysomeconstant??
N
isa
LegendrePolynomial?
?
N
?x??
bN??c
X
n??
????
n
?
N??n
?
?
?
?
?N?n
?
?
?
?
?
?
?n???N??n??
?x
N??n
??
Page 18
General Case?Assumethat?
N
is ofunitnorm?Then?
N
isa
normalizedGegenbauerpolynomial?
?
N
?x??
?
K
?
???
N
C
???
N
?x
?
?
whereC
???
N
?x??
bN??c
X
n??
????
n
?
N??n
?c
???
N
k
?n
?x
N??n
withc
???
N?n
?
?????N?n??
?????n???N??n??
andwhere
?
K
???
N
?
?
?
Z
S
D??
?
?
?
C
???
N
?x
?
?
?
?
?
?
dx
?
??
?D?????
????
?
?
??bN??c
X
k??
????
k
??
?N??k
?
?
?
N?k?
?
?
?
?
?
N?k?
D
?
?
?
?
X
n??
c
???
N?n
c
???
N ??k?n?
?
?
where???
D??
?
?
?
Pro of?
Proof ofCharacterizationof Eigenvalues?See?forexample?
?????chapter???????chapter??or??? ??
ProofofCaseD???Itis clearfromthede?nitionofthe
LaplacianonS
?
thatthefunction?
N
isaneigenfunctionof?S
?
?
ThesubgroupofSO
?
?R??xingeisjustthetwo?elementgroupof
maps?x
?
?x
?
????x
?
??x
?
??sincethefunction?
N
issymmetric
relativeto thexvariable?itis zonalrelativetothesemaps?
Theformula ???is thenjust astandardtrigonometric iden tity?
where weiden tifyx
?
?cos????see?for example?????x???????
p????
Proof of CaseD???ThisisjusttheGegenbauerpolynomial
whenD???F oradirectproof?see?forexample????Theorem
?? x????p????wherethereisunfortunately anerror inthe de??
nitionofthe Legendrefunctions?see ?????x?? p??? foracorrect
de?nition??
ProofofGeneralCase?Thisisjustabigcomputation?See????
or ?????
??
Page 19
??Proposition???
?SphericalFourier Series
Theorem ????Spheric alF ourierAnalysis?
For alln?N?let?
n
?S
D??
??Cbethe zonal harmonicp olynomi?
alsde?nedbyProposition??? andthen de?neZ
n
?S
D??
?S
D??
??C
by
Z
n
?x?y???
n
?e???
n
?hx?yi?
ThenZ
n
isrotationallyequivariant?
Now? suppose??L
?
?S
D??
?C??Ifwe de?ne?
n
??Z
n
??then
?
n
?V
??
n
?
?and?hastheorthogonaldecomp osition?
??
?
X
n??
?
n
????
Proof?This followsfromTheorem??andCorollary??? usingthe
zonalfunctions pro videdbyProp osition???
??Theorem???
Corollary?????De?convolutiononSpheres?
Supp ose??S
D??
?S
D??
??C isrotationally equiv ariant?and
supposethatg??????If?
?
for
?
alln?
g
N??
n
andZ
n
areasin
Theorem ???andwede?ne
g
n
??Z
n
?g andA
n
??
????
n
??e
?
?
?
n
?e
?
?
theng
n
?A
n
??
n
?
Conversely? supposethat?isunknown?butweknow?andg?We
canreconstruct?viatheformula?
?
X
n??
?
A
n
n
Proof?Thisfollows fromtheprevious theorem?andProp osition ???
??Corollary ???
??
Page 20
De?nition??? SphericalF ourierCoe?cients
If??L
?
?S
D??
?? then thesphericalFourier Co e?cients of
?are the functions?
n
??Z
n
???forn?N? ?Notice that
these?coe?cients?are themselvesfunctions?notn umbers??The
sphericalF ourierseriesfor? is thentheorthogonaldecomp o?
sition
??
?
X
n??
?
n
?
Example????Spheric alFourierseriesonS
?
?
LetforN?N?let?
N
?S
?
??Cbeas inPart?ofProposition
???
?
N
???? cos?N???
?
?
?
E
N
????E
??N?
???
?
whereweidentifyS
?
?
?
???????andde?neE
K
????? exp?K??i??
LetZ
N
?S
?
?S
?
??Cbede?nedfrom?
N
asin Theorem??? Then?
forany??S
?
??R?
?
N
?Z
N
??
?
???
???
N
?
?
?
?
??E
N
???E
??N?
?
?
???
?
?
?
b??N??E
N
?
b
f??N??E
??N?
?
?
???
?
?
?
b??N??E
N
?b??N??E
N
?
? re?b??N??E
N
??
???wheretheconvolution is now meant in the ?usual?sense on
the groupS
?
?T
?
?
??? here?b?isthe?classical?Fouriertransformof?asafunction
onthecircle?
???because?isreal?valued?
Now?ifwewriteb??N??r
N
exp??
N
?i??where r
N
??????and
?
N
???????? then?forany??S
?
?
?
???????wehave?
??
Page 21
?
N
????re?r
N
?exp??
N
i??E
N
????
?r
N
? re ?exp??
N
i?? exp??N???i??
?r
N
?re
?
exp
?
N?
?
??
?
N
N
?
?i
??
?r
N
?re
?
E
N
?
??
?
N
N
??
?r
N
??
N
?
??
?
N
N
?
?
Inotherwords?convolving?
N
by?isequivalent tom ultiplying the
magnitude of?
N
byr
N
? androtating the phaseby?
N
?N?
?AsymptoticDecayandConvergence
Rates
Inclassicalharmonicanalysis? the in?nitesimalpropertiesofa func?
tionfarein manywaysre?ectedintheasymptoticbeha viourofits
Fouriertransform?andviceversa?Generally?thesmootherfis?the
more rapidly
b
fdeca ysnearin?nity?Conversely?iffisvery?jaggy??
undi?erentiable?ordiscontinuous?then
b
fdecays slowlyornotatall
nearin?nity?re?ectingaconcentration ofthe?energy?offinhigh
frequencyFouriercomponents?
Hence?when approximatingfbyapartialFourier sum?themore
jaggyf is?themore slowly thesum converges?and the moretermswe
m ustincludetoensureourselvesofa
?
good approximation?
Asimilarphenomenonmanifests when approximating afunctions
??S
D??
??Cbyaspherical Fourierseries?Byrelatingthedecay
rateofthesphericalFourierseriestothesmoothnessof??wewill
beable toestimatethe errorin troducedby approximating?with a
partial sphericalFourier sum?
Sa ythat asequenceof functions??
n
j
?
n??
?isoforderlessthanor
equaltoO?n
??
?if
lim
n??
k?
n
k
n
?
??
?withthelimitpossiblyzero??
??
Page 22
Theorem???Let??S
D??
??C? andsupposethat?is con?
tinuously?M ?di?erentiable?Then thesequence??
n
j
?
n??
? isof order
lessthanorequal toO?n
???M ???
?
Proof? Firstsupp osethat? istwice continuously di?erentiable?
Th us?usingformula ??? onpage ??we canapply?
S
D??
to??Let
???
S
D??
??Since?isa contin uousfunction? it isinL
?
?S
D??
??
andwecan computethesphericalFouriercoe?cien ts?
n
?Z
n
???
for alln?andconclude?
??
?
X
n??
?
n
?
Inparticular? sincethis sumconvergesabsolutely inL
?
?S
D??
??
weknow that thesequence??
n
j
?
n??
? isoforderlessthanO?n
??
??
Byconstruction?weknowthat?
n
?Z
n
??isaneigenfunction
of?
S
D??
? witheigenvalue?
n
?n?n?D? ???By Claim?of
Proposition???theLaplacianop eratorcommuteswithconvolu?
tionoperators?Thus?
n?n?D????
n
??
S
D??
?
n
??
S
D??
?Z
n
???
?Z
n
???
S
D??
??
?Z
n
??
??
n
Sincethis istrue foralln?weconcludethat??
n
j
?
n??
?isoforder
lessthanorequaltoO
?
?
n?n?D???
?
?O?n
??
??O?n
??
??
Proceed inductivelytoprovethegeneralcase?
??Theorem???
Conclusion
Byexpressingthelog characteristicfunctiongofequation???as a
sphericalFourierseriesviaTheorem???andthenapplyingthe?de?
convolution?formulaprovidedbyCorollary???wecanreconstructa
??
Page 23
sphericalF ourierseriesforthespectral measure??Of course? forprac?
ticalpurp oses?wecanonlyevercomputea ?nitenumberoftermsof
thisseries? The degreeofapproximation errorintro ducedby?nitely
truncatingthesphericalFourier seriesisdeterminedby theasymp?
toticdecayrate of thecoe?cients?Aswith classicalFourierseries?
thisdecayrate isa function of the?smoothness?of? ?Theorem????
F or anextremelysingular ??unsurprisingly?the coe?cientsmaydecay
insizeslo wly?sotheserieswilltak ealong timetoconvergetoagood
appro ximation?
Theadvantageofthisapproachisthat?oncewehaveexpressed
gin termsofitssphericalFourierseries?computing?isextremely
straigh tforward?weneedonlydividetheFourierseriesofgbythecon?
stantsA
n
ofCorollary???ComputationoftheFourierseries? in turn?
involvesconv olutionwith Gegen bauerpolynomials? A closed?formex?
pressionfor thesepolynomialsisgiven?Theorem????andtheconvolu?
tioncanbe computedbyn umericalintegrationov erS
D??
?a task with
complexityO
?
N
??D???
?
? tobe contrastedwiththeO
?
N
??D???
?N
??D???
?
required byanexplicitmatrix?inversionapproach?whereN????re?
?ectsaprecisionof?inour approximation??
Througha linearcom binationofsphericalharmonics?wecanex?
plicitlyrepresen t?as acontinuousobject onS
D??
?ratherthanasa
sumofatoms? Ofcourse?if?inrealitywasdiscrete?thisrepresenta?
tionmightbemisleading?andadiscreterepresentationmightactually
bepreferable?However?inmanycases?isabsolutelycontinuous?rel?
ativetotheLebesguemeasure?for example?ifthestable distribution
is sub?Gaussian?see??? ??x????? Inthesecases?an explicitlycontin u?
ous representation maybepreferable toav oidin troducinganomalous
asymptoticbehaviourtothedistribution?
MarcusPivato
DepartmentofMathematics?
UniversityofToronto?
???St?GeorgeStreet?
Toronto?Ontario?
M?S?G?
CANADA
Email?pivato?math?toronto?edu
URL? http???www?math?toronto?edu??pivato?
??
Page 24
Phone?????????????
Fax?????????????
LuisSeco
Universityof Toronto?
??? St?GeorgeStreet?
Toronto?Ontario?
M?S?G?
CANADA
Email?seco?math?toronto?edu
URL?www?risklab?erin?utoronto?ca?seco?index? htm
Phone? ????????????
Fax?????? ???????
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