DiscreteAnalogsofWignerTransforms andWeylTransforms
ShahlaMolahajlooandM.W.Wong
Abstract WefirstintroducethediscreteFourier–Wignertransformandthediscrete Wignertransformactingonfunctionsin L2 (Z).Weprovethatpropertiesofthe standardWignertransformoffunctionsin L2 (Rn ) suchastheMoyalidentity,the inversionformula,time-frequencymarginalconditions,andtheresolutionformula holdfortheWignertransformsoffunctionsin L2 (Z).UsingthediscreteWigner transform,wedefinethediscreteWeyltransformcorrespondingtoasuitablesymbol on Z × S1 .Wegiveanecessaryandsufficientconditionfortheself-adjointnessof thediscreteWeyltransform.Moreover,wegiveanecessaryandsufficientcondition foradiscreteWeyltransformtobeaHilbert–Schmidtoperator.Thenweshowhow wecanreconstructthesymbolfromitscorrespondingWeyltransform.Weprove thattheproductoftwoWeyltransformsisagainaWeyltransformandanexplicit formulaforthesymboloftheproductoftwoWeyltransformsisgiven.Thisresult givesanecessaryandsufficientconditionfortheWeyltransformtobeinthetrace class.
Keywords Fourier–Wignertransform·Wignertransform·Weyltransform· Moyalidentity·Time-frequencymarginalconditions·Wignerinversion formula·Weylinversionformula·Kernels·Hilbert–Schmidtoperators·Trace classoperators·Twistedconvolution·Weylcalculus
MathematicsSubjectClassification(2000) 47F05,47G30
ThisresearchhasbeensupportedbytheNaturalSciencesandEngineeringResearchCouncil ofCanadaunderDiscoveryGrant0008562.
S.Molahajloo DepartmentofMathematics,InstituteforAdvancedStudiesinBasicSciences,Zanjan,Iran
M.W.Wong( ) DepartmentofMathematicsandStatistics,YorkUniversity,Toronto,ON,Canada e-mail: mwwong@mathstat.yorku.ca
©SpringerNatureSwitzerlandAG2019
S.Molahajloo,M.W.Wong(eds.), AnalysisofPseudo-DifferentialOperators, TrendsinMathematics, https://doi.org/10.1007/978-3-030-05168-6_1
1Introduction
Toputthispaperinperspective,wefirstrecalltheWignertransformandtheWeyl transformmappingfunctionsin L2 (Rn ) intofunctionson,respectively, Rn × Rn and Rn
Let σ ∈ L2 (Rn × Rn ).ThentheWeyltransform Wσ : L2 (Rn ) → L2 (Rn ) correspondingtothesymbol σ isdefinedby
(Wσ f,g)L2 (Rn ) = (2π) n/2 Rn Rn σ(x,ξ)W(f,g)(x,ξ)dxdξ
forall f and g in L2 (Rn ), where W(f,g) istheWignertransformof f and g definedby
W(f,g)(x,ξ) = (2π) n/2
.
CloselyrelatedtotheWignertransform W(f,g) of f and g in L2 (Rn ) isthe Fourier–Wignertransform V(f,g) givenby
V(f,g)(q,p) = (2π) n/2 Rn e iq y f y + p 2 g y p 2 dy,q,p ∈ Rn
WeyltransformsandWignertransformson Rn havebeenextensivelystudiedin [5, 13]amongothers.
WeyltransformsongroupssuchastheHeisenberggroup,theupperhalfplane, andthePoincaréunitdiskareinvestigatedin[8, 10–12].CloselyrelatedtoWeyl transformsarepseudo-differentialoperatorsongroups.See,forinstance,[4, 7, 9, 15].
ThestrategythatweusetodeveloptheWeyltransformon Z istohavealook atthecaseof Rn ,wherethesymbol σ isafunctionon Rn × Rn .Recentworksin pseudo-differentialoperatorsandWeyltransformsontopologicalgroups G suggest thatthecorrectphasespacetoworkinis G × G,where G isthedualgroupof G. Thatthedualgroupof Rn isthesameas Rn isthereasonwhythephasespaceon whichsymbolsaredefinedis Rn × Rn .
Inthecaseofthegroup Z inthispaper,thedualgroupistheunitcircle S1 centeredattheoriginandthephasespace G × G isthen Z × S1 .
For1 ≤ p< ∞, thesetofallmeasurablefunctions F on Z suchthat
isdenotedby Lp (Z).Wedefine Lp (S1 ) tobethesetofallmeasurablefunctions f ontheunitcircle S1 withcenterattheoriginforwhich f p Lp (S1 ) = 1 2π π π |f(θ)|p dθ< ∞.
WedefinetheFouriertransform FZ F of F ∈ L1 (Z) tobethefunctionon S1 by
(FZ F ) (θ) = n∈Z e inθ F(n),θ ∈[−π,π ].
If f isasuitablefunctionon S1 ,thenwedefinetheFouriertransform FS1 f of f to bethefunctionon Z by
FS1 f (n) = 1 2π π π e inθ f(θ)dθ,n ∈ Z.
Notethat FZ : L2 (Z) → L2 (S1 ) isasurjectiveisomorphism.Infact, FZ = F 1 S1 = F ∗ S1 and FZ F L2 (S1 ) = F L2 (Z) ,F ∈ L2 (Z).
Let H beasuitablefunctionon S1 × Z.ThenwedefinetheFouriertransform FS1 ×Z H of H tobethefunctionon Z × S1 by
FS1 ×Z H (m,θ) = 1 2π π π n
Z e imφ +inθ H(φ,n)dφ,(m,θ) ∈ Z × S1
Similarly,forallsuitablefunctions K on Z × S1 ,wedefinetheFouriertransform
FZ×S1 K of K tobethefunctionon S1 × Z by
(FZ×S1 K)(θ,m) = 1 2π π π n∈Z e imφ +inθ K(n,φ)dφ,(θ,m) ∈ S1 × Z.
For1 ≤ p< ∞,wedefine Lp (Z× S1 ) tobethespaceofallmeasurablefunctions h on Z × S1 suchthat h p Lp (Z×S1 ) = 1 2π n∈Z π π |h(n,θ)|p dθ< ∞
InSect. 2,wedefinetheFourier–WignertransformandtheWignertransformas mappingsfrom L2 (Z) into,respectively, L2 (Z × S1 ) and L2 (S1 × Z).Thenwe showthatthediscreteFourier–WignertransformandthediscreteWignertransform satisfytheMoyalidentity.Wegiveaninversionformulatoreconstructafunction fromitsdiscreteWignertransformuptoaconstantfactor.Thenwegivethetimeand frequencymarginalconditionsandaconvolutiontheoremforthediscreteWigner transform.TheresultsinthissectionareanalogsoftheresultsfortheWigner transformson Rn givenin[1, 13].InSect. 3,weusethediscreteWignertransform todefinetheWeyltransformon Z.AcharacterizationofHilbert–Schmidtdiscrete Weyltransformsisalsogiven.TheWeylinversionformularecoveringasymbol fromthecorrespondingdiscreteWeyltransformisgiven.InSect. 4,wepresentthe WeylcalculusgivingthesymboloftheadjointofadiscreteWeyltransformon L2 (Z) andthesymboloftheproductoftwodiscreteWeyltransforms.Theadjoint formulagivesacharacterizationofself-adjointdiscreteWeyltransformsandthe productformulagivesacharacterizationoftraceclassdiscreteWeyltransforms.
Weuse Ze and Zo todenote,respectively,thesetofallevenintegersandtheset ofalloddintegers.
2DiscreteFourier–WignerTransformsandDiscreteWigner
Transforms
Let F ∈ L2 (Z).Thenforall (n,θ) ∈ Z × S1 ,wedefine ρ(n,θ)F tobethefunction on Z by
i(k + n 2 )θ F(k + n),
(ρ(n,θ)F ) (k) =
i(k + n 1 2 )θ F(k + n), n ∈ Ze , n ∈ Zo ,
forall k ∈ Z. Notethatforall (n,θ) ∈ Z × S1 , ρ(n,θ) : L2 (Z) → L2 (Z) isa unitaryoperatorand
ρ(n,θ)∗ = ρ( n, θ).
Forallfunctions F and G in L2 (Z),wedefinetheFourier–Wignertransform V(F,G) of F and G tobethefunctionon Z × S1 by
V(F,G)(n,θ) = (ρ(n,θ)F,G)L2 (Z) ,(n,θ) ∈ Z × S1
Thereforeforall (n,θ) ∈ Z × S1 ,
V(F,G)(n,θ) = k ∈Z e i(k + n 2 )θ F(k + n)G(k), k ∈Z e i(k + n 1 2 )θ F(k + n)G(k), n ∈ Ze , n ∈ Zo
Bythechangeofvariablesfrom k to m using m = k + n 2 , m = k + n 1 2 , n ∈ Ze , n ∈ Zo ,
weget
V(F,G)(n,θ) = m∈Z e imθ F(m + n 2 )G(m n 2 ), m∈Z e imθ F(m + n+1 2 )G(m n 1 2 ), n ∈ Ze , n ∈ Zo .
Infact,ifwelet
Hn (m) = F(m + n 2 )G(m n 2 ), F(m + n+1 2 )G(m n 1 2 ), n ∈ Ze , n ∈ Zo
Then
V(F,G)(n,θ) = (FZ Hn ) (θ). (2.1)
WehavethefollowingMoyalidentityforthediscreteFourier–Wignertransform.
Theorem2.1 Let F1 ,F2 ,G1 ,and G2 befunctionsin L2 (Z).Then V(F1 ,G1 ),V(F2 ,G2 ) L2 (Z×S1 ) = F1 ,F
Proof For j = 1, 2,welet
(m)
Thenby(2.1)andtheParsevalidentity,
Therefore (V(F1 ,G1 ),V(F2 ,G2 ))L
If n ∈ Ze ,thenwemakethechangeofvariablesfrom (m,n) to (k1 ,l1 ) by k1 = m + n 2 and l1 = m n 2 .If n ∈ Zo ,thenthechangeofvariablesfrom (m,n) to (k2 ,l2 ) isgivenby k2 =
.Weget
Let F and G befunctionsin L2 (Z).ThenwedefinetheWignertransform W(F,G) of F and G tobethefunctionon S1 × Z by W(F,G) = FZ×S1 V(F,G).
Theorem2.2 Forall (φ,m) ∈ S1 × Z,
Proof WebeginwiththedefinitionofthediscreteWignertransformtotheeffect that W(F,G)(θ,m)
(FZ×S1 V(F,G))(θ,m)
Wecarryoutthesumover n ∈ Z byfirstperformingthesumover n ∈ Ze andthen over n ∈ Zo Summingoverallevenintegersgives
forall (θ,m) ∈ S1 × Z Thesumover n ∈ Zo canbecalculatedsimilarly. Similarly,wehavetheMoyalidentityfortheWignertransform.
Theorem2.3 Let F1 ,F2 ,G1 ,and G2 befunctionsin L2 (Z).Then W(F1 ,G1 ),W(F2 ,G2 ) L2 (S1 ×
AsinthecaseofWignertransformson Rn ,thefollowingpropositionguarantees thatforall F ∈ L2 (Z), W(F,F) isreal.
Proposition2.4 Let F and G befunctionsin L2 (Z).Then
W(F,G) = W(G,F).
Inparticular, W(F,F) isareal-valuedfunctionon S1 × Z.
Proof Forall (φ,m) ∈ S1 × Z, wegetby(2.2)
W(F,G)(φ,m)
Ifwechangetheindexofsummationfrom n to k by n =−k, thenforall (φ,m) ∈ S1 × Z, W(F,G)(φ,m)
= W(G,F)(φ,m).
Thiscompletestheproof.
Forsimplicity,wedenote W(F,F) by W(F) forallfunctions F ∈ L2 (Z).The followingtheoremstatesthatwecanreconstructtheoriginalfunction F fromits Wignertransform W(F) uptoaconstantfactor.
Theorem2.5 Let F ∈ L2 (Z).Thenforall n ∈ Z, F(n)F(0) =
W(F)(φ, n 2 )dφ,n ∈ Ze ,
W(F)(φ, n 1 2 )dφ,n ∈ Zo .
Proof By(2.1)andthedefinitionoftheWignertransform,forall m and n in Z,we have Hn (m) = (FS1 (W(F)( ,m)) )(n).
First,weassumethat n ∈ Ze .Thenforall m ∈ Z,weget
Now,let m = n 2 .Then
F(n)F(0) =
inφ W(F) φ, n 2 dφ.
Similarly,weobtain F(n)F(0),for n ∈ Zo byletting m = n 1 2 . WehavethetimeandfrequencymarginalconditionsforthediscreteWigner transform.
Proposition2.6 Let F ∈ L2 (Z).Then
(i) Forall m ∈ Z,
= 2π |F(m)|2 .
(ii) Forall φ ∈[−π,π ], m∈Z W(F)(φ,m) =| (FZ F ) (φ)|2 .
Proof Let n = 0in(2.3).Thenwegetpart(i).Toprovepart(ii),wehaveforall φ ∈[−π,π ],
W(F)(φ,m)
m∈Z
= (FZ (W(F)(φ, ·)) )(0)
Forall n ∈ Ze , wemakethechangeofvariablesfrom (m,n) to (k1 ,l1 ) by k1 = m + n 2 and l1 = m n 2 .Thenweget
(2.4)
andforall n ∈ Zo ,usingthechangeofvariablesfrom (m,n) to (k2 ,l2 ) givenby k2 = m + n+1 2 and l2 = m n 1 2 ,weget
(2.5)
Thereforeweget
W(F)(φ,m)dφ
m∈Z
k ∈Z l ∈Z e i(k l)φ F(k)F(l) =| (FZ F ) (φ)|2 andtheproofiscomplete.
F(k
)F(l
Let T : L2 (Z × Z) → L2 (Z × Z) bethetwistingoperatordefinedby (TF)(n,m) = F(m + n
forall F ∈ L2 (Z × Z) andall (n,m) ∈ Z × Z.Infact, T : L2 (Z × Z) → L2 (Z × Z) isaunitaryoperatoranditsinverse T 1 isgivenby
(T 1 F)(n,m) = F(n m, m+n 2 ),n + m ∈ Ze , F(n m, m+n 1 2 ),n + m ∈ Zo
Moreover,forall F and G in L2 (Z),
W(F,G)(φ,m) = F1,Z T(F ⊗ G) (φ,m),(φ,m) ∈ S1 × Z, (2.6)
where F ⊗ G isthetensorproductof F and G givenby
(F ⊗ G)(n,m) = F(n)G(m),(n,m) ∈ Z × Z, and F1,Z T(F ⊗ G) isthepartialFouriertransformof T(F ⊗ G) withrespectto thefirstvariable.Thefollowingpropositiongivestheshift-invarianceoftheWigner transformandtheproofisstraightforward.
Proposition2.7 Let F ∈ L2 (Z).For θ ∈[−π,π ] and k ∈ Z,wedefinethefunction G on Z by
G(n) = e inθ F(n k),n ∈ Z
Then
W(G)(φ,m) = W(F)(φ + θ,m k),(φ,m) ∈ S1 × Z
WecannowgivearesultontheWignertransformoftheproductoftwofunctions on Z
Proposition2.8 Let F and G befunctionsin L2 (Z).Thenforall (φ,m) in S1 × Z,
W(FG)(φ,m) = W(F)( ,m) ∗ W(G)( ,m) (φ),
where ∗ istheconvolutionon S1 definedby (f ∗ g)(φ) = 1 2π π π f(φ θ)g(θ)dθ
forall f and g in L2 (S1 ).
Proof Let (φ,m) ∈ S1 × Z. Then
(W(F)(·,m) ∗ W(G)(·,m))(φ) = 1 2π π π W(F)(φ θ,m)W(G)(θ,m)dθ
itfollowsthat (W(F)(·,m) ∗ W(G)(·,m))(φ)
= W(FG)(φ,m) forall (φ,m) ∈ S1 × Z.
3DiscreteWeylTransforms Let σ ∈ L2 (Z×S1 ).Thenforallfunctions F in L2 (Z),wedefinetheWeyltransform Wσ F correspondingtothesymbol σ by (Wσ F,G) L
forall G ∈ L2 (Z).Infact,
(Wσ F,G) L2 (Z)
1
If n ∈ Ze ,thenwemakethechangeofvariablesfrom (m,n) to (k1 ,l1 ) by k1 = m + n 2 and l1 = m n 2 .If n ∈ Zo , thechangefrom (m,n) to (k2 ,l2 ) iseffectedby k2 = m + n+1 2 and l2 = m n 1 2 .(See(2.4)and(2.5)inthisconnection.)Therefore weobtain
2π (Wσ F,G) L2 (Z)
l2 ∈Z
Thereforeforall l ∈ Z, (Wσ F ) (l) = 1
(k l)+1)φ σ(k,φ)F(2k + 1 l)G(l)dφ.
i(2(k l)+1)φ σ(k,φ)F(2k + 1 l)dφ.
Byanotherchangeofvariables,weget (Wσ F ) (l) = m∈Z kσ (l,m)F(m),
where kσ isthekernelof Wσ givenby
(l,m) =
Therefore
where F2,S1 σ istheFouriertransformon S1 of σ withrespecttothesecondvariable. ThefollowingtheoremisaninversionformulafordiscreteWeyltransforms.It showshowwecanreconstructthesymbolfromthecorrespondingWeyltransform. ForWeyltransformson Rn ,thecorrespondingformulaandotherrelatedformulas canbefoundin[2, 3, 6].
Theorem3.1 Let σ ∈ L2 (Z × S1 ) besuchthat ρ(n,θ)Wσ isatraceclassoperator forall (n,θ) ∈ Z × S1 .Thenforall (θ,n) ∈ S1 × Z,wehave (FZ×S1 σ)(θ,n) = tr(ρ(n,θ)Wσ ).
Proof Let F ∈ L2 (Z).Firstweassumethat n ∈ Ze .Thenforall l ∈ Z,
(ρ(n,θ)Wσ F ) (l) = e i(l + n 2 )θ (Wσ F ) (l + n) = e i(l + n 2 )θ m∈Z kσ (l + n,m)F(m),
where kσ isthekernelof Wσ . So,forall l ∈ Z,
(ρ(n,θ)Wσ F ) (l) = m∈Z k θ (l,m)F(m), where k θ (l,m) = e i(l + n 2 )θ kσ (l + n,m).
Hence
tr(ρ(n,θ)Wσ ) = l ∈Z k θ (l,l)
Bythechangeofvariablesfrom l to k by k = l + n 2 ,wegetforall (θ,n) ∈ S1 × Z, tr(ρ(n,θ)Wσ ) = FZ×S1 σ (θ,n).
Similarly,theaboveformulaholdsforall n ∈ Zo
4Hilbert–SchmidtDiscreteWeylTransforms
Thefollowingpropositiongivesaclass ofboundedandHilbert–SchmidtWeyl transformson L2 (Z).
Proposition4.1 Let σ ∈ L2 (Z × S1 ).Then Wσ : L2 (Z) → L2 (Z) isabounded linearoperatorand
where · ∗ isthenorminthe C ∗ -algebraofallboundedlinearoperatorson L2 (Z) Moreover, Wσ isaHilbert–Schmidtoperatoron L2 (Z) and Wσ HS(L
Proof Ifwedefinethefunction ˜ σ on S1 × Z by ˜ σ(θ,n) = σ(n,θ),(θ,n) ∈ S1 × Z, then ˜ σ ∈ L2 (S1 × Z) and
Let F and G befunctionsin L2 (Z).ThenbySchwarz’sinequalityandtheMoyal identityfortheWignertransform,wehave |(Wσ F,G)L2 (Z) |=|( ˜ σ,W(G,F))L2 (S1 ×Z) |
Hence
Therefore
By(3.1),
BythechangeofvariablesandtheParsevalidentity,weget
AHilbert–Schmidtoperatoron A : L2 (Z) → L2 (Z) isoftheform
(AF)(n) = m∈Z h(n,m)F(m),F ∈ L2 (Z),
where h isafunctionin L2 (Z × Z).Thefunction h iscalledthekerneloftheHilbert–Schmidtoperator A on L2 (Z). ThefollowingtheoremstatesthateveryHilbert–Schmidtoperatoron L2 (Z) isaWeyltransformwithsymbolin L2 (Z × S1 ).
Theorem4.2 Let A : L2 (Z) → L2 (Z) beaHilbert–Schmidtoperator.Thenthere existsauniquesymbol σ in L2 (Z × S1 ) suchthat A = Wσ .
Proof Let h ∈ L2 (Z × Z) besuchthat
(AF ) (n) = m∈Z
h(n,m)F(m),F ∈ L2 (Z).
Thenforall F,G ∈ L2 (Z),
(AF,G) L2 (Z) = n∈Z m∈Z
h(n,m)F(m)G(n) = n∈Z m∈Z ˜ h(m,n) F ⊗ G (m,n) = (F ⊗ G, h)L2 (Z×Z) ,
where ˜ h isthefunctionin L2 (Z × Z) suchthat
h(m,n) = h(n,m),(m,n) ∈ Z × Z.
Wedefinethefunction σ on Z × S1 by σ = F1,Z T ˜ h ∼ ,
where T : L2 (Z × Z) → L2 (Z × Z) isthetwistingoperatordefinedearlier.Then ˜ h = T 1 F1,S1 ˜ σ .Hencewehave
(AF,G)L2 (Z) = F ⊗ G,T 1 F1,S1 σ L2 (Z×Z) = F1,Z T(F ⊗ G), ˜ σ L2 (S1 ×Z) = W(F,G), ˜ σ L2 (S1 ×Z) = (Wσ F,G)L2 (Z) .
5TheWeylCalculus
ThefollowingpropositionontheadjointofadiscreteWeyltransformon L2 (Z) followsdirectlyfromthedefinitionoftheWeyltransformandProposition 2.4.
Proposition5.1 Let σ ∈ L2 (Z × S1 ).Then W ∗ σ = Wσ ,where W ∗ σ : L2 (Z) → L2 (Z) istheadjointof Wσ : L2 (Z) → L2 (Z).Inparticular, Wσ isself-adjointif andonlyif σ isreal-valued.
Proof Let σ ∈ L2 (Z × S1 ). Then (W ∗ σ F,G)L2 (Z) = (F,Wσ G)L2 (Z) = (Wσ G,F)L2 (Z) = 1
ByProposition 2.4, W(G,F) = W(F,G), andhenceby(5.1), (W ∗ σ F,G)L2 (Z) = 1
So, (W ∗ σ F,G)L2 (Z) = (Wσ F,G)L2 (Z) andtheproofiscomplete.
Let σ ∈ L2 (Z × S1 ).Forsimplicity,wedenote FZ×S1 σ by σ .
Lemma5.2 Let σ ∈ L2 (Z × S1 ).Thenforall F ∈ L2 (Z), (Wσ F ) (m) = 1
(ρ(n,θ)F ) (m)dθ,m ∈ Z.
Proof Let F and G bein L2 (Z). Thenusingtheadjointformula, (Wσ F,G) L2 (Z) =
(ρ(n,θ)F,G) L2 (Z) dθ = 1
σ(θ,n) (ρ(n,θ)F ) (m)G(m)dθ.
(5.1)
Hence
(Wσ F ) (m) = 1 2π π π n∈Z σ(θ,n) (ρ(n,θ)F ) (m)dθ,m ∈ Z.
Lemma5.3 Forall (n,θ) and (k,φ) in Z × S1 ,wehave
ρ(n,θ)ρ(k,φ) = e i [(n,θ);(k,φ)] ρ(n + k,θ + φ), where [(n,θ); (k,φ)] =
Theproofofthelemmaisstraightforward.Let F and G besuitablefunctionson S1 × Z.Thenwedefinethetwistedconvolution F G of F and G tobethefunction on S1 × Z by (F G) (γ,l) = 1 2π π π k ∈Z e i [(l k,γ φ);(k,φ)] F(γ φ,l k)G(φ,k)dφ,(γ,l) ∈ S1 × Z.
Let σ ∈ L2 (Z × S1 ).Thefollowingtheoremguaranteesthattheproductoftwo WeyltransformsisstillaWeyltransform.
Theorem5.4 Let σ and τ besymbolsin L2 (Z × S1 ). Then Wσ Wτ = Wω , where
Proof Let F ∈ L2 (Z). Thenforall m ∈ Z,wegetbyLemma 5.2 (Wσ Wτ F ) (m) = 1 2π π π n∈Z σ(θ,n) (ρ(n,θ)Wτ F ) (m)dθ = 1 (2π)2 π π n∈Z π π k ∈Z σ(θ,n)τ(φ,k) (ρ(n,θ)ρ(k,φ)F ) (m)dφdθ.
Now,byLemma 5.3,wehave
(Wσ Wτ F ) (m) = 1 (2π)2
σ(θ,n)τ(φ,k)e i [(n,θ);(k,φ)] × (ρ(n + k,θ + φ)F ) (m)dφdθ.
Let l = n + k and γ = θ + φ .Thenweget
(Wσ Wτ F ) (m) = 1 (2π)2
Let ω ∈ L2 (Z × S1 ) besuchthat
φ,l k)τ(φ,k)e i [(l k,γ φ);(k,φ)] × (ρ(l,γ)F ) (m)dφdγ.
Then (Wσ Wτ F ) (m) = 1 2
forall m ∈ Z.
(ρ(l,γ)F ) (m)dγ = (Wω F ) (m)
Asanapplicationoftheproductformula,wegiveacharacterizationoftrace classdiscreteWeyltransforms.ItisananalogforthediscreteWeyltransformof thecharacterizationoftraceclassWeyltransformson Rn in[14].Let W betheset definedby W = FS1 ×Z ( ˆ
ˆ τ) : σ,τ ∈ L2 (Z × S1 ) .
Let S1 (L2 (Z)) bethespaceofalltraceclassoperatorson L2 (Z).Thefollowing theoremgivesacharacterizationoftraceclassdiscreteWeyltransformson L2 (Z)
Theorem5.5 Let σ ∈ L2 (Z × S1 ).Then Wσ : L2 (Z) → L2 (Z) isin S1 (L2 (Z)) if andonlyif σ ∈ W .Moreover,if σ = FS1 ×Z ( ˆ α ˆ β) with α and β in L2 (Z × S1 ), then
S1 (L2 (Z))
L2 (Z×S1 )
L2 (Z×S1 )
Proof Firstweassumethat σ ∈ W .Then
= FS1 ×Z (
forsome α and β in L2 (Z × S1 ). ByTheorem 5.4,
Wσ = Wα Wβ
Moreover,byProposition 4.1, Wα and Wβ areHilbert–Schmidtoperators.Hence Wσ istheproductoftwoHilbert–Schmidtoperatorson L2 (Z) andthereforeisin S1 (L2 (Z)).Conversely,assumethat Wσ ∈ S1 (L2 (Z)). Then Wσ istheproduct oftwoHilbert–Schmidtoperatorson L2 (Z).HencebyTheorem 4.2,thereexist symbols α and β in L2 (Z × S1 ) suchthat
Wσ = Wα Wβ
So,byTheorem 5.4, σ ∈ W.
References
1.L.Cohen, Time-FrequencyAnalysis (Prentice-HallPTR,UpperSaddleRiver,1995)
2.L.Cohen, TheWeylOperatorandItsGeneralization (Birkhäuser,Basel,2013)
3.L.Cohen,InverseWeyltransform/operator.J.Pseudo-Differ.Oper.Appl. 8,661–678(2017)
4.A.Dasgupta,M.W.Wong,Pseudo-differentialoperatorsontheaffinegroup,in PseudoDifferentialOperators:Groups,GeometryandApplications.TrendsinMathematics (Birkhäuser,Basel,2017),pp.1–14
5.M.deGosson, TheWignerTransform (WorldScientific,Singapore,2017)
6.X.Duan,M.W.Wong,Pseudo-differentialoperatorsforWeyltransforms. Politehn.Univ. BucharestSci.Bull.Ser.A:Appl.Math.Phys. 75,3–12(2013)
7.S.Molahajloo,Pseudo-differentialoperatorson Z,in NewDevelopmentsinPseudoDifferentialOperators.OperatorTheory:AdvancesandApplications,vol.205(Birkhäuser, Basel,2009),pp.213–221
8.S.Molahajloo,Pseudo-differentialoperators,WignertransformsandWeyltransformsonthe Poincaréunitdisk.Compl.Anal.Oper.Theory 12,811–833(2018)
9.S.Molahajloo,M.W.Wong,Ellipticity,Fredholmnessandspectralinvarianceofpseudodifferentialoperatorson S1 .J.Pseudo-Differ.Oper.Appl. 1,183–205(2010)
10.L.Peng,J.Zhao,WeyltransformsassociatedwiththeHeisenberggroup.Bull.Sci.Math. 132, 78–86(2008)
11.L.Peng,J.Zhao,Weyltransformsontheupperhalfplane.Rev.Mat.Complut. 23,77–95 (2010)
12.T.Tate,Weylpseudo-differentialoperatorandWignertransformonthePoincarédisk.Ann. GlobalAnal.Geom 22,29–48(2002)
13.M.W.Wong, WeylTransforms (Springer,NewYork,1998)
14.M.W.Wong,TraceclassWeyltransforms,in RecentAdvancesinOperatorTheoryand itsApplications.OperatorTheory:AdvancesandApplications,vol.160(Birkhäuser,Basel, 2005),pp.469–478
15.M.W.Wong, DiscreteFourierAnalysis (Birkhäuser,Basel,2011)
