Signal Communication and Modular Theory

We propose a conceptual frame to interpret the prolate differential operator, which appears in Communication Theory, as an entropy operator; indeed, we write its expectation values as a sum of terms, each subject to an entropy reading by an embedding suggested by Quantum Field Theory. This adds meaning to the classical work by Slepian et al. on the problem of simultaneously concentrating a function and its Fourier transform, in particular to the “lucky accident” that the truncated Fourier transform commutes with the prolate operator. The key is the notion of entropy of a vector of a complex Hilbert space with respect to a real linear subspace, recently introduced by the author by means of the Tomita-Takesaki modular theory of von Neumann algebras. We consider a generalization of the prolate operator to the higher dimensional case and show that it admits a natural extension commuting with the truncated Fourier transform; this partly generalizes the one-dimensional result by Connes to the effect that there exists a natural selfadjoint extension to the full line commuting with the truncated Fourier transform.


Introduction
The aim of this paper is to provide an interpretation of the prolate operator, which plays an important role in the theory of signal transmission, as an entropy operator, by means of the modular theory of von Neumann algebras, following recent concepts and abstract analysis of entropy in the framework of Quantum Field Theory.We begin with a brief account of the background of our work.Band limited signals.Suppose Alice sends a signal to Bob that is codified by a function of time f .Bob can measure the value f only within a certain time interval; moreover, the frequency of f is filtered by the signal device within a certain interval.For simplicity, let us assume these intervals are both equal to the interval B = (−1, 1).As is well known, if a function f and its Fourier transform f are both supported in bounded intervals, then f is the zero function.So one is faced with the problem of simultaneously maximizing the portions of energy and amplitude spectrum within the intervals where || • ||, || • || B denote the L 2 -norms on R and B, the concentration problem.
The problem of best approximating, with support concentration, a function and its Fourier transform is a classical problem; in particular, it lies behind Heisenberg uncertainty relations in Quantum Mechanics and is studied in Quantum Field Theory too, see [10].
In the '60ies, this problem was studied in seminal works by Slepian, Pollak and Landau [22,12], see also [21].The functions that best maximize (1) are eigenfunctions of the angle operator associated with the truncated Fourier transform.This is a Hilbert-Schmidt integral operator whose spectral analysis is not easily doable a priori.However, by the lucky accident figured out in [22], this integral operator commutes with a linear differential operator, the prolate operator that shares its eigenfunctions with the angle operator, so these eigenfunctions were computed.
W is a classical operator, it arises by separating the 3-dimensional scalar wave equation in a prolate spheroidal coordinate system.More recently, Connes has reconsidered and raised new interest in this operator [5].The papers [6,7] show an impressive relation of the prolate spectrum with the asymptotic distribution of the zeros of the Riemann ζ-function.Our paper is not related to this point; however, our Sect.3 is inspired and generalizes a small part of the analysis in [7].
Our purpose is to understand the role of the prolate operator on a conceptual basis, in relation to the mentioned lucky accident.We shall argue that the prolate operator gives rise to an entropy operator, in a sense that will be explained.Within our aim, we shall generalize the prolate operator in higher dimensions and analyze it guided by the Quantum Field Theory context.
We shall consider the prolate operator on the Schwarz space S(R d ), with r the radial coordinate in R d , and show it admits a natural closed extension W that commutes with the truncated Fourier transform.We shall see that the expectation values of πE B W on L 2 (R d ), with E B the orthogonal projection onto L 2 (B), is positive, selfadjoint and its expectation values are indeed entropy quantities.
In the one-dimensional case, W itself is selfadjoint [7], and this is probably true also in higher dimensions; however, for our aim, it suffices to know that E B W is selfadjoint.
Modular theory, the entropy of a vector.In the '70ies, Tomita and Takesaki uncovered a fundamental, deep operator algebraic structure.In particular, associated with any faithful, normal state ϕ of a von Neumann algebra M, there is a canonical one-parameter automorphism group σ ϕ of M, the modular group, see [23].The relevance of this intrinsic evolution in Physics was soon realized in the framework of Quantum Statistical Mechanics since σ ϕ is characterized by the KMS thermal equilibrium condition, see [8].Now, part of the modular theory shows up at a more elementary level, with potential points of contact with contexts not immediately related to Operator Algebras: the general framework is simply provided by a real linear subspace of complex Hilbert space, cf.[13,18].
Let H be a complex Hilbert space and H a real linear subspace of H; by considering its closure, we may assume that H is closed.H is said to be a standard subspace if H is closed and H + iH = H , H ∩ iH = {0}.Every closed real linear subspace H has a standard subspace direct sum component and we may assume that H is standard by restricting to this component.
With H standard, the anti-linear operator S : H then gives an anti-linear, involutive unitary J H and a positive, non-singular, selfadjoint operator ∆ H on H, the modular conjugation and the modular operator, such that H, the orthogonal of iH with respect to the real scalar product ℜ(•, •).We refer to [14] for the modular theory and basic results on standard subspaces.We say that the standard subspace H is factorial if H ∩ H ′ = {0}.Thus H + H ′ is dense in H and H + H ′ is the direct sum (as linear space) of H and H ′ .Again, we may assume that H is factorial by restricting to the factorial component.Our abstract results have an immediate extension to the non-factorial, non-standard case.
The cutting projection relative to H is the real linear, densely defined projection The entropy of a vector Φ ∈ H with respect to a standard subspace H ⊂ H is defined by this notion was introduced in [15,3].A first way to realize the entropy meaning of S Φ is to consider the von Neumann algebra R(H) associated with H by the second quantization on the Fock Hilbert space over H; then S Φ is Araki's relative entropy [1] between the coherent state associated with Φ and the vacuum state on R(H).However, in this paper, this fact does not play any direct role.Note that iP H i log ∆ H is a real linear operator.This is our first instance of an entropy operator, namely a real linear, positive, selfadjoint operator whose expectation values give the entropy of states.In concrete situations, the subspace H may correspond to a region of a manifold and Φ to signal, then S Φ acquires the meaning of local entropy of Φ.
Entropy density of a wave packet.The local entropy of a wave packet has been studied in [15,3,4] for the case of a half-space, and in [17] for the space ball case, which is directly related to the present paper; these works were motivated by Quantum Field Theory.
Let T be the real linear space of wave packets, that is Φ ∈ T if Φ is a real function on R 1+d that satisfies the wave equation x Φ, with Cauchy data in the real Schwarz space S r (R d ).Quantum Relativistic Mechanics tells us that T is equipped with a natural (Lorentz invariant) complex pre-Hilbert structure so, by completion, we get a complex Hilbert space H. Wave packets with Cauchy data supported in the open, unit ball B of R d form a real linear subspace of H denoted by H = H(B) (after closure).The entropy of Φ in B is given by Here D = (d − 1)/2 and T 00 Φ = 1 2 (∂ t Φ) 2 + |∇ x Φ| 2 is the energy density of Φ.We discuss here d > 1 case; the case d = 1 is similar but requires modifications due to infrared singularities, which is not important for our discussion.
The two terms in T 00 Φ have separate meanings, they correspond to the kinetic and to the potential energy of the wave packet.H is naturally a direct sum of the two real Hilbert subspaces associated with the Cauchy data.
In terms of the Cauchy data f, g of Φ, the modular Hamiltonian log ∆ B relative to B is given by [17].Here, is a higher-dimensional Legendre operator.Each of the two terms in the expression of S Φ , L D ≡ L−2D, have an entropy meaning.As we will discuss on general grounds, −π(f, L D f ) B is the field entropy of f , and π(g, M g) B is the momentum, or parabolic, entropy of g, in B.
We infer that also −π(f, Lf ) B is an entropy quantity, the Legendre entropy of f in B.
The measure of concentration.We now return to the Communication Theory setting.The truncated Fourier transform operator is obviously defined in any space dimension.Indeed, the concentration problem often arises in higher dimensions too.It is also studied in [20], although with a point of view different from the one in this paper.
As said, the higher dimensional prolate operator (3) extends to a natural operator W on L 2 (R d ), that commutes both with the Fourier and the truncated Fourier transforms; W also commutes with the orthogonal projection E B onto L 2 (B) and its Fourier conjugate ÊB . As that is, −π(f, W f ) B is the sum of the Legendre entropy of f and π||f || 2 B (that we call the Born entropy), minus the parabolic entropy of f , i.e.
We conclude that −π(f, W f ) B is an entropy quantity, i.e. a measure of information, that we call the prolate entropy of f w.r.t.B. In other words, −πE B W is an entropy operator.The lucky accident [22], that W commutes with the truncated Fourier transform, finds a conceptual clarification in this fact.
Based on the ordering of eigenvalues result in [22], we then have lower prolate entropy ←→ higher concentration where the concentration is both on space and in Fourier modes as above.This is intuitive since information is the opposite of entropy.The above correspondence holds in the onedimensional case, and we expect it to hold in general.
In other words, in order to maximize simultaneously both quantities in (1), we have to minimize the prolate entropy.

Higher-dimensional Legendre operator
The Legendre operator is the one-dimensional linear differential operator It is a Sturm-Liouville operator, probably best known because its eigenfunctions on L 2 (−1, 1) are the Legendre polynomials.In the following, we consider a natural higher-dimensional generalization of this operator.
Let S(R d ) be the Schwartz space of smooth, rapidly decreasing functions, d ≥ 1.For the moment, we deal with complex-valued functions; the corresponding results for realvalued functions are obtained by restriction.We denote by L min the d-dimensional Legendre operator, acting on S(R d ), that we define by namely, L min is the divergence of the vector field (1 − r 2 )∇, where ∇ denotes the gradient and r the radial coordinate in R d .L min can be written as indeed We consider L min as a linear operator on the Hilbert space because, by integration by parts, we have Lemma 2.1.L min is a Hermitian operator.
Proof.Equation (10) shows that where the latter means the value of the distribution ∇(1 − r 2 )∇f on the test function g.
Proof.Taking into account that the vector field G = (1 − r 2 )∇g vanishes on ∂B, we have L min does not commute with E B , however, the following holds.
Moreover, L max is Hermitian on Proof.To prove the first part of the statement, namely eq. ( 13), we must check that, for every Taking into account that the vector field G = (1 − r 2 )∇g vanishes on ∂B, by eq. ( 12) we have thus ( 14) holds because the last term in the above equality is symmetric in f and g.
We shall denote by L the closure of the restriction of L max to S(R d ) + χ B S(R d ).By Prop.13, L is Hermitian and commutes with E B .Given f ∈ L 2 (R d ), we denote by f its Fourier transform and by F the Fourier transform operator: Ff = f .By Plancherel theorem, F is a unitary operator on L 2 (R d ).
In Fourier transform, L min is given by the operator Lmin = FL min F −1 ; clearly D( Lmin ) = S(R d ).We denote by the multiplication operator by (1 − r 2 ) on L 2 (R d ).
Lemma 2.5.Lmin = −r 2 (1 + ∇ 2 ) − 2r∂ r on S(R d ), where r denotes the radial coordinate |p| also in the dual space R d .Therefore Proof.With f ∈ S(R d ), we have On the other hand, hence, accordingly with the expression (9), 3 Higher-dimensional prolate operator We now extend to the higher dimension some results in [7,Sect. 1].Let W min be the operator on L 2 (R d ) given by with D(W min ) = S(R d ).W min is a higher-dimensional generalisation of the prolate operator.By Prop.2.1, W min is a Hermitian, being a Hermitian perturbation of L min on S(R d ); moreover, , so −W min is a positive operator on this domain.
We explicitly note the equality on S(R d ) and that because 0 ≤ M ≤ 1 on L 2 (B).
Proposition 3.1.W min commutes with the Fourier transformation F: Any linear combination of L min and M commuting with F is proportional to W min .
Proof.We have M = 1 + ∇ 2 , therefore (17) gives Lmin = L min + M − M , thus . By (20), we then have so W min = FW min F −1 , as desired.Finally, if a ∈ R, we have Let ÊB = FE B F −1 be the Fourier transform conjugate of the orthogonal projection where tilde denotes the Fourier anti-transform and * the convolution product.We put W max = W * min .We have and in the distributional sense.Clearly, by Prop.3.1, also W max commutes with F Proof.Clearly M commutes with E B .Since The second equation then follows from the first one by applying the Fourier transform because W min , W max commute with F, ÊB = FE B F −1 , and By the above proposition, we have and where the last equality follows because W max acts locally on f by (23), so W max f = W min f 0 on a neighbourhood of B. Lemma 3.4.For every g ∈ S(R d ), E B ÊB g belongs to D(W max ) and we have Proof.We may apply Lemma 3.3 with f = ÊB g; indeed f = χB * g is a smooth function because g is smooth, and f in the domain of W max by Prop.3.2.
Recall that a closed linear operator Z on a Hilbert space H commutes with the orthogonal projection If D ⊂ D(Z) is a core for Z, then it suffices to verify the above condition for all u ∈ D.
Denote by F B = E B FE B the truncated Fourier transform.Note that is the angle operator.
Proposition 3.5.The restriction of W max to D is Hermitian.Its closure is Hermitian and commutes with F and E B , thus with ÊB and F B too.
Proof.W max | D commutes with F because W max commutes with F, and D is globally F-invariant.
We now show that W max | D is Hermitian.First, note that W max is Hermitian on S(R d ) + χ B S(R d ) by the eq.( 22), because L max is Hermitian on S(R d ) + χ B S(R d ) by Prop.2.4, and M is Hermitian too on this domain.It then follows that W max is Hermitian on S(R d ) + χ B S(R d ) too due to (24).
So we have to show that W max is symmetric on mixed terms in (26).By (12), we are indeed left to check that ( ÊB g, W max E B h) = (W max ÊB g, E B h), for all g, h ∈ S(R d ).
Now, by Prop.3.2 and Lemma 3.4, we have So W max | D is Hermitian, hence its closure W is Hermitian too.It remains to show that W commutes with E B .We need to check that By linearity, we can check the above condition for each of the three terms individually.Concerning the first term, that is the case g = h = 0, we have f ∈ D ⊂ D(W ) and by Prop.
Consider now the last term.With k ≡ ÊB h, we have to show that E B k belongs to D(W ) and , and by Lemma 3.3 we have where the equality E B W max k 0 = E B W max k follows as in the proof of Lemma 3.3 because W max acts locally.
Concerning the remaining second-term case, take then g ∈ S(R d ); clearly and the proof is complete.
W is the minimal closed extension of W min that commutes both with E B and ÊB .Indeed, if W is an extension of W min with this property, then D( W ) must contain E B D and Note that the angle operator shows that the normalized k-th eigenfunctions of T B are concentrated at level λ k in an appropriate sense.Note that, on the even function subspace, F is a unitary involution, thus F B is selfadjoint; so F B and F * B F B = E B ÊB E B share the same eigenfunctions.We now show that −W is positive on B, namely −E B W is positive.Proposition 3.6.For every u ∈ D(W ), we have Proof.As D is a core for W , it suffices to check (30) with u = f + E B g + ÊB h, with f, g, h ∈ S(R d ).Now, χ B u is a smooth function on B; choose u 0 ∈ S(R d ) that agrees with u on B. By eq. ( 29), we have As seen, both W and L commute with E B , and we consider now their restrictions to L 2 (B), which we denote by W B and L B .
Let C ∞ ( B) be the space of smooth function on B, up to the boundary; we may regard C ∞ ( B) as a subspace of L 2 (B) ⊂ L 2 (R d ).As is known, χ B S(R d ) = C ∞ ( B).We now show that W B and L B are essentially selfadjoint on C ∞ ( B).We will also denote by C ∞ 0 (B) the space of smooth functions on B with compact support contained in B. Since L B is a bounded perturbation of W B on L 2 (B), also L B is selfadjoint with core C ∞ ( B). L B is then positive by Lemma 2.3.
In the one-dimensional case, the essentially selfadjointness of L B on C ∞ [−1, 1] (thus of its bounded perturbation W B ) follows by the well-known fact that the Legendre polynomials form a complete orthogonal family of L B -eigenfunctions.Note that L B is not essentially selfadjoint on C ∞ 0 (−1, 1), see [11].
. Now, q is given by ( 12) on C ∞ 0 (B).By [19,Prop. 10.1], it suffices to show that, given u ∈ C ∞ ( B), there exists a sequence of functions u n ∈ C ∞ 0 (B) such that u n → u and ) be even such that h n = 1 on (0, 1 − 1 n ) and |h ′ n | bounded by 2n and set u n (x) = h n (r).Then u n → χ B and the sequence is bounded.The case of a general u ∈ C ∞ ( B) follows on the same lines by replacing u n by See e.g.[19] for the Friedrichs extension.

Modular theory and entropy of a vector
In this section, we recall the basic structure concerning the modular theory of a standard subspace H, the entropy of a vector relative to H, and their applications to the entropy density of a wave packet.

Entropy operators
Let H be a complex Hilbert space and H ⊂ H a standard subspace, i.e.H is a real linear, closed subspace of H such that H ∩ iH = {0} and H + iH = H, with H ′ the symplectic complement of H, The Tomita operator (in a quadratic form sense), where P H is the cutting projection [15,3], the semigroup generator d ds ∆ −is H | s=0 of the modular unitary group.
We have P * H = −iP H i and the formula in [3] (P H is the closure of the right hand side of (32)).
The entropy operator E H is defined by (closure of the right-hand side).We have Here, S Φ is defined for any vector Φ ∈ H as follows.S Φ = q(Φ, Φ) with q the closure of the real quadratic form ℜ(Φ, E H Ψ), Φ, Ψ ∈ D(E H ). So S Φ = +∞ if Φ is not in the domain of q.
Proposition 4.1.The entropy operator E H is real linear, positive, and selfadjoint w.r.t. to the real part of the scalar product.
Proof.E H is clearly real linear, and positive because the entropy of a vector is positive [3, Prop.2.5 (c)].The selfadjointness of E H follows by the formula (32), see [17,Lemma 2.3].
In our view, an entropy operator E is a real linear operator on a real or complex Hilbert space H, such E is positive, selfadjoint and its expectation values (f, Ef ), f ∈ H, correspond to entropy quantities (w.r.t.B).E may be unbounded, and (f, Ef ) is understood in the quadratic form sense, so it takes values in [0, ∞].It is convenient to consider more entropy operators by performing operations, that preserve our demand, on the entropy operators.
Basic.If E is a real linear operator on a real Hilbert space H of the form (33), we say that E is an entropy operator.
Restriction and direct sum.
then E is an entropy operator on H, iff both E ± are entropy operators.
Change of metric.Suppose that S ⊂ H is a core for the entropy E on H and (•, •) ′ is a scalar product on S; denote by H ′ the corresponding real Hilbert space completion and by  : S ⊂ H ′ → H the identification map.If  * E is densely defined, its Friedrichs extension E ′ is an entropy operator on H ′ .Note that Sum, difference.If E 1 , E 2 are entropy operators and E = E 1 ± E 2 is densely defined and positive, the Friedrichs extension E is an entropy operator.
Born entropy.πE B , with E B the orthogonal projection onto L 2 (B), is an entropy operator on L 2 (R d ).
In order to justify the last item, note that (f, B is the particle probability to be localized in Ω, accordingly to Born's interpretation.Moreover, in Communication Theory, ||f || 2 B represents the part of energy of f contained in B [22].We thus define The π normalization is chosen by compatibility reasons (Sect.5);

Abstract field/momentum entropy
We consider two real linear spaces S + and S − and a duality f, g ∈ S + × S − → f, g ∈ R.
A real linear, invertible operator µ : S + → S − is also given; we assume that µ is symmetric and positive with respect to the duality, i.e.
So S ± are real pre-Hilbert spaces with scalar products and µ is a unitary operator.Let H ± be the real Hilbert space completion of S ± .Then µ extends to a unitary operator H + → H − , still denoted by µ.Moreover, the duality between S + and S − extends to a duality between , is symplectic and non-degenerate (the coefficient 1 2 is to conform with the next section case).This will be the imaginary part of the complex scalar product of H: ℑ(Φ, Ψ) = β(Φ, Ψ).Now, the operator By (36), ı preserves β, that is β(ıΦ, ıΨ) = β(Φ, Ψ).As ı 2 = −1, the unitary ı defines a complex structure (multiplication by the imaginary unit) on H that becomes a complex Hilbert space with a scalar product (Φ, Ψ) = β(Φ, ıΨ) + iβ(Φ, Ψ) Suppose now K ± ⊂ H ± are closed, real linear subspaces.The symplectic complement where K o ± denote the annihilatotors of K ± in H ∓ under the duality •, • .Let us consider the case K is standard and factorial.Then the cutting projection Proposition 4.2.The modular Hamiltonian log ∆ K and conjugation J K are diagonal; so with M and L operators H ± → H ∓ .The entropy of Φ ≡ f ⊕ g ∈ H with respect to K is given by In particular, if Φ ∈ K, S Φ = −π f, Lf + π g, Mg .
Proof.As K = K + ⊕ K − and ıK = µ −1 K − ⊕ µK + are direct sum subspaces, the Tomita operator S K is clearly diagonal, and so is its adjoint S * K .The modular operator ∆ K = S * K S K is thus diagonal.Since the logarithm function is real on (0, ∞), by functional calculus the modular Hamiltonian log ∆ K is diagonal too.Also J K is diagonal due to formula (32).
So A K is off-diagonal because ı is off-diagonal and we may write A K as in (39).We have thus the entropy of Φ is given by The fact that log ∆ K is diagonal was shown in [2], based on the the formula P K − ıP K ı = 2(1 − ∆ K ) −1 , which follows from (32).The entropy operator is given by Note that, since A K is skew-selfadjoint and complex linear on H, we have the relations Clearly, −π f, P − Lf = S f ⊕0 , π g, P + Mg = S 0⊕g .
(quadratic form sense).Note that only the duality, not the Hilbert space structure, enters directly into the definitions of the above entropies.

Local entropy of a wave packet
The above structure concretely arises in the wave space context; namely, in the free, massless, one-particle space in Quantum Field Theory.Denote by S r (R d ) the real Schwarz space.As is known, if f, g ∈ S r (R d ), there is a unique smooth real function Φ(t, x) on R 1+d which is a solution of the wave equation (a wave packet or, briefly, a wave) with Cauchy data Φ| t=0 = f , ∂ t Φ| t=0 = g.We set Φ = w(f, g) and denote by T the real linear space of these Φ's; we will often use the identification We thus deal directly with S r (R d ) ⊕ S r (R d ) and consider the symplectic form on it where the scalar product in (45) is the one in L 2 (R d ).
We set S + = S r (R d ), S − = µS r (R d ), with µ the given by The duality between S + and S − is given by the L 2 scalar product.Let H ± be the real Hilbert space of tempered distributions f ∈ S r (R d ) ′ such that f is a Borel function with The complex Hilbert space is H is the real Hilbert space H = H + ⊕ H − equipped with complex structure given ı (38).With for the modular operator associated with H(B), and namely The cutting projection w.r.t.H(B) is P B = χ B 0 0 χ B , therefore the entropy operator So each of the two components of E ′ B is an entropy operator on L 2 (R d ); and so is This is equal to the entropy S Φ of the flat wave Φ = w(0, f ).Similarly, we set This is equal to the entropy S Ψ of the stationary wave Ψ = w(f, 0).Now, so πW E B is an entropy operator too; we thus define: We summarize our discussion in the following theorem.Proof.The first statement is immediate from our discussion and the relation The parabolic distribution (1 − r 2 ) appears in both the parabolic and the Legendre entropy expression.Near the center of B, the parabolic entropy is close to the Born entropy.On the other hand, near the boundary of B, the prolate entropy gets close to the Born entropy.
Let us specialize now on the one-dimensional case as studied in [22] (on the even functions subspace of L 2 (B)).As T B is strictly positive and Hilbert-Schmidt, its eigenvalues can be ordered as λ 1 > λ 2 > • • • > 0; moreover, they are simple; the eigenvalues of −W B can be ordered as α 1 < α 2 < • • • < ∞; they correspond to the λ k 's in inverse order, that is T B and −W B share the same k-the eigenfunction f k , which is unique up to a phase once we normalize it as ||f k || 2 B = 1.Then and πα k is the prolate entropy of f k .As the information is the opposite of the entropy, the above relations show the intuitive fact that the functions with lower prolate entropy, thus higher information in B, are the ones with better support concentration in B in space and Fourier modes.f 1 carries the best information as it is optimally concentrated.
We expect the ordering correspondence between the eigenvalues of T B and W B to hold in the higher dimensional case too.

Concentration in balls of arbitrary radius
We briefly indicate here how the results in this paper easily extend to the case of localization in balls of any radius.The more general prolate operator is studied in [22], c > 0. We consider W min (c) as an operator on L 2 (R d ) with domain S(R d ).Denote by δ λ , λ > 0, the dilation operator on L 2 (R d ) (δ λ f )(x) = λ −d/2 f (λ −1 x) , so δ λ is a unitary operator.We also set F λ = δ −1 λ F; in particular, F 2π is the commonly used Fourier transform in Communication Theory and elsewhere.Proposition 6.1.W min (c) commutes with F c .
Proof.Since δ −1 λ rδ λ = λr and δ −1 λ ∇δ λ = λ −1 ∇, we have The analysis of W min (c) is now the same as in the case c = 1.W min (c) admits a natural extension W c that commutes with F c , E B , ÊBc , where B c denotes the ball of radius c centered at the origin.
The prolate operator corresponding to the localization in balls B λ , and B λ ′ in Fourier transform, is obtained by conjugating W c by the dilation operator, that is an entropy relation that generalizes Thm.5.1.The first term on the left of (56) is the prolate entropy of f w.r.t.B λ and B λ ′ .Note however that the Legendre entropy −π(f, L λ f ) B λ does not depend on λ ′ ; as λ ′ → 0, the prolate entropy approaches the Legendre entropy.

2 π sin p p if d = 1 . 3 . 3 .
R d ) ; (26) recall that χB is a smooth L 2 -function vanishing at infinity, χB (p) = Lemma Let f ∈ D(W max ) be a smooth function.Then, also the function χ B f ∈ D(W max ), and

Corollary 3 . 7 .
Both W B and L B are selfadjoint, positive operators on L 2 (B).C ∞ ( B) is a core for both W B and L B .Proof.As W B is Hermitian and commutes with the positive Hilbert-Schmidt operator T B , it follows that W B is selfadjoint.Since W commutes with E B , D is a core of W and E B D ⊂ D, it follows that E B D is a core for W B .On the other hand, E B D = χ B S(R d ) because functions in S(R d ) + ÊB S(R d ) are smooth; so χ B D = C ∞ ( B). Therefore C ∞ ( B) is a core for W B .W B is then positive by Prop.3.6.

Theorem 5 . 1 .
−πW E B is an entropy operator on L 2 (R d ).The sum of the prolate entropy and the parabolic entropy is equal to the sum of the Legendre entropy and the Born entropy, all with respect to B. Namely, the relation (55) holds for every f ∈ L 2 (R d ), in the quadratic form sense.W E B commutes with the truncated Fourier transform F B .Let V be a real linear combination of LE B , M E B and E B commuting with F B ; then V = aW E B +bE B for some a, b ∈ R. If V is also positive, and the spectral lower bound of V | L 2 (B) is zero, then V = aW E B , a ≥ 0.

cf. ( 22 )
, taking into account that the Friedrichs extensions of −E B L on S(R d ) is equal to −L B E B by Prop.3.8.W E B commutes with F B by Prop.3.5.The characterization of V follows by an argument similar to the one in the proof of Prop.3.1.
H H = H , s ∈ R .For simplicity, let us assume that H is factorial, namely H ∩ H ′ = {0}, see e.g.[3, Sect.2.1] for the general case of a closed, real linear subspace.The entropy of a vector Φ ∈ H with respect to a standard subspace H ⊂ H is defined by