1

Quantum Harmonic Analysis and its Applicationsjointly with Eirik SkrettinglandNordfjordeid Summer School 2019Analysis, Geometry and PDE

Franz LuefDepartment of mathematical sciences, NTNU

July 1-5, 2019

References

• F. Luef, E. Skrettingland: Convolutions for localizationoperators. J. Math. Pures Appl.:118(9), 288–316, 2018.• F. Luef, E. Skrettingland: Convolutions for Berezin quantization

and Berezin-Lieb inequalities. J. Math. Phys. 59, 023502,2018.• F. Luef, E. Skrettingland: Mixed-state localization operators:

Cohen’s class and trace class operators. J. Fourier Anal. App.,to appear.• F. Luef, E. Skrettingland: On accumulated Cohen’s class

distributions and mixed-state localization operators. Const.Approx., to appear.• R. Werner. Quantum harmonic analysis on phase space. J.

Math. Phys. 25(5):1404–1411, 1984.

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Harmonic Analysis

Harmonic analysis is a branch of mathematics concerned with therepresentation of functions or signals as the superposition of basicwaves, and the study of and generalization of the notions of Fourierseries and Fourier transforms (i.e. an extended form of Fourieranalysis).

Quantum mechanicsQuantum mechanics describes the physics of atoms andmolecules. His mathematical formulation is linked with therepresentation theory of the Heisenberg group and the theory ofstates and density matrices.

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Harmonic analysis in a nutshell

Basic notions and facts

• Translation Tx f (t) = f (t − x)

• Convolution (f ∗ g)(x) =∫

f (y)g(x − y)dy =∫

f (y)Ty g(x) dy

• Fourier transform f (ω) = F f (ω) =∫

f (t)e−2πiωtdt• F(f ∗ g) = F(f )F(g)

Algebraic structure

• L1(Rd ) is a Banach convolution algebra: ‖f ∗ g‖1 ≤ ‖f‖1‖g‖1.• For 1 ≤ p <∞ the space Lp(Rd ) is a Banach module over

L1(Rd ) for the module action (f ,g) 7→ f ∗ g for f ∈ Lp(Rd ) andf ∈ L1(Rd ).

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Riemann-Lebesgue

The Fourier transform of an L1(Rd ) function vanishes at∞.

Hausdorff-Young

For p ∈ [1,2] and q = p/(p − 1) we have ‖f‖q ≤ p1/2pq−1/2q‖f‖p.

Wiener Tauberian Theorem

• span{Tz f : z ∈ R} is norm dense in L1(R) if and only if{z ∈ R : f (z) = 0} is empty.• span{Tz f : z ∈ R} is norm dense in L2(R) if and only if{z ∈ R : f (z) = 0} has Lebesgue measure zero.

• span{Tz f : z ∈ R} is a weak* dense subspace of L∞(R) if andonly if {z ∈ R : f (z) = 0} has dense complement.

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Spectrogram

6

Blurred image

7

Basic building blocks

−20 0 20 40

−40

−20

0

20

40

nz = 3

three points in TF−plane

−200 −100 0 100 2000

0.1

0.2

0.3

0.4Gabor atom

−200 −100 0 100 2000

0.1

0.2

0.3

0.4shifted version of atom

−100 0 100−0.2

−0.1

0

0.1

0.2

0.3

modulated version of atom

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Basic notions – Time-frequency analysis

Given φ, ψ : Rd → C and z = (x , ω) ∈ R2d .• translation operator Tx by Txψ(t) = ψ(t − x),• modulation operator Mω by Mωψ(t) = e2πiω·tψ(t)• time-frequency shifts π(z) by π(z) = MωTx

• symmetric time-frequency shifts Mω/2TxMω/2 = e−πix ·ωπ(z)

Commutation relation

TxMω = e2πixωMωTx

TxMωψ(t) = e2πi(t−x)ψ(t − x)

= e−2πixωe2πi(t−x)ψ(t − x)

= e−2πixωMωTxψ(t)9

Basic notions – Time-frequency analysis

Time-frequency representations

• short-time Fourier transform (STFT)Vφψ(z) = 〈ψ, π(z)φ〉 =

∫Rd ψ(t)φ(t − x)e−2πiωtdt

• ambiguity function A(φ, ψ)(z) = 〈ψ, π(z)φ〉 = eπixωVφψ(z)

• spectrogram |Vφψ|2

• cross-Wigner distributionW (ψ, φ)(x , ω) =

∫Rd ψ

(x + t

2

)φ(x − t

2

)e−2πiω·t dt .

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Basic notions – Time-frequency analysis

Symplectic Fourier transform

Standard symplectic form σ is defined for (x1, ω1), (x2, ω2) ∈ R2d byσ(x1, ω1; x2, ω2) = ω1 · x2 − ω2 · x1 and the symplectic Fouriertransform Fσf of f by

Fσf (z) =

∫∫R2d

f (z ′)e−2πiσ(z,z′) dz ′

• The symplectic Fourier transform and the regular Fouriertransform F f (z) =

∫∫R2d f (z ′)e−2πiz·z′ dz ′ are related by

Fσf (x , ω) = F f (ω,−x).• F2

σ = I and F−1σ = Fσ

• W (ψ, φ) = FσA(ψ, φ)

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Shift for operators

One can obtain a unitary representation α of R2d on theHilbert-Schmidt operators T 2 by defining

αz(A) = π(z)Aπ(z)∗ for z ∈ R2d , A ∈ B(L2(Rd )).

αzαz′ = αz+z′ , and thus α is a shift or translation of operators.

RemarkSince we defined α by αzT = π(z)Tπ(z)∗ for z = (x , ω) ∈ R2d , wecan modify π by any phase factor without affecting α. In particularthe family of representations πλ(z) = TλxMωT(1−λ)x would all givethe same α for λ ∈ [0,1].

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Observations

• The map z 7→ αzT is continuous from R2d to K (L2(Rd )) forany fixed T ∈ K (L2(Rd )).

• The map z 7→ αzA is weak*-continuous from R2d to B(L2(Rd ))for any fixed A ∈ B(L2(Rd )).

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Schrödinger representation

Time-frequency shifts – Basic identities

• π(z)∗ = e2πixωπ(−z)

• π(z + z ′) = e2πixω′π(z)π(z ′)• π(z)π(z ′) = e2πiσ(z,z′)π(z ′)π(z), i.e. αzπ(z ′) = e2πiσ(z,z′)π(z ′)

Projective representation

In other words, the time-frequency shifts π(z) give a projectiverepresentation of R2d on L2(Rd ) with respect to the cocyclec(z, z ′) = e−2πiω′·x .

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Moyal’s identity

If ψ1, ψ2, φ1, φ2 ∈ L2(Rd ), then Vφiψj ∈ L2(R2d ) for i , j ∈ {1,2}, andthe relation

〈Vφ1ψ1,Vφ2ψ2〉 = 〈ψ1, ψ2〉〈φ1, φ2〉

holds, where the leftmost inner product is in L2(R2d ) and those onthe right are in L2(Rd ).

Moyal’s identity implies a reconstruction formula:

Reconstruction formulaSuppose φ, ψ ∈ L2(Rd ). Then for any ξ ∈ L2(Rd ) we have

ξ = 〈ψ, φ〉−1∫∫

RdVφξ(z)π(z)ψ dz.

Consequence

The Schrödinger representation is irreducible.15

Spectral decomposition of compact operators

Let S be a compact operator on L2(Rd ). There exist twoorthonormal sets {ψn}n∈N and {φn}n∈N in L2(Rd ) and a sequence{sn(S)}n∈N of positive numbers with sn(S)→ 0, such that S maybe expressed as

S =∑n∈N

sn(S)ψn ⊗ φn,

with convergence in the operator norm. The numbers {sn(S)}n∈Nare called the singular values of S, and are the eigenvalues of theoperator |S| = (S∗S)1/2.

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Schatten classesFor 1 ≤ p <∞ we define the Schatten class T p of operators by

T p = {T compact : (sn(T ))n∈N ∈ `p}.

We will also write T ∞ = B(L2(Rd )) with ‖ · ‖T∞ given by theoperator norm to simplify the statement of some results.

Basic properties

The Schatten class T p becomes a Banach space under pointwiseaddition and scalar multiplication in the norm

‖S‖T p =

(∑n∈N

sn(S)p

)1/p

.

We have ‖ · ‖B(L2(Rd )) ≤ ‖ · ‖p ≤ ‖ · ‖1 for 1 ≤ p ≤ ∞. Furthermore,the spaces T p are ideals in B(L2(Rd )), i.e. for A ∈ B(L2(Rd )) andT ∈ T p we have that AT ,TA ∈ T p

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Recall that an operator S ∈ B(L2(Rd )) is positive if 〈Sψ,ψ〉 ≥ 0 forany ψ ∈ L2(Rd ).

Trace class operators

For a positive operator S ∈ B(L2(Rd )), the trace of S is defined tobe

tr(S) =∑n∈N〈Sen,en〉, (1)

where {en}n∈N is an orthonormal basis for L2(Rd ).If S ∈ T 1, then tr(S) is well-defined and a simple calculation showsthat

tr(S) =∑n∈N

sn(S),

where the sum of singular values converges by the definition of T 1.For this reason the class T 1 is often referred to as trace classoperators.

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RemarkThis definition is independent of the orthonormal basis used, andthe trace is linear and satisfies tr(ST ) = tr(TS). However, theexpression in the defintion may well be infinite, and is notwell-defined for a general non-positive operator S.

Duality

Let 1 ≤ p <∞, and let q be the number determined by 1p + 1

q = 1.The dual space of T p is T q, and the duality may be given by

〈T ,S〉T q ,T p = tr(TS)

for S ∈ T p and T ∈ T q.

Hilbert-Schmidt operators

Another well-known Schatten class is T 2, known as theHilbert-Schmidt operators. T 2 is a Hilbert space under the innerproduct 〈S,T 〉T 2 := tr(ST ∗) for S,T ∈ T 2.

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RemarkThe Schatten classes behave analogously to the Lp-spaces offunctions – the duality relations are the same, and both L1(R2d )and T 1 have a natural linear functional given by the integral andtrace, respectively. The intuition that Lp corresponds to T p willoften be useful.

Trace-Properties

Let S ∈ T 1(H), A ∈ B(H).1. S∗ ∈ T 1(H), and tr(S∗) = tr(S).2. tr(AS) = tr(SA).3.∑

n∈N|〈ASen,en〉| ≤ ‖A‖B(L2)‖S‖T 1 .

4. |tr(AS)| ≤ ‖A‖B(L2)‖S‖T 1 .

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Parity operator

we define the analogue of f 7→ f for an operator A ∈ B(L2(Rd )) by

A = PAP,

where P is the parity operator.

LemmaLet A ∈ B(L2(R2d )) and z, z ′ ∈ R2d .

1. If T ∈ T p for 1 ≤ p ≤ ∞, then ‖αzT‖T p = ‖T‖T p and‖T‖T p = ‖T‖T p .

2. (αzA)∗ = αzA∗ and(A)∗

= (A∗)ˇ.

3. π(z)P = Pπ(−z), ˇπ(z) = π(−z) and (αzA)ˇ = α−zA.

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Vector-valued integration

We will need to integrate operator-valued functionsG : R2d → B(L2(Rd )) of the form G(z) = g(z)F (z), whereg ∈ L1(R2d ) and F : R2d → B(L2(Rd )) is measurable, bounded andstrongly continuous. The operator-valued integral∫∫

R2d g(z)F (z) dz ∈ B(L2(Rd )) is defined in a weak and pointwisesense: for any ψ ∈ L2(Rd ) we define

(∫∫R2d g(z)F (z) dz

)ψ by

〈(∫∫

R2dg(z)F (z) dz

)ψ, φ〉 =

∫∫R2d

g(z)〈F (z)ψ, φ〉 dz

for any φ ∈ L2(Rd ). This defines an operator∫∫

R2d g(z)F (z) dz,and we get the norm estimate‖∫∫

R2d g(z)F (z) dz‖B(L2(Rd )) ≤ ‖g‖L1 supz∈R2d ‖F (z)‖B(L2(Rd )). Forfixed ψ ∈ L2(Rd ) the L2(Rd )-valued function z 7→ g(z)F (z)ψ iseven Bochner integrable.

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Quantum variants of convolutions

• For S,T ∈ T 1 we define the convolution of S and T

S ∗ T (z) := tr(Sαz(T )),

where T := PTP is the parity operator.• For f ∈ L1(R2d ) and S ∈ T 1 we define the convolution of f and

S by

f ∗ S :=

∫∫R2d

f (y)αy (S) dy

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Rank-one caseGiven ϕ1, ϕ2 ∈ L2(Rd ) and let f be a function on R2d . Then

f ∗ (ϕ2 ⊗ ϕ1) =

∫∫R2d

f (z)Vϕ1(z)π(z)ϕ2 dz.

is the a localization operator, denoted by Aϕ1,ϕ2f .

ProofLet ψ ∈ L2(Rd ) and set S = φ2 ⊗ φ1.

(f ∗ S)(ψ) =

∫∫R2d

f (z)(αzS)(ψ) dz

=

∫∫R2d

f (z)〈π(z)∗ψ,ϕ1〉π(z)ϕ2 dz

=

∫∫R2d

f (z)Vϕ1ψ π(z)ϕ2 dz = Aϕ1,ϕ2f ψ.

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Rank-one caseThe Berezin transform of T with windows ϕ1 and ϕ2 is given by

Bϕ1,ϕ2T (z) = T ∗ (ϕ1 ⊗ ϕ2).

For T = ψ1 ⊗ ψ2 we get that

(ψ1 ⊗ ψ2) ∗ (ϕ1 ⊗ ϕ2) = Vϕ1ψ1Vϕ2ψ2.

We also note that the compact operators K (L2(Rd )) and theuniformly continuous functions vanishing at infinity, C0(R2d ), arecorresponding under convolutions with trace class operators.

LemmaLet T ∈ T 1. If f ∈ C0(R2d ) and S ∈ K (L2(Rd )), thenf ∗ T ∈ K (L2(Rd )) and S ∗ T ∈ C0(R2d ).

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General Moyal idenity

Let S,T ∈ T 1. The function z 7→ tr(SαzT ) for z ∈ R2d is integrableand

‖tr(SαzT )‖L1 ≤ ‖S‖T 1‖T‖T 1 .

Furthermore, ∫∫R2d

tr(SαzT ) dz = tr(S)tr(T ).

RemarkFor rank-one operators ψ1 ⊗ ψ2 and ϕ1 ⊗ ϕ2 with ψi , ϕj in L2(Rd )for i , j ∈ {1,2} the preceding equation gives the Moyal identity.

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ProofWe start by showing the norm-inequality. First use the singularvalue decomposition of the operators S and T to write

S =∑m∈N

smψm ⊗ φm T =∑n∈N

tnηn ⊗ ξn,

where {sm}m∈N and {tn}n∈N are the singular values of S and T ,respectively, and the sets {ψm}m∈N, {φm}m∈N, {ηn}n∈N and {ξn}n∈Nare orthonormal in L2(Rd ).Then extend the set {ψm}m∈N to an orthonormal basis {ei}i∈N ofL2(Rd ). Using this basis to calculate the trace, we find that

tr(SαzT ) =∑i∈N〈Sπ(z)Tπ(z)∗ei ,ei〉

=∑

i,m,n∈Nsmtn〈π(z)∗ei , ξn〉〈π(z)ηn, φm〉〈ψm,ei〉

=∑

m,n∈Nsmtn〈π(z)∗ψm, ξn〉〈π(z)ηn, φm〉

=∑

m,n∈NsmtnVξnψm(z)Vηnφm(z). (2)

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Proof-ctd.By Moyal’s identity, Vξnψm,Vηnφm ∈ L2(R2d ), and soVξnψmVηnφm ∈ L1(R2d ) by Hölder’s inequality. The followingcomputation shows that the series above converges absolutely inL1(Rd ) with the desired norm estimates.

‖∑

m,n∈NsmtnVξnψmVηnφm‖L1 ≤

∑m,n∈N

smtn‖VξnψmVηnφm‖L1

≤∑

m,n∈Nsmtn‖Vξnψm‖L2‖Vηnφm‖L2

=∑

m,n∈Nsmtn‖ξn‖L2‖ψm‖L2‖ηn‖L2‖φm‖L2

=∑

m,n∈Nsmtn = ‖S‖T 1‖T‖T 1 .

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Proof-ctd.The equality

∫∫R2d tr(SαzT ) dz = tr(S)tr(T ) now follows easily

from Moyal’s identity.∫∫R2d

tr(SαzT ) dz =

∫∫R2d

∑m,n∈N

smtnVξnψmVηnφm dz

=∑

m,n∈Nsmtn

∫∫R2d

VξnψmVηnφm dz

=∑

m,n∈Nsmtn〈ψm, φm〉〈ηn, ξn〉

= tr(S)tr(T ),

where the last equality follows from an easy calculation of tr(S)and tr(T ).

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TheoremThe convolution operations are commutative and associative.

proof

Commutativity: Let S,T ∈ T 1. We find that

S ∗ T (z) = tr(Sαz T )

= tr(Sπ(z)PTPπ(z)∗)

= tr(T (α−zS)ˇ)

= tr(TαzS) = T ∗ S(z)

We have made extensive use of the property tr(AB) = tr(BA).

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Associativity–Proof

The most interesting case is the convolution of three operators. Wewill need the general Moyal identity in addition to some moretechnical calculations. Let T1,T2,T3 ∈ T 1.To show that T1 ∗ (T2 ∗ T3) = (T1 ∗ T2) ∗ T3 it will be helpful toassume an arbitrary operator T0 ∈ T 1. If we can show that the dualspace actions 〈T1 ∗ (T2 ∗ T3),T0〉 = 〈(T1 ∗ T2) ∗ T3,T0〉 for any T0,we will have shown that the two expressions define the sameelement in the dual space B(L2(Rd )), and therefore the sameoperator. It will suffice to show that

tr [T0(T1 ∗ (T2 ∗ T3))] = tr [T0((T1 ∗ T2) ∗ T3)] .

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Associativity–Proof

Writing out the left side of the equation, we find that

tr [T0(T1 ∗ (T2 ∗ T3))] = tr[T0

∫∫R2d

tr(T2αx T3)αxT1 dx]

=

∫∫R2d

tr[T2αx T3

]tr [(αxT1)T0] dx

=

∫∫R2d

∫∫R2d

tr[(αxT1)T0αy (T3αx T2)

]dy dx .

The last equality uses the general Moyal identity to introduce thesecond integral, and also exploits the commutativity of convolutionsto switch the order of T2 and T3. It is a simple exercise to checkthat αy (AB) = (αyA)(αyB) for operators A and B, henceαy (T3αx T2) = (αyT3)(αxαy T2).

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Associativity–ProofUsing this in our calculation we get that∫∫

R2d

∫∫R2d

tr[(αx T1)T0αy (T3αx T2)

]dy dx

=

∫∫R2d

∫∫R2d

tr[(αx T1)T0(αy T3)(αxαy T2))

]dy dx

=

∫∫R2d

∫∫R2d

tr[(T0αy T3)(αxαy T2)(αx T1)

]dy dx .

As above, (αxαy T2)(αxT1) = αx ((αy T2)T1). We may use Fubini’stheorem to change the order of integration, and then invoke thegeneral Moyal idenity again to reduce the expression to a form thatwe recognize as the desired equality.

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Associativity–Proof

∫∫R2d

∫∫R2d

tr[(T0αyT3)αx ((αy T2)T1)

]dx dy

=

∫∫R2d

tr [T0αyT3] tr[(αy T2)T1

]dy

= tr [T0((T1 ∗ T2) ∗ T3)] .

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Using duality we can extend the domains of the convolutionsintroduced above, by allowing one factor to belong to the dualspace. For instance, if h ∈ L∞(R2d ) and S ∈ T 1, we defineh ∗ S ∈ B(L2(Rd )) by 〈h ∗ S,T 〉 = 〈h,T ∗ S∗〉 for every T ∈ T 1. Astandard interpolation argument then gives the following result,since (L1(Rd ),L∞(Rd ))θ = Lp and (T 1, T ∞)θ = T p with 1

p = 1− θ.

Proposition

Let 1 ≤ p,q, r ≤ ∞ be such that 1p + 1

q = 1 + 1r . If

f ∈ Lp(R2d ),g ∈ Lq(R2d ),S ∈ T p and T ∈ T q, then the followingconvolutions may be defined and satisfy the norm estimates

‖f ∗ T‖T r ≤ ‖f‖Lp‖T‖T q ,

‖g ∗ S‖T r ≤ ‖g‖Lq‖S‖T p ,

‖S ∗ T‖Lr ≤ ‖S‖T p‖T‖T q .

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Here is a more abstract aspect of the convolution of a function withan operator.

TheoremT p is a Banach module over L1(R2d ) for the module action given by(f ,S) 7→ f ? S for 1 ≤ p ≤ ∞ since ‖f ∗ T‖T p ≤ ‖f‖L1‖T‖T p .

Cohen-Hewitt factorization yields

FactorizationFor 1 ≤ p <∞, every element of T p can be written as f ∗ S forf ∈ L1(R2d ), S ∈ T p.

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We will now introduce an analogue of the Fourier transform for atrace class operator S.

Fourier-Wigner transform

The Fourier- Wigner transform FW S of S is the function given by

FW S(z) = e−πix ·ωtr(π(−z)S)

for z ∈ R2d .

Example

Let S = ϕ2 ⊗ ϕ1 with ϕ1, ϕ2 ∈ L2(Rd ). The Fourier-Wignertransform of S is given by

FW (ϕ2 ⊗ ϕ1)(z) = A(ϕ2, ϕ1)(z),

where A(ϕ2, ϕ1)(z) is the cross-ambiguity function.

37

GaussianConsider the Gaussian ϕ(t) = 2d/4e−πt ·t for t ∈ Rd and theoperator S = ϕ⊗ ϕ. We know that FW S = eπix ·ωVϕϕ(z), and thenfind that FW (ϕ⊗ ϕ)(z) = e2πix ·ωe−

12πz·z .

38

Another way of associating an operator to a function is to define theoperator as a superposition of time-frequency shifts using thetheory of vector-valued integration. The integrated Schrödingerrepresentation is the map ρ : L1(R2d )→ B(L2(Rd )) given by

ρ(f ) =

∫∫R2d

f (z)e−πix ·ωπ(z) dz,

where the integral is defined in the weak and pointwise sense. Wesay that f is the twisted Weyl symbol of ρ(f ).We will use the important product formula ρ(f )ρ(g) = ρ(f \g), wherethe product \ is the twisted convolution, defined by

f \g(z) =

∫∫R2d

f (z − z ′)g(z ′)eπiσ(z,z′) dz ′

for f ,g ∈ L1(R2d )

39

Factρ may be extended to a unitary operator from L2(R2d ) to T 2, andthat the twisted convolution f \g may be defined for f ,g ∈ L2(R2d )with norm estimate ‖f \g‖L2 ≤ ‖f‖L2‖g‖L2 .

Proposition

The Fourier-Wigner transform extends to a unitary operatorFW : T 2 → L2(R2d ). This extension is the inverse operator of theintegrated Schrödinger representation ρ, andFW (ST ) = FW (S)\FW (T ) for S,T ∈ T 2.

40

The Fourier-Wigner transform shares several properties with theFourier transform of functions.

Riemann-Lebesgue lemma

If S ∈ T 1, the Fourier-Wigner transform FW (S) is continuous andvanishes at infinity, i.e. lim

|z|→∞|FW (z)| = 0.

Mapping properties

Let f ∈ L1(R2d ) and S,T ∈ T 1.1. Fσ(S ∗ T ) = FW (S)FW (T ).2. FW (f ∗ S) = Fσ(f )FW (S).

41

Proof of item 1By definition,Fσ(S ∗ T )(z) =

∫∫R2d tr

[Sπ(z ′)Tπ(z ′)∗

]e−2πiσ(z,z′) dz ′. Since

e−2πiσ(z,z′)π(z ′) = α−zπ(z ′), the integrand may be written in a waythat will allow us to use the general Moyal identity:

tr[Sπ(z ′)Tπ(z)∗

]e−2πiσ(z,z′) = tr

[Se−2πiσ(z,z′)π(z ′)Tπ(z ′)∗

]= tr

[Sπ(−z)π(z ′)π(−z)∗Tπ(z ′)∗

].

The general Moyal identity andtr(π(z)T ) = tr(π(z)PTP) = tr(Pπ(z)PT ) = tr(π(−z)T ) gives

Fσ(S ∗ T )(z) =

∫∫R2d

tr[Sπ(−z)αz′(π(−z)∗T )

]dz ′

= tr(Sπ(−z))tr(π(−z)∗T )

= tr(Sπ(−z))tr(e−2πix ·ωπ(z)T )

= tr(e−πix ·ωSπ(−z))tr(e−πix ·ωπ(−z)T )

= FW (S)(z)FW (T )(z). 42

Proof of item 2By taking the trace inside the integral:

FW (f ∗ S)(z) = e−πix ·ωtr(π(−z)

∫∫R2d

f (z ′)π(z ′)Sπ(z ′)∗ dz ′)

= e−πix ·ω∫∫

R2df (z ′)tr

[π(−z)π(z ′)Sπ(z ′)∗

]dz ′.

A simple manipulation of the integrand yields thattr [π(−z)π(z ′)Sπ(z ′)∗] = e−2πiσ(z,z′)tr(π(−z)S). Inserting thisexpression into our calculation concludes the proof, since

FW (f ∗ S)(z) = e−πix ·ω∫∫

R2df (z ′)e2πiσ(z,z′)tr(π(−z)S) dz ′

= e−πix ·ωtr(π(−z)S)

∫∫R2d

f (z ′)e−2πiσ(z,z′) dz ′

= Fσ(f )FW (S).

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Hausdorff-Young inequality

Let 1 ≤ p ≤ 2 and let q be the conjugate exponent determined by1p + 1

q = 1. If S ∈ T p, then FW (S) ∈ Lq(R2d ) with norm estimate

‖FW (S)‖Lq ≤ ‖S‖T p .

Lieb’s inequality

If we pick S = ψ ⊗ φ for ψ, φ ∈ L2(Rd ) in the Hausdorff-Younginequality, we obtain for 2 ≤ q <∞ that∫∫

R2d|Vφψ(z)|q dz ≤ ‖ψ‖qL2‖φ‖

qL2 .

This is Lieb’s inequality, except for the constant(

2q

)dthat makes

Lieb’s inequality sharp. Hence we can consider Lieb’s uncertaintyprinciple to be a sharp version of the Hausdorff-Young inequality forrank-one operators.

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As a corollary, we note an extension of Lieb’s uncertainty principleto trace class operators.

Hausdorff-Young–sharp

Let 2 ≤ q <∞. If S ∈ T 1, then

‖FW (S)‖Lq ≤(

2q

)d/q

‖S‖T 1 .

ObservationLieb’s inequality expresses localization results of a function φ withrespect to a window function ψ in terms of summability conditionson the the STFT, a time-frequency representation.

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Modulation spaces

The modulation spaces are a class of spaces of functions anddistributions introduced by Feichtinger in a series of papers startingwith the introduction of the so-called Feichtinger algebra in , andthey have since been recognized as a suitable setting fortime-frequency analysis.

STFT-extenstionThe STFT can be extended to other spaces by interpreting thebracket 〈·, ·〉 as a duality bracket. This allows us to consider Vφψ forφ ∈ S(Rd ) and ψ ∈ S ′(Rd ).

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Modulation spaces - definition

Fix a window φ ∈ S(Rd ) \ {0}. For 1 ≤ p,q ≤ ∞, the modulationspace Mp,q

m (Rd ) is the set of tempered distributions ψ such that

‖ψ‖Mp,qm

=

(∫Rd

(∫Rd|Vφψ(x , ω)|pm(x , ω) dx

)q/p

dω

)1/q

<∞,

where m is positive function (weight) on R2d .In the special cases where p or q is∞, the integral is replaced byan essential supremum. When p = q, we will denote the spaceMp,p

m (Rd ) by Mpm(Rd ).

RemarkThe modulation spaces are Banach spaces with the norms‖ψ‖Mp,q

m, and using a different window φ ∈ S(Rd ) \ {0} in the

definition yields the same spaces with equivalent norms.

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Modulation spaces - examples

• p = q = 1 and m = 1 is Feichtinger’s algebra M1(Rd ) alsodenoted by S0(Rd ), a useful space of test functions.• p = q = 2 and m(x , ω) = (1 + |ω|2)s/2 gives the Sobolev

spaces H2s (Rd ).

• p = q = 2 and m(x , ω) = (1 + |x |2)s/2 gives the Besselpotential spaces W 2

s (Rd ).• p = q = 2 and m(x , ω) = (1 + |x |2 + |ω|2)s/2 gives the Shubin

class Q2s (Rd ).

• p =∞ and q = 1 and m(x , ω) = (1 + |x |2 + |ω|2)s/2 is oftencalled Sjöstrand’s class.• p = q =∞ and m = 1 is the dual of Feichtinger’s algebra often

denoted by S′0(Rd ), a useful space of distributions.

Modulation spaces have turned out to be of good use in the theoryof pseudodifferential operators, Schrödinger equations, Fourierintegral operators, Fourier multipliers, etc. . 48

Cross-Wigner distribution

The cross-Wigner distribution, W (ψ, φ), of two functions ψ and φon Rd is by definition given by

W (ψ, φ)(x , ω) =

∫Rdψ

(x +

t2

)φ

(x − t

2

)e−2πiω·t dt .

This expression is similar to the definition of the cross-ambiguityfunction, and in fact W (ψ, φ) = FσA(ψ, φ).

Weyl calculus

Our main motivation for studying the cross-Wigner distribution is itsconnection with the Weyl calculus. For σ in the space of tempereddistributions S ′(R2d ) and ψ, φ ∈ S(Rd ), we define the Weyltransform Lσ of σ to be the operator given by

〈Lσψ, φ〉 = 〈σ,W (φ, ψ)〉.

σ is called the Weyl symbol of the operator Lσ. 49

Recall the integrated Schrödinger representation, which is themap ρ : L1(R2d )→ B(L2(Rd )) given by

ρ(f ) =

∫∫R2d

f (z)e−πix ·ωπ(z) dz,

ObservationThe relationship between the Weyl calculus and the integratedSchrödinger representation is given by Lf = ρ(Fσf ) for a symbol f .

One may assign to a function k on R2d a so-called integraloperator Ak on L2(Rd ) by

Akψ(s) =

∫Rd

k(s, t)ψ(t) dy (3)

for ψ ∈ L2(Rd ). k is called the kernel of Ak .

Useful class of Ak ’s

We will letM denote the set of integral operators Ak with kernel kin M1(R2d ).

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LemmaM is also the set of operators with Weyl symbol or twisted Weylsymbol in M1(R2d )

Concrete realizationMLet k ∈ M1(R2d ) and let Ak be the integral operator with kernel k .Let {wn}n∈N be a Wilson basis for L2(Rd ), and denote by Wmn thecorresponding Wilson basis for L2(R2d ) given byWmn(x , y) = wm(x)wn(y).Then Ak ∈ T 1 with ‖Ak‖T 1 ≤ C‖k‖M1 for some constant C, andAk =

∑m,n∈N〈k ,Wmn〉wm ⊗wn where the sum converges in the T 1-

norm

51

LemmaLet f ∈ L1(R2d ), and let Lf be the Weyl transform of f .• αz(Lf ) = LTz f for z ∈ R2d .• Lf = Lf .• L∗f = Lf∗ .

In particular, if S ∈M, then αz(S), S,S∗ ∈M.

RemarkHence, applying αz to a pseudodifferential operator amounts to atranslation of its symbol.

52

LemmaLet f ∈ L1(R2d ) and S ∈ T 1. The twisted Weyl symbol of f ∗ S isthe function Fσ(f )FW (S). In particular, if ϕ1, ϕ2 ∈ L2(Rd ), then thetwisted Weyl symbol of the localization operator Aϕ1,ϕ2

f is thefunction Fσ(f ) · A(ϕ2, ϕ1).

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