WHAT IS THE WIGNER FUNCTION CLOSEST TO A GIVEN SQUARE

SIAM J. MATH. ANAL. Vol. 50, No. 5, pp. 5161--5197

c 2018 Society for Industrial and Applied Mathematics \bigcirc

WHAT IS THE WIGNER FUNCTION CLOSEST TO A GIVEN SQUARE INTEGRABLE FUNCTION?\ast J. S. BEN-BENJAMIN\dagger , L. COHEN\ddagger , N. C. DIAS\S , P. LOUGHLIN\P , AND J. N. PRATA\P Abstract. We consider an arbitrary square integrable function F on the phase-space and look for the Wigner function closest to it with respect to the L2 norm. It is well known that the minimizing solution is the Wigner function of any eigenvector associated with the largest eigenvalue of the Hilbert--Schmidt operator with Weyl symbol F . We solve the particular case of radial functions on the two-dimensional phase space exactly. For more general cases, one has to solve an infinite dimensional eigenvalue problem. To avoid this difficulty, we consider a finite dimensional approximation and estimate the errors for the eigenvalues and eigenvectors. As an application, we address the so-called Wigner approximation suggested by some of us for the propagation of a pulse in a general dispersive medium. We prove that this approximation never leads to a bona fide Wigner function. This is our prime motivation for our optimization problem. As a by-product of our results, we are able to estimate the eigenvalues and Schatten norms of certain Schatten-class operators. The techniques presented here may be potentially interesting for estimating eigenvalues of localization operators in time-frequency analysis and quantum mechanics. Key words. Weyl quantization, Wigner functions, Wigner approximation, Hilbert--Schmidt operators AMS subject classifications. 41A99

Primary, 35S05, 81Q99, 42B35; Secondary, 42B10, 35P15,

DOI. 10.1137/18M116633X

1. Introduction. Given two functions \psi , \phi \in L2 (\BbbR d ), the cross-Wigner function W (\psi , \phi ) is given by [12, 17, 35, 53]: \int 1 \psi (x + y/2)\phi (x - y/2)e - iy\cdot k dy. (1) W (\psi , \phi )(x, k) = (2\pi )d \BbbR d Here z = (x, k) \in \BbbR 2d is interpreted as a phase-space (time-frequency or position-wave number) variable. If \psi = \phi , we shall simply write (with some abuse of notation) W \psi , meaning W (\psi , \psi ) [53]: \int 1 \psi (x + y/2) \psi (x - y/2)e - iy\cdot k dy. (2) W \psi (x, k) := W (\psi , \psi )(x, k) = (2\pi )d \BbbR d The Wigner distribution W \psi for a signal \psi \in L2 (\BbbR d ) is interpreted as a joint phase space representation of the signal. \ast Received by the editors January 22, 2018; accepted for publication (in revised form) July 11, 2018; published electronically September 25, 2018. http://www.siam.org/journals/sima/50-5/M116633.html Funding: The work of the first author was supported by the Robert A. Welch Foundation (grant A-1261), the Office of Naval Research Award (N00014-16-1-3054), and the Air Force Office of Scientific Research (FA9550-18-1-0141). The work of the third and fifth authors was supported by the Portuguese Science Foundation (FCT) grant PTDC/MAT-CAL/4334/2014. \dagger Institute for Quantum Science and Engineering, Texas A\&M University, College Station, TX 77843-4242 ([email protected]). \ddagger Department of Physics, Hunter College of the City University of New York, New York, NY 10021 ([email protected]). \S Grupo de F\' {\i}sica Matem\' atica, Departamento de Matem\' atica, Universidade de Lisboa, 1749-016 Lisboa, Portugal, and Escola Superior N\' autica Infante D. Henrique, 2770-058 Pa\c co de Arcos, Portugal ([email protected], [email protected]). \P Departments of Bioengineering and Electrical \& Computer Engineering, University of Pittsburgh, Pittsburgh, PA 15261 ([email protected]).

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BEN-BENJAMIN, COHEN, DIAS, LOUGHLIN, PRATA

In the present work, we intend to develop a systematic method for solving the following problem: Given some measurable function F : \BbbR 2d \rightarrow \BbbR , which is not a Wigner function, what is the Wigner function W \psi 0 closest to it with respect to the L2 norm? In other words, we want to determine \psi 0 \in L2 (\BbbR d ), such that (3)

\| F - W \psi 0 \| L2 (\BbbR 2d ) =

inf

\psi \in L2 (\BbbR d )

\| F - W \psi \| L2 (\BbbR 2d ) .

This problem and the methods we present may be useful in various contexts. But let us briefly explain our particular motivation for addressing it. In [15, 36, 37, 38] some of us considered the evolution of the Wigner function of a pulse \psi (in d = 1) given by \int 1 \psi (x + y/2, t) \psi (x - y/2, t)e - iy\cdot k dy, (4) W \psi (x, k, t) = 2\pi \BbbR where \int (5)

\psi (x, t) =

G(x - x\prime , t)\psi 0 (x\prime )dx\prime ,

\BbbR

\psi 0 (x) = \psi (x, 0) is the pulse at t = 0, and \int 1 eikx - i\omega (k)t dk (6) G(x, t) = 2\pi \BbbR is the propagator. In the previous formula \omega (k) is the dispersion relation (7)

\omega (k) = \omega R (k) + i\omega I (k),

which connects the wave number k and the frequency \omega . One should understand (5) in the distributional sense G \in \scrS \prime (\BbbR \times \BbbR ) and \psi \in \scrS (\BbbR ). In [37, 38] the following approximation---called Wigner approximation---was derived: (8)

W \psi (x, k, t) \sim e2t\omega I (k) W \psi 0 (x - \nu (k)t, k) ,

where (9)

\prime \nu (k) = \omega R (k)

is the group velocity. The advantage of considering (8) instead of the exact (4) is obvious. In (8), we have a local, computable expression, which has a simple interpretation. Each mode k evolves along a ``classical"" trajectory with velocity given by the group velocity. However, with this approximation, one faces a difficulty. As we shall prove in section 6, the expression on the right-hand side of (8) is never the Wigner function W \phi of a signal \phi \in L2 for t > 0. In this case, we say that that expression is not representable. However, it may still be a good approximation. Nonrepresentable functions may also appear, when one conducts ``time-varying filtering"" [9, 28] by multiplying a Wigner distribution W \psi by a weighting function of position and wave-number: (10)

W\Gamma \psi (x, k) = W \psi (x, k)\Gamma (x, k).

THE CLOSEST WIGNER FUNCTION

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The weighting function \Gamma is chosen so that W\Gamma \psi has some optimal time-frequency concentration. The Wigner transform is a fundamental instrument in the spectral estimation of nonstationary signals. In some situations a nonrepresentable phase-space function may appear, for instance, in multitaper estimation [7], especially when combined with reassignment [50], and in the Wigner distribution of linear signal spaces [24, 25]. Motivated by these three situations, we intend to study the problem stated above. If a given real-valued function F \in L2 (\BbbR 2d ) is not representable, that is, if there is no \psi \in L2 (\BbbR d ) such that F = W \psi , then what is the Wigner function W \psi 0 ``closest"" to F ? Since, via Moyal's identity [39], Wigner functions belong to L2 (\BbbR 2d ), it seems natural to require proximity in the L2 -norm. This least squares problem has emerged in other contexts, such as Bessel multipliers [5, 6], and time-varying filtering and signal estimation using Wigner functions [9, 28], and short-time Fourier transforms [21]. In a companion paper [8] we proved that such a minimizer always exists, although it may not be unique. Moreover, we give an explicit construction of the minimizers. Nevertheless, it may be difficult to obtain it. This is because the construction requires the computation of the spectrum and the eigenspace associated with the largest eigenvalue of the self-adjoint Hilbert--Schmidt operator F\widehat with Weyl symbol F . Since, in general, the spectrum of F\widehat may be infinite, albeit countable, this may prove to be a difficult task. If such is the case, we choose to replace the infinite dimensional eigenvalue problem by a finite dimensional one. We then give precise estimates for the errors of the eigenvalues and eigenvectors of the truncated problem. As a by-product of these estimates, we can approximate the eigenvalues and Schatten norms of certain Schatten-class operators [10]. These techniques may be potentially interesting in other contexts. For instance, in quantum mechanics, mixed states are represented by positive trace-class operators--the so-called density matrices. In general, it is very difficult to assess whether a given operator acting on an infinite dimensional Hilbert space is positive. The techniques developed here allow us to iteratively compute a sequence of positive trace-class operators which approximate the given operator. Also, as we will point out, this optimization problem is intimately related to localization (Toeplitz) operators [11, 16, 34, 42]. Here is a brief summary of the paper. In the next section, we introduce the main concepts related to the spectrum and Weyl transform of Hilbert--Schmidt operators. In section 3, we present the solution for the optimization problem, and we solve exactly a particular case in d = 1 in section 4. (This is roughly speaking the case of ``radial"" functions.) In section 5, we present the main results of this work. We consider the truncated eigenvalue problem and derive precise estimates for the errors of the eigenvalues and eigenvectors. In section 6, we go back to the Wigner approximation. We show that the Wigner approximation is never representable. We illustrate our results with a simple example. In section 7, we address the problem of obtaining approximately the spectrum and the Schatten norm for some Schatten-class operators. Finally, in section 8, we present our conclusions and discuss the possibility of applying our results to quasi-distributions other than the Wigner distribution. \widehat is a linear Notation. The complex conjugate of a number c is written c. If A \widehat operator acting on some Hilbert space, then we denote by Ker(A) its kernel. The inner product and the norm on L2 (\BbbR d ) are \int (11)

\langle \psi , \phi \rangle L2 (\BbbR d ) =

\psi (x)\phi (x)dx \BbbR d

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and \biggr) 1/2 , | \psi (x)| dx

\biggl( \int

2

| | \psi | | L2 (\BbbR d ) =

(12)

\BbbR d

respectively. We denote by \scrS (\BbbR d ) the Schwartz class of test functions and by \scrS \prime (\BbbR d ) its dual---the tempered distributions. We shall denote by | | \cdot | | l2 the norm for the spaces of square-summable sequences, \Biggl\{ \Biggr\} \infty \sum 2 2 2 (13) l (\BbbN ) = c = \{ cn \} n : | | c| | l2 = | cn | < \infty n=1

and \infty \sum

\Biggl\{ (14)

2

2

l (\BbbN ) =

\BbbF = \{ fn,m \} n,m :

| | \BbbF | | 2l2

=

\Biggr\} 2

| fn,m | < \infty .

n,m=1

The Fourier--Plancherel transform of f \in L1 (\BbbR d ) \cap L2 (\BbbR d ) is defined by \int 1 f (x)e - ik\cdot x dx. (15) (\scrF f )(k) := (2\pi )d/2 \BbbR d 2. Hilbert--Schmidt operators and Weyl transform. In this section we review some well known definitions and results of Hilbert--Schmidt operators and the Weyl transform. For more details, the reader should refer to [10, 55]. \widehat : \scrH \rightarrow \scrH 2.1. Hilbert--Schmidt operators. A Hilbert--Schmidt operator A on a separable Hilbert space \scrH is a linear operator such that [10] \sum \widehat i | | 2\scrH < +\infty (16) | | Ae i

for any orthonormal basis \{ ei \} i . We denote by S2 (\scrH ) the set of Hilbert--Schmidt operators. This is a Hilbert space with inner product \sum \widehat B\rangle \widehat S (\scrH ) := \widehat i , Be \widehat i \rangle \scrH (17) \langle A, \langle Ae 2 i

and norm (18)

\widehat 2 | | A| | S2 (\scrH ) :=

\sum

\widehat i | | 2\scrH . | | Ae

i

It can be shown that the previous expressions do not depend on the orthonormal basis. \widehat \in S2 (\scrH ) is selfHilbert--Schmidt operators are compact operators [10, 43]. If A adjoint, then it admits the spectral decomposition (19)

\widehat = A \widehat + + A \widehat - , A

where (20)

\widehat \pm := A

\sum j\in \BbbT \pm

\lambda j P\widehat j


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are the positive (+) and the negative ( - ) parts, \BbbT + and \BbbT - are (possibly finite) sets of integer indices labelling the positive and the negative eigenvalues, respectively, \{ \lambda j \} j\in \BbbT + are the positive eigenvalues, written as a decreasing sequence \lambda 1 > \lambda 2 > \lambda 3 > \cdot \cdot \cdot > 0,

(21)

\{ \lambda j \} j\in \BbbT - are the negative eigenvalues, written as an increasing sequence \lambda - 1 < \lambda - 2 < \lambda - 3 < \cdot \cdot \cdot < 0,

(22)

and P\widehat j : \scrH \rightarrow \scrH j is the orthogonal projection onto the eigenspace \scrH j associated with the eigenvalue \lambda j . Each eigenspace is finite dimensional: nj = dim(\scrH j ) < +\infty . The Hilbert space splits into the Hilbert sum: (23)

\widehat \oplus (\scrH - 1 \oplus \scrH - 2 \oplus \cdot \cdot \cdot ) \oplus (\scrH 1 \oplus \scrH 2 \oplus \cdot \cdot \cdot ) . \scrH = Ker(A)

By choosing orthonormal basis in each eigenspace \scrH j , we can rewrite (20) as (24)

\widehat \pm := A

\sum

\mu \pm j P\widehat \pm j ,

j\in \BbbU \pm

where P\widehat \pm j is the projector in the direction of the vector e\pm j of the orthonormal set \widehat \alpha = \mu \alpha e\alpha . of eigenvectors with \langle e\alpha , e\beta \rangle \scrH = \delta \alpha ,\beta for all \alpha , \beta \in \BbbU = \BbbU + \cup \BbbU - , and Ae The eigenvalues \{ \mu \alpha \} \alpha \in \BbbU are the same as \{ \lambda \alpha \} \alpha \in \BbbT , but they are not all necessarily distinct. This happens whenever some eigenvalue is degenerate (nj > 1). 2.2. The Weyl transform. In this work we deal with the case \scrH = L2 (\BbbR d ). A \bigl( \bigr) \widehat \in S2 L2 (\BbbR d ) is given by Hilbert--Schmidt operator A \int (25)

\widehat (A\psi )(x) :=

KA (x, y)\psi (y)dy,

\psi \in L2 (\BbbR d ),

\BbbR d

with a kernel KA \in L2 (\BbbR d \times \BbbR d ). The Weyl transform [12, 55] is a linear map \bigl( \bigr) (26) \scrW : S2 L2 (\BbbR d ) \rightarrow L2 (\BbbR 2d ) defined by \int (27)

\widehat \scrW (A)(x, k) :=

KA (x + y/2, x - y/2)e - iy\cdot k dy.

\BbbR d

\widehat \widehat \scrW is a bijection with inverse The function \scrW (A)(x, k) is called the Weyl symbol of A. - 1 2 2d \scrW . Thus, given a symbol A(x, k) \in L (\BbbR ), the associated Weyl operator is the Hilbert--Schmidt operator \biggr) \biggl( \int \int \bigl( - 1 \bigr) x+y 1 A , k eik\cdot (x - y) \psi (y)dkdy, (28) (\scrW A)\psi (x) = (2\pi )d \BbbR d \BbbR d 2 \widehat is self-adjoint if and only defined for all \psi \in L2 (\BbbR d ). A Hilbert--Schmidt operator A \widehat has the spectral if its symbol A(x, k) is a real function. In this case, the operator A

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\widehat + ) and a negative part (A \widehat - ). Using decomposition (19), (20), (24) with a positive (A the Weyl transform, we may thus define \widehat = A+ + A - , A = \scrW (A)

(29) where (30)

\widehat \pm ). A\pm = \scrW (A

\widehat \psi ,\phi (25) is the rank one operator with kernel An important case is when the operator A (31)

K\psi ,\phi (x, y) = (\psi \otimes \phi )(x, y) = \psi (x)\phi (y)

with \psi , \phi \in L2 (\BbbR d ). The associated Weyl symbol is (up to a multiplicative constant) the nondiagonal Wigner function (1): \int d \widehat \psi (x + y/2)\phi (x - y/2)e - iy\cdot k dy. (32) \scrW (A\psi ,\phi )(x, k) = (2\pi ) W (\psi , \phi )(x, k) = \BbbR d

Weyl operators and Wigner functions are also related via the following remarkable formula. Let F\widehat be some Hilbert--Schmidt operator with Weyl symbol F = \scrW (F\widehat ), and let \psi , \phi \in L2 (\BbbR d ). Then we have [20] (33)

\langle F\widehat \psi , \phi \rangle L2 (\BbbR d ) = \langle F, W (\phi , \psi )\rangle L2 (\BbbR 2d ) .

\widehat \psi ,\psi as in (31), (32): In the previous identity, let us choose F\widehat = A 1 2 \Bigl( \Bigr) \widehat \psi ,\psi \eta (x) = \langle \eta , \psi 2 \rangle L2 (\BbbR d ) \psi 1 (x) (34) A 1 2 for all \eta \in L2 (\BbbR d ). From (33), (34), we have \widehat \psi ,\psi \phi 2 , \phi 1 \rangle L2 (\BbbR d ) = \langle A\psi ,\psi , W (\phi 1 , \phi 2 )\rangle L2 (\BbbR 2d ) \langle A 1 2 1 2 (35) \leftrightarrow \langle \phi 2 , \psi 2 \rangle L2 (\BbbR d ) \langle \psi 1 , \phi 1 \rangle L2 (\BbbR d ) = (2\pi )d \langle W (\psi 1 , \psi 2 ), W (\phi 1 , \phi 2 )\rangle L2 (\BbbR 2d ) , and we obtain Moyal's identity [39]: (36)

\langle W (\psi 1 , \psi 2 ), W (\phi 1 , \phi 2 )\rangle L2 (\BbbR 2d ) =

1 \langle \psi 1 , \phi 1 \rangle L2 (\BbbR d ) \langle \phi 2 , \psi 2 \rangle L2 (\BbbR d ) . (2\pi )d

As a consequence of this, we have the following. \bigl\{

Lemma 1. Let \bigr\} \{ en \} n be an orthonormal basis of L2 (\BbbR d ). Then the functions (2\pi )d/2 W (en , em ) n,m form an orthonormal basis of L2 (\BbbR 2d ). Proof. From Moyal's identity, we have \langle W (en , em ), W (ek , el )\rangle L2 (\BbbR 2d ) =

1 \langle en , ek \rangle L2 (\BbbR d ) \langle el , em \rangle L2 (\BbbR d ) (2\pi )d

(37) 1 \delta n,k \delta l,m , (2\pi )d \bigl\{ \bigr\} which shows that (2\pi )d/2 W (en , em ) n,m are an orthonormal set. =


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Next, assume that F \in L2 (\BbbR 2d ) is such that \langle F, W (en , em )\rangle L2 (\BbbR 2d ) = 0

(38)

for all n, m. Let F\widehat = \scrW - 1 (F ) be the Hilbert--Schmidt operator with Weyl symbol F . From (33), we have for all n, m 0 = \langle F\widehat en , em \rangle L2 (\BbbR d ) .

(39)

Since \{ en \} n is an orthonormal basis of L2 (\BbbR \bigl\{ d ), this is possible \bigr\} if and only if F\widehat = 0 and d/2 F \equiv 0. Consequently the orthonormal set (2\pi ) W (en , em ) n,m is complete. 3. The optimization problem. Let F \in L2 (\BbbR 2d ) and \scrE = \{ en \} n\in \BbbN be some orthonormal basis of L2 (\BbbR d ). We may thus write \sum (40) F (z) = fn,m W (en , em )(z), n,m\in \BbbN

where the coefficients fn,m are given by fn,m = (2\pi )d \langle F, W (en , em )\rangle L2 (\BbbR 2d ) .

(41)

If F is a real function, then for all n, m \in \BbbN .

fn,m = fm,n ,

(42)

Moreover, we have (cf. (36)) | | F | | 2L2 (\BbbR 2d ) =

(43)

\sum 1 | fn,m | 2 < \infty . (2\pi )d n,m\in \BbbN

2

d

Given some \psi \in L (\BbbR ), we can also expand it in the basis \scrE : \sum (44) \psi (x) = cn en (x) n\in \BbbN

with (45)

| | \psi | | 2L2 (\BbbR d ) =

\sum

| cn | 2 < \infty .

n\in \BbbN

This entails (46)

W \psi (z) =

\sum

cn cm W (en , em )(z).

n,m\in \BbbN

In what follows, \BbbF denotes the infinite matrix with coefficients \{ fn,m \} n,m\in \BbbN and c is the column vector cT = (c1 , c2 , c3 , . . .). In view of (43), (45), we have that \BbbF \in l2 (\BbbN 2 ) and c \in l2 (\BbbN ). We may regard \BbbF as a bounded linear operator l2 (\BbbN ) \rightarrow l2 (\BbbN ), c \mapsto \rightarrow \BbbF c with operator norm (47)

| | \BbbF | | Op := supc\in l2 (\BbbN )\setminus \{ 0\}

| | \BbbF c| | l2 | | c| | l2

with | | \cdot | | l2 the l2 norm. The boundedness is easily established by the fact that the operator norm is dominated by the l2 -norm: (48)

| | \BbbF | | Op \leq | | \BbbF | | l2 = (2\pi )d/2 | | F | | L2 (\BbbR 2d ) .

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Remark 2. It is interesting to remark that we have three representations of the same object. First, we have a self-adjoint Hilbert--Schmidt operator F\widehat = F\widehat + + F\widehat - acting on the Hilbert space L2 (\BbbR d ). From the Weyl transform, we obtain its counterpart in phase space \scrW (F\widehat ) = F = F+ + F - . Finally, using the expansion (40), (41) of F in some orthonormal basis of Wigner functions, we obtain yet another representation of F\widehat ---the matrix \BbbF \in l2 (\BbbN 2 ). The important thing is that they all have the same spectrum. That is, the eigenvalues of the operator F\widehat are the same as those of the matrix \BbbF . Moreover, they also coincide with the eigenvalues of the function F regarded as a 2 pseudodifferential operator (\BbbR 2d ) \bigr) \rightarrow L2 (\BbbR 2d ), which acts on G \in L2 (\BbbR 2d ) as \bigl( - 1 F\ast : L - 1 F\ast (G) = F \ast G := \scrW \scrW (F ) \cdot \scrW (G) , where \ast is the Moyal star product. Recall that we want to find the Wigner function W \psi (0) closest to F in L2 (\BbbR 2d ). This amounts to minimizing the following functional: \scrL (c) := | | F - W \psi | | 2L2 (\BbbR 2d ) .

(49)

From (40), (46), and Moyal's identity, we obtain (50)

\scrL (c) =

| | F | | 2L2 (\BbbR 2d )

\sum 1 2 cn fn,m cm + - d (2\pi ) (2\pi )d

\Biggr) 2

\Biggl(

n,m\in \BbbN

\sum

2

| cn |

.

n\in \BbbN

The following can be found in [28] for nondegenerate spectrum. We consider it here for completeness. Theorem 3. If F+ \equiv 0, then the minimizing function of (49) is W \psi (0) \equiv 0. Otherwise, it is given by (46), where c(0) is any eigenvector of \BbbF associated with the largest eigenvalue \lambda max : \BbbF c(0) = \lambda max c(0) .

(51)

Moreover, the normalization of \psi (0) is such that (52)

| | \psi (0) | | 2L2 (\BbbR d ) = | | c(0) | | 2l2 = \lambda max .

The minimal distance is then min\psi \in L2 (\BbbR d ) | | F - W \psi | | 2L2 (\BbbR 2d ) = \scrL (c(0) ) (53) = | | F | | 2L2 (\BbbR 2d ) -

| | \psi (0) | | 4L2 (\BbbR d ) \lambda 2max 2 = | | F | | - . L2 (\BbbR 2d ) (2\pi )d (2\pi )d

Proof. In [8] we proved the existence of a global minimizer. In the calculus of variations, if c(0) is a minimizer, then the functional \scrL has to be stationary at c(0) [26, 30]. In other words, the Fr\'echet derivative of (49) with respect to the real and imaginary parts of cn , or equivalently with respect to cn and cn , has to vanish identically. Imposing a vanishing derivative with respect to cn is equivalent to doing the same with respect to cn : one equation is obtained from the other by complex conjugation. In particular, the Fr\'echet derivative of (49) with respect to cn yields (54)

\partial \scrL 2 \sum 2 | | c| | 2l2 cn . = - fn,m cm + \partial cn (2\pi )d (2\pi )d m\in \BbbN


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Thus the stationarity condition at c(0) becomes \sum (0) 2 (0) (55) fn,m c(0) | | l2 cn m = | | c m\in \BbbN

for all n \in \BbbN . In other words, c(0) is an eigenvector of \BbbF with eigenvalue | | c(0) | | 2l2 \geq 0. If F+ \equiv 0, then | | c(0) | | 2l2 = 0 and the minimizing solution is W \psi (0) \equiv 0. Alternatively, if F+ is not identically zero, then it follows that (56)

\scrL (c(0) ) = | | F | | 2L2 (\BbbR 2d ) -

| | c(0) | | 4l2 , (2\pi )d

where | | c(0) | | 2l2 is one of the eigenvalues of \BbbF . Obviously, (56) is minimal if | | c(0) | | 2l2 is equal to the largest eigenvalue \lambda max . The problem we want to address is how to compute \lambda max and c(0) . If the matrix \BbbF is finite dimensional, there are good approximation techniques to compute eigenvalues and eigenvectors, such as the Rayleigh--Ritz method [29]. Here however, we want to focus on the infinite dimensional case. Several approaches can be considered. In [9, 28] the authors considered the ``weighted Wigner distribution"" W\Gamma \psi (x, k) (10). To deal with the infinite dimensional problem, they chose to use a discrete-time Wigner function. Our strategy is different and consists of truncating the infinite eigenvalue problem at some finite order N \in \BbbN . We cannot hope to obtain (in general) the exact solution, but we can derive precise estimates for the truncated version. Let us then explain our approach in more detail. We are given some real-valued function F \in L2 (\BbbR 2d ). We evaluate its norm | | F | | L2 (\BbbR 2d ) , choose a particular orthonormal basis \scrE , and compute the expansion coefficients (41) for n, m = 1, 2, . . . , N . We next obtain the truncated N \times N matrix \BbbF (N ) . To fix the order N , we use the following criterion. Let (57)

F (N ) (z) =

N \sum

fn,m W (en , em )(z)

n,m=1

be the truncated function. The relative error is given by (58)

0
0 and n = 0, 1, 2, . . . . The Wigner functions W (ej , ej+k ) can be obtained from (72), by noticing that (75)

W (f, g)(z) = W (g, f )(z).

We should add a word of caution concerning the notation of (73). In complex analysis the letter z is used to denote complex numbers such as the one in (73), \surd 2(x+ik) \in \BbbC . This is also the notation in analytic (or poly-analytic) time-frequency representations, such as the Bargmann (or poly-Bargmann) transforms [1, 4, 23]. However, here we have reserved the letter z to denote the real phase space point z = (x, k) \in \BbbR 2 . This is the reason for choosing the notation a. Notice that this is closer to the physicists notation, where a, a can be seen as the Weyl symbols of the annihilation and creation operators, respectively. Before we continue, let us make the following observation. The diagonal Wigner functions associated with the Hermite functions (72) are radial: 2 ( - 1)n 0 Ln (2| z| 2 )e - | z| . \pi To stress this fact, we will rewrite (76) as

(76)

(77)

W en (z) =

W en (z) = Fn (\eta (z)),

where (78)

Fn (\eta ) =

2 ( - 1)n 0 Ln (2\eta 2 )e - \eta , \pi

and (79)

\eta (z) = | z| .

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Theorem 4. Let F \in L2 (\BbbR 2 ) be a function of \rho (z) only, (80)

F (z) = G(\rho (z)),

where

\surd

\rho G(\rho ) \in L2 (0, +\infty ), and \rho (z) := ((z - z0 ) \cdot A(z - z0 ))

(81)

1/2

.

Here z0 \in \BbbR 2 and A is a real, symmetric, positive-definite 2 \times 2 matrix with (82)

det A = 1.

Then, we have (83)

F (z) =

\infty \sum

\mu n Fn (\rho (z)),

n=0

where Fn is the Wigner function associated with the nth eigenstate of the harmonic oscillator given by (78), and \int \infty 2 (84) \mu n = 4\pi ( - 1)n G(\rho )Ln (2\rho 2 )e - \rho \rho d\rho . 0

Proof. A function G with the conditions stated in the theorem admits the following ``diagonalization"" (see [55, section 24] and [27]): (85)

G(\rho ) =

\infty \sum

\mu n Fn (\rho ),

n=0

where the \mu n are given by (84), and Fn is given by (78). From (80) the result follows. Before we proceed, let us make the following remarks. Remark 5. First, if we set z0 = 0 and A = Id in (81), then \rho (z) = | z| and the function F (z) = G(| z| ) is radial. By considering an arbitrary positive matrix A, we can solve more problems, such as the one in Example 8 below. Second, let us point out that (82) does not pose any serious restriction. Indeed, suppose that det(A) \not = 1. Define B = \surd A . Then det B = 1, and since det(A)

(86)

\rho =

\surd 4

det A\widetilde \rho ,

where (87)

1

\rho \widetilde = ((z - z0 ) \cdot B(z - z0 )) 2 ,

we conclude that F can also be regarded as a function of \rho \widetilde only and the same results follow. Remark 6. A function with the ``radial"" property stated in Theorem 4 seems to be diagonalized in (83). Although this is true, some care is required to make this assertion. Indeed, we have to make sure that Fn (\rho (z)) are Wigner functions. To show that this is indeed the case, we recall the following symplectic covariance property of Wigner distributions [19, 20, 22, 33, 44, 51, 55] and Williamson's theorem [54]. Let


5173

Sp2 (d) be the metaplectic group, i.e., the two fold cover of Sp(d). For each S \in Sp(d), there exist \pm S\widetilde \in Sp2 (d) which project onto S. The metaplectic representation M p(d) \widetilde with the property that is a unitary representation of Sp2 (d), Sp2 (d) \ni S\widetilde \mapsto \rightarrow \mu (S), Weyl \widetilde - 1 A\mu ( \widehat S) \widetilde \leftarrow \mu (S) \rightarrow a \circ S

(88)

\widehat : \scrS (\BbbR d ) \rightarrow \scrS \prime (\BbbR d ) a Weyl operator with Weyl symbol a \in \scrS \prime (\BbbR 2d ). In particular, for A for Wigner functions, we have \Bigl( \Bigr) \widetilde \mu (S)g \widetilde (89) W \mu (S)f, (z) = W (f, g)(S - 1 z) for all f, g \in \scrS (\BbbR d ). By usual density arguments this extends to L2 (\BbbR d ). Moreover, the set of Wigner functions is left invariant under phase space translations. Altogether, if W \psi (z) is a Wigner function, then under an affine symplectic transformation W \psi (Sz - z0 ) (S \in Sp(d), z0 \in \BbbR 2d ) we obtain another Wigner function. Now, let us go back to the 2\times 2 matrix A in (81). Williamson's theorem [54] states that there exists S \in Sp(1) and a positive number \lambda (called a Williamson invariant) such that A = \lambda S T S. By assumption det(A) = 1 and thus \lambda = 1. Hence A = S T S.

(90) We thus have that (91)

\rho 2 (z) = (z - z0 ) \cdot A(z - z0 ) = (S(z - z0 )) \cdot (S(z - z0 )) .

It follows that Fn (\rho (z)) is obtained from W en (z) by the affine symplectic transformation: z \mapsto \rightarrow Sz - Sz0 .

(92)

Thus Fn (\rho (z)) is again a Wigner function. Remark 7. If a function F is a function of \rho (81) only, as in Theorem 4, and F+ is not identically zero, then we can solve the optimization problem exactly by the following (finite) iterative procedure. First notice that from Remarks 2 and 6, the function F is diagonalized in (83) and thus the coefficients \mu n are in fact the eigenvalues of F\widehat , \BbbF , and F\ast . We then proceed as follows. (1) Compute the eigenvalues (84) until you find the first positive one, say, \mu k1 . Define (93)

F (k1 ) (z) :=

k1 \sum

\mu n Fn (\rho (z)).

n=0

If (94)

\mu k | | F - F (k1 ) | | L2 (\BbbR 2d ) \leq \surd 1 , 2\pi

then we conclude that \sqrt{} (95)

| \mu l | \leq

+\infty \sum n=k1 +1

| \mu n | 2 < \mu k1

5174


for all l = k1 + 1, k1 + 2, . . . . Consequently, \mu k1 is the largest eigenvalue of F\widehat and the optimal solution is (96)

W \psi (0) (z) = \mu k1 Fk1 (\rho (z)).

(2) If (94) does not hold, then look for the next positive eigenvalue \mu k2 (k2 > k1 ) and set (97)

F (k2 ) (z) :=

k2 \sum

\mu n Fn (\rho (z)).

n=0

Define K \in \{ k1 , k2 \} , such that (98)

\mu K = max \{ \mu k1 , \mu k2 \} .

If (99)

\mu K | | F - F (k2 ) | | L2 (\BbbR 2d ) \leq \surd , 2\pi


| \mu l | \leq

+\infty \sum

| \mu n | 2 < \mu K

n=k2 +1

for all l = k2 + 1, k2 + 2, . . . . Consequently, \mu K is the largest eigenvalue of F\widehat , and the optimal solution is (101)

W \psi (0) (z) = \mu K FK (\rho (z)).

(3) If (99) is still not valid, then we proceed in the same fashion and obtain a set of positive eigenvalues \mu k1 , \mu k2 , . . . , \mu kn (0 \leq k1 < k2 < \cdot \cdot \cdot < kn ). As before, we set (102)

F (kn ) (z) :=

kn \sum

\mu n Fn (\rho (z)),

n=0

and define K \in \{ k1 , k2 , . . . , kn \} such that (103)

\mu K = max \{ \mu k1 , \mu k2 , . . . , \mu kn \} .

If (104)

\mu K | | F - F (kn ) | | L2 (\BbbR 2d ) \leq \surd , 2\pi


| \mu l | \leq

+\infty \sum

| \mu n | 2 < \mu K ,

n=kn +1

for all l = kn + 1, kn + 2, . . . . Consequently, \mu K is the largest eigenvalue of F\widehat and the optimal solution is given by (101). Notice that condition (104) will eventually be satisfied for some kn \in \BbbN , since \sum +\infty 2 n=k | \mu n | \rightarrow 0 as k \rightarrow \infty .

5175


Example 8. A particular instance of the previous construction is a Gaussian of the form F (z) = N exp ( - \alpha (z - z0 ) \cdot A(z - z0 )) ,

(106)

where N and \alpha are arbitrary positive constants, and where we assume that A is a real, symmetric, positive-definite 2 \times 2 matrix with det(A) = 1. A straightforward calculation yields for n \geq 1 \int \infty 2 e - (1+\alpha )\rho \rho Ln (2\rho 2 )d\rho \mu n = 4\pi ( - 1)n N 0

n

(107)

= 4\pi ( - 1) N

\int n \biggl( \biggr) \sum n ( - 2)k k=0

k

k!

\infty

2

e - (1+\alpha )\rho \rho 2k+1 d\rho

0

\biggr) k \biggl( \biggr) n \biggl( \biggr) n n \biggl( \biggr) \biggl( 2 ( - 1)n 2\pi N 2 2\pi N 1 - \alpha ( - 1)n 2\pi N \sum n = = , - 1 - = k 1 + \alpha 1 + \alpha 1 + \alpha 1 + \alpha 1 + \alpha 1 + \alpha k=0

and (108)

\mu 0 =

2\pi N . 1 + \alpha

Clearly, if \alpha = 1, then \mu n = \pi N \delta n,0 , and the largest eigenvalue is \mu 0 . If \alpha < 1, then all the eigenvalues are strictly positive. Notice that in this case, the Gaussian satisfies the Robertson--Schr\"odinger uncertainty principle [41]: A - 1 + i\alpha J \geq 0,

(109) where

\biggl( (110)

J=

0 - 1

1 0

\biggr)

is the standard symplectic matrix. The uncertainty principle (109) is well known to be a necessary and sufficient condition for a Gaussian measure to be the Weyl symbol of a positive trace-class operator [40, 41]. Thus, if \alpha < 1, the sequence (107) is strictly decreasing. Hence, the largest eigenvalue is \mu 0 . Finally, if \alpha > 1, then we have an alternating sequence \biggl( \biggr) n ( - 1)n 2\pi N \alpha - 1 (111) \mu n = , n \geq 0. 1 + \alpha 1 + \alpha But again, since the moduli sequence | \mu n | =

(112)

2\pi N 1 + \alpha

\biggl(

\alpha - 1 1 + \alpha

\biggr) n

is strictly decreasing, \mu 0 is again the largest positive eigenvalue. Consequently, for any \alpha > 0, the Wigner function closest to the Gaussian measure (106) is (113)

W \psi (0) (z) =

2\pi N 2N - (z - z0 )\cdot A(z - z0 ) F0 (\rho (z)) = e . 1 + \alpha 1 + \alpha 2

Thus in particular, if we have F (z) = W e0 (z) = \pi 1 e - | z| (that is, N = A = Id, and z0 = 0), then we obtain W \psi (0) (z) = W e0 (z) as expected.

1 \pi ,

\alpha = 1,

5176


Remark 9. Before we conclude this section, we remark that the optimization problem considered in this paper is intimately related with the so-called localization or Toeplitz operators in time-frequency analysis [16, 18, 42, 45, 46] and quantum mechanics [11, 34]. In [11] the authors addressed the optimization problems \int W \psi (z)dz \int D (114) min \psi \in L2 (\BbbR )\setminus \{ 0\} W \psi (z)dz \BbbR 2 and \int W \psi (z)dz D \int max , \psi \in L2 (\BbbR )\setminus \{ 0\} W \psi (z)dz \BbbR 2

(115)

where D \subset \BbbR is some bounded domain whose boundary \partial D is a regular curve. If we define F (z) = \chi D (z) \in L2 (\BbbR 2 ) to be the characteristic function of D, then it is straightforward to prove that the Wigner function W \psi (0) closest in L2 to F is the optimal solution of (115). In [11, 18, 34] the authors proved that Gaussians maximize (115), when D is a disk, a poly-disk, or a ball. 5. The general approximation procedure. A crucial point in our derivation will be the Courant--Fischer min-max theorem, which we recapitulate here for completeness. Theorem 10 (Courant--Fischer min-max theorem). Let A be an N \times N Hermitian matrix and write its eigenvalues as a decreasing sequence \alpha 1 \geq \alpha 2 \geq \cdot \cdot \cdot \geq \alpha N . Then we have (116)

\alpha j =

v \cdot Av,

sup

inf

dim(V )=j

v\in V,| | v| | =1

and (117)

\alpha j =

inf

sup

dim(V )=N - j+1 v\in V,| | v| | =1

v \cdot Av

for all j = 1, 2, . . . , N and V ranges over all subspaces of \BbbC N with the indicated dimension. Here | | v| | 2 = | v1 | 2 + \cdot \cdot \cdot + | vN | 2 . Before we proceed, let us recall that \{ \lambda j \} j are the distinct eigenvalues of the matrix \BbbF , whereas \{ \mu j \} j are its eigenvalues with multiplicities. With our previous (N ) notation (60)--(67) the eigenvalues of \BbbF (N ) are \{ \mu j \} j . If we rearrange the eigenval(N )

ues of \BbbF (N ) as a decreasing sequence \{ \alpha j \} j (as described in the Courant--Fischer theorem), we may write \left\{ (N ) \mu j , j = 1, . . . , N+ , (118)

(N )

\alpha j

=

0,

j = N+ + 1, . . . , N+ + NK ,

(N ) \mu j - N - 1 ,

j = N+ + NK + 1, . . . , N.

We follow closely [31] for the estimates of the eigenvalues. We just have to make some adaptations to complex-valued matrices and to the facts that we have the L2 bound (59) and that the matrix \BbbF is not necessarily positive. (N ) (N ) We start by proving that each sequence \{ \mu j \} N \in \BbbN and \{ - \mu - j \} N \in \BbbN with fixed j \geq 1 of nonzero eigenvalues is nondecreasing with respect to N and, since they are bounded, they are convergent.


5177

But before we do that, we remark that we may at various places denote the vectors (x1 , . . . , xN ) \in \BbbC N and (x1 , . . . , xN , 0, 0, 0, . . .) \in l2 (\BbbN ) by the same symbol x(N ) according to our convenience. Proposition 11. With the previous notation, we have that for 1 \leq j \leq N+ (N )

(119)

\mu j

\nearrow \mu j ,

N \rightarrow \infty ,

while (N )

\mu - k \searrow \mu - k ,

(120)

N \rightarrow \infty ,

for 1 \leq k \leq N - . Proof. By the Courant--Fischer theorem, we have for 1 \leq j \leq N (N )

(121)

\alpha j

=

sup

inf

dim V =j

x(N ) \in V,| | x(N ) | | =1

x(N ) \cdot \BbbF (N ) x(N ) ,

and (122)

(N +1)

\alpha j

=

sup

inf

dim V =j

x(N +1) \in V,| | x(N +1) | | =1

x(N +1) \cdot \BbbF (N +1) x(N +1) .

Notice that we can rewrite (121) as (123)

(N )

\alpha j

=

sup

x(N +1) \cdot \BbbF (N +1) x(N +1) ,

inf

dim V \prime =j

x(N +1) \in V \prime ,| | x(N +1) | | =1

where the supremum is taken over all subspaces V \prime of \BbbC N +1 with dimension j, such that x(N +1) \in V \prime if and only if its (N + 1)th coordinate is zero. From (122) and (123) it is then obvious that (N )

(124)

\alpha j

(N +1)

\leq \alpha j

.

Thus, for fixed j this is a nondecreasing sequence. In particular, we have (N + 1)+ \geq N+ .

(125) Thus for 1 \leq j \leq N+ , we have (126)

(N )

\mu j

(N )

= \alpha j

(N +1)

\leq \alpha j

(N )

Next, if we replace \BbbF (N ) by - \BbbF (N ) , then \alpha j previous conclusions to - \BbbF (N ) , it follows that (127)

(N +1)

= \mu j

.

(N )

\rightarrow - \alpha N - j+1 . If we then apply the

(N + 1) - \geq N - ,

and (128)

(N )

(N +1)

\mu - j \geq \mu - j

for 1 \leq j \leq N - . Concerning convergence, we invoke a familiar theorem for the spectral radius of bounded linear operators on Banach spaces. For any element \mu (N ) in the spectrum of \BbbF (N ) and for fixed N \in \BbbN , we have (129)

| \mu (N ) | \leq | | \BbbF (N ) | | Op \leq | | \BbbF (N ) | | l2 \leq | | \BbbF | | l2 = (2\pi )d/2 | | F | | L2 (\BbbR 2d ) .

5178

BEN-BENJAMIN, COHEN, DIAS, LOUGHLIN, PRATA (N )

(N )

We conclude that the sequences \{ \mu j \} N \in \BbbN and \{ \mu - k \} N \in \BbbN for fixed j, k are bounded and monotone, and hence convergent. (N ) It remains to prove that the limit of \mu j for fixed j \not = 0 is \mu j as N \rightarrow \infty . We prove the result for the positive eigenvalues j \geq 1. The proof for negative eigenvalues is identical. (N ) So assume that the limit as N \rightarrow \infty of some \mu j is not in the spectrum of \BbbF . (N )

Let j0 be the smallest j \geq 1 for which \mu j \rightarrow \mu \ast j , and \mu \ast j is not in the spectrum of \BbbF . Recalling the notation (62)--(69) for the orthonormal eigenvectors of \BbbF and \BbbF (N ) , we have (N )

(N )

(N )

(N )

(N )

| | \BbbF cj0 - \mu \ast j0 cj0 | | l2 = | | (\BbbF - \BbbF (N ) )cj0 + \BbbF (N ) cj0 - \mu \ast j0 cj0 | | l2 (N )

(N )

(N )

\leq | | (\BbbF - \BbbF (N ) )cj0 | | l2 + | | (\mu j0 - \mu \ast j0 )cj0 | | l2 (130) (N )

(N )

(N )

\leq | | (\BbbF - \BbbF (N ) )| | Op | | cj0 | | l2 + | \mu j0 - \mu \ast j0 | | | cj0 | | l2 (N )

\leq (2\pi )d/2 | | F - F (N ) )| | L2 (\BbbR 2d ) + | \mu j0 - \mu \ast j0 | . And thus (131)

(N )

(N )

| | \BbbF cj0 - \mu \ast j0 cj0 | | l2 \rightarrow 0,

as N \rightarrow \infty . (N ) Next we expand cj0 in the basis \scrB (68) of l2 (\BbbN ): (132)

(N )

cj0 =

\sum

a\alpha e\alpha

\alpha \in \scrA

with (133)

(N )

| | cj0 | | l2 =

\sum

| a\alpha | 2 = 1.

\alpha \in \scrA

On the other hand, if \mu \ast j0 is not in Spec(\BbbF )---the spectrum of \BbbF ---then \bigl\{ \bigr\} (134) 0 < Kj0 := inf | \mu \ast j0 - \eta | : \eta \in Spec(\BbbF ) . It follows from (132) that \sum \bigl( \bigr) (N ) \bigl( \bigr) \sum (135) | | \BbbF - \mu \ast j0 Id cj0 | | l2 = | | \BbbF - \mu \ast j0 Id a\alpha e\alpha | | l2 = | | a\alpha (\beta \alpha - \mu \ast j0 )e\alpha | | l2 , \alpha \in \scrA

\alpha \in \scrA

where \beta \alpha is the eigenvalue of \BbbF associated with the eigenvector e\alpha . From Pythagoras's theorem and (133), (134), (135), we have \sqrt{} \sum \sqrt{} \sum \bigl( \bigr) (N ) \ast \ast 2 2 | a\alpha | | \beta \alpha - \mu j0 | > Kj0 | a\alpha | 2 = Kj0 > 0, (136) | | \BbbF - \mu j0 Id cj0 | | l2 = \alpha \in \scrA

\alpha \in \scrA

which contradicts (131). (N ) This proves that \mu \ast j = limN \rightarrow \infty \mu j is in the spectrum of \BbbF for all j \geq 1. But it still remains to prove that \mu \ast j = \mu j .

5179


(N ) Suppose that \=j is the smallest j \geq 1, such that \mu j \rightarrow \mu \ast j \not = \mu j . From this \ast assumption and the monotonicity, we must have in fact \mu \=j < \mu \=j . From the min-max principle and the monotonicity, we have for N > \=j (N )

\mu \=j

=

x(N ) \cdot \BbbF (N ) x(N ) | | x(N ) | | 2 x(N ) \in V ;x(N ) \not =0

sup

inf

dim V =\= j

(137) x(N ) \cdot \BbbF x(N ) \leq \mu \=\ast j < \mu \=j , 2 x(N ) \in V \prime ;x(N ) \not =0 | | x(N ) | | l2

= sup

inf

imV \prime =\= j

where V ranges over all \=j-dimensional subspaces of \BbbC N and V \prime ranges over all \=jdimensional subspaces of l2 (\BbbN ) such that x(N ) = (x1 , x2 , . . .) \in V \prime implies xN +1 = xN +2 = xN +3 = \cdot \cdot \cdot = 0. But as N \rightarrow \infty , the spaces V \prime become dense in l2 (\BbbN ) and, from the last inequality in (137), we have a contradiction with (138)

\mu \=j =

sup

inf

dim W =\= j x\in W ;x\not =0

x \cdot \BbbF x , | | x| | 2l2

where W ranges over all \=j-dimensional subspaces of l2 (\BbbN ). In the next theorem, we obtain an estimate for the approximate eigenvalues. In particular, the estimate stated in (141) can be regarded as an infinite dimensional version of the Weyl [52] or the Wielandt--Hoffman [47] inequalities. In [5, 6], the authors considered the multiplication of a fixed multiplier pattern (Bessel multiplier), which is inserted between the analysis and the synthesis operator. They then considered the perturbation of Bessel sequences and obtain results which have some resemblance with our next theorem. Theorem 12. With the assumption (59), we have \Bigl\{ \Bigr\} (N ) (N ) (139) \mu j \leq \mu j \leq max \mu j + (2\pi )d/2 \epsilon | | F | | L2 (\BbbR 2d ) , 2(2\pi )d/2 \epsilon | | F | | L2 (\BbbR 2d ) , for j = 1, 2, . . . , N+ , and \Bigr\} \Bigl\{ (N ) (N ) (140) min - \mu - j - (2\pi )d/2 \epsilon | | F | | L2 (\BbbR 2d ) , - 2(2\pi )d/2 \epsilon | | F | | L2 (\BbbR 2d ) \leq \mu - j \leq \mu - j , for j = 1, 2, . . . , N - . Consequently, (N )

- \mu j | < 2(2\pi )d/2 \epsilon | | F | | L2 (\BbbR 2d ) ,

| \mu j

(141)

for all j = - N - , . . . , - 1, 1, . . . , N+ . Proof. Choose j \leq N < M . Again, by the min-max theorem (142)

(M )

\alpha j

=

inf

sup

dim V =M - j+1 x(M ) \in V \setminus \{ 0\}

x(M ) \cdot \BbbF (M ) x(M ) . | | x(M ) | | 2

For x(M ) = (x1 , . . . , xN , . . . , xM ), set \Biggl( (143)

y1 =

N \sum i=1

\Biggr) 1/2 2

| xi |

\Biggl( ,

y2 =

M \sum

i=N +1

\Biggr) 1/2 2

| xi | ,

,

5180


so that | | x(M ) | | 2 =

(144)

M \sum

| xi | 2 = y12 + y22 .

i=1

Moreover, x(M ) \cdot \BbbF (M ) x(M ) =

M \sum

xi fik xk

i,k=1

(145) =

x(N )

(N ) (N )

\cdot \BbbF

x

+ 2Re

\Biggl( N M \sum \sum

\Biggr) xi fik xk

+

i=1 k=N +1

M \sum

xi fik xk .

i,k=N +1

Applying the Cauchy--Schwarz inequality twice to the second term on the right-hand side of the previous equation, \bigm| \bigm| \Biggl( N \Biggr) M N M \bigm| \sum \bigm| \sum \sum \sum \bigm| \bigm| 2Re xi fik xk \leq 2 | xi | \bigm| fik xk \bigm| \bigm| \bigm| i=1 k=N +1 i=1 k=N +1 \right) \left( \bigm| M \bigm| 2 1/2 N \bigm| \sum \bigm| \sum \bigm| \bigm| fik xk \bigm| \leq 2y1 \bigm| \bigm| \bigm| i=1 k=N +1 (146) \Biggl[ \Biggl( M \Biggr) \Biggr] 1/2 N \sum \sum 2 2 \leq 2y1 y2 | fik | i=1

= 2y1 y2

k=N +1

\Biggr) 1/2

\Biggl( N M \sum \sum

2

| fik |

\leq 2\epsilon N y1 y2 | | F | | L2 (\BbbR 2d ) ,

i=1 k=N +1

where \Bigl( \sum (147)

\epsilon N :=

N i=1

\sum \infty

2 k=N +1 | fik |

\Bigr) 1/2

| | F | | L2 (\BbbR 2d )

.

Next we consider the third term on the right-hand side of (145): \bigm| \bigm| \sum M M \bigm| \bigm| \sum x f x i ik k \bigm| \bigm| i,k=N +1 (148) xi fik xk \leq y22 \bigm| \sum M \bigm| \leq \rho N y22 , 2 \bigm| \bigm| | x | l l=N +1 i,k=N +1 where (149)

\rho N

\bigm| \sum \infty \bigm| \bigm| \bigm| \bigm| \sum i,k=N +1 xi fik xk \bigm| := supx \bigm| \bigm| \infty 2 \bigm| \bigm| l=N +1 | xl |

and the supremum is taken over all x = (0, . . . , 0, xN +1 , xN +2 , . . .) \in l2 (\BbbN ). Finally, consider the first term in (145): \Biggl( \Biggr) (N ) \cdot \BbbF (N ) x(N ) x (N ) (N ) 2 (150) x(N ) \cdot \BbbF x = y1 . \sum N 2 i=1 | xi | Recall from (142) that we are considering planes of dimension M - j + 1. Their codimension in \BbbC M is j - 1. But, if we set the coordinates xN +1 = \cdot \cdot \cdot = xM = 0


5181

in these planes, then the resulting codimension in \BbbC N is lower or equal to j - 1. For some planes it is strictly smaller. We conclude that the set of all subspaces of \BbbC N with dimension N - j + 1 is a proper subset of the set of subspaces of \BbbC N with codimension j - 1 obtained in this fashion. It follows by the min-max principle that (151)

dim V

inf

\prime =M - j+1

x(N ) \cdot \BbbF (N ) x(N ) (N ) \leq \alpha j , \sum N 2 | x | x(M ) \in V \prime \setminus \{ 0\} i i=1 sup

where, as before, if x(N ) = (x1 , x2 , . . . , xM ) \in V \prime , then xN +1 = \cdot \cdot \cdot = xM = 0. Consequently from (142), (145), (146), (148), (150), (151) we have that (N )

(152)

(M )

\alpha j

\leq

y12 \alpha j

max

(y1 ,y2 )\in (\BbbR + )2

+ 2\epsilon N | | F | | L2 (\BbbR 2d ) y1 y2 + \rho N y22 . y12 + y22

In particular, for the sequence of positive eigenvalues (j = 1, . . . , N+ ), we have (N )

(153)

(M ) \mu j

\leq

y12 \mu j

max

(y1 ,y2 )\in (\BbbR + )2

+ 2\epsilon N | | F | | L2 (\BbbR 2d ) y1 y2 + \rho N y22 . y12 + y22

The maximum on the right-hand side of (153) is easily computed and we obtain \sqrt{} (N ) (N ) \mu + \rho + (\mu j - \rho N )2 + 4\epsilon 2N | | F | | 2L2 (\BbbR 2d ) N j (M ) . (154) \mu j \leq 2 Since the right-hand side is independent of M > N , we have from Proposition 11 \sqrt{} (N ) (N ) \mu + \rho + (\mu j - \rho N )2 + 4\epsilon 2N | | F | | 2L2 (\BbbR 2d ) N j (N ) \mu j \leq \mu j \leq 2 (155) \Bigl\{ \Bigr\} (N ) \leq max \mu j + \epsilon N | | F | | L2 (\BbbR 2d ) , \rho N + \epsilon N | | F | | L2 (\BbbR 2d ) , \surd where we used the inequality a2 + b2 \leq | a| + | b| . If we apply the Cauchy--Schwartz inequality twice as in (146), we conclude that \right) 1/2 \bigm| \sum \infty \bigm| \left( \infty \bigm| \bigm| \sum x f x \bigm| \sum i,k=n+1 i ik k \bigm| | fik | 2 . \bigm| \bigm| \leq \infty 2 \bigm| \bigm| | x | l l=n+1

(156)

i,k=n+1

Thus, in particular \left( \infty \sum (157) \rho N \leq

\right) 1/2 | fik | 2

\leq (2\pi )d/2 | | F - F (N ) | | L2 (\BbbR 2d ) < (2\pi )d/2 \epsilon | | F | | L2 (\BbbR 2d ) .

i,k=N +1

Likewise (158)

\epsilon N \leq (2\pi )d/2

| | F - F (N ) | | L2 (\BbbR 2d ) < (2\pi )d/2 \epsilon . | | F | | L2 (\BbbR 2d )

From (155), (157), (158), we recover (139). The result for the negative eigenvalues can be easily obtained by considering the matrix - \BbbF (N ) as before.

5182


We now turn to the eigenspaces. In the next theorem, we denote by dist(b, A) := inf \{ | | b - a| | l2 : a \in A\}

(159)

the distance of b \in l2 (\BbbN ) to the set A \subset l2 (\BbbN ). We also make the following observation. For fixed j \in \{ - N - , . . . , - 1, 1, . . . , N \} , let \Bigl\{ \Bigr\} (N ) (N ) (N ) (N ) (N ) (N ) (160) mj := min | \mu j | , | \mu k - \mu j | : \mu k \not = \mu j . (N )

(N )

Thus, mj measures the distance between \mu j it. Of course, as N \rightarrow \infty , this converges to

and the eigenvalue of \BbbF (N ) closest to

mj := min \{ | \mu j | , | \mu k - \mu j | : \mu k \not = \mu j \} > 0.

(161)

Clearly, for fixed j, we may choose \epsilon > 0 sufficiently small and N = N (\epsilon ) sufficiently large so that (59) holds and (N )

(162)

\epsilon
1. Thus cj may converge to some other eigenvector with the same eigenvalue \mu j other than cj or it may not even converge at all. However, what does happen is that its distance to the eigenspace \scrH j tends to zero. We also note that the eigenvalues converge in a uniform way. By this we mean that the estimates in (141) are independent of j. On the contrary our estimates in the next theorem for the eigenvector are not uniform. But since we are interested in estimating only one eigenvalue (the largest) and the corresponding eigenspace, that is fine. Theorem 13. For fixed j \in \BbbZ \setminus \{ 0\} , choose \epsilon and N = N (\epsilon ) \in \BbbN such that for the truncated matrix \BbbF (N ) we have j \in \{ - N - , . . . , - 1, 1, . . . , N+ \} ,

(163) and

(N )

(164)

0
0. (N )

Let \scrH j denote the eigenspace of \BbbF associated with the eigenvalue \mu j , and let cj (N )

a normalized eigenvector of \BbbF (166)

associated with the eigenvalue

\Bigl( \Bigr) 3(2\pi )d/2 \epsilon | | F | | 2 2d L (\BbbR ) (N ) dist cj , \scrH j < . (N ) Mj

(N ) \mu j .

be

We then have

5183


Proof. From (130) with j0 and \mu \ast j0 replaced by j and \mu j , we have (167)

(N )

| | \BbbF cj

(N )

- \mu j cj

| | l2 \leq 3(2\pi )d/2 \epsilon | | F | | L2 (\BbbR 2d ) ,

where we used (141). This equation is roughly equivalent to saying that eigenfunctions of F (N ) become 3(2\pi )d/2 \epsilon -pseudoeigenfunctions of the operator F with pseudoeigenvalue \mu j . (Pseudospectra is more often associated with nonnormal operators [49], but it can also be useful in the analysis of normal operators. There are several definitions and the above has been introduced by Landau inside the proof of a Szeg\" o theorem in [32]. Similar heuristics have been used in [2].) (N ) As in (132), we expand cj in the orthonormal basis \scrB (68) of l2 (\BbbN ) formed by the eigenvectors of \BbbF : \sum (N ) (168) cj = a\alpha e\alpha \alpha \in \scrA

with (N )

| | cj

(169)

| | l2 =

\sum

| a\alpha | 2 = 1.

\alpha \in \scrA

Let \scrA j \subset \scrA denote the set of indices such that \scrH j = Span \{ e\alpha : \alpha \in \scrA j \} .

(170)

These are the indices associated with the eigenvectors in the basis \scrB which have eigenvalue \mu j . Clearly, if \mu j has multiplicity one, then \scrA j = \{ j\} , and if it has multiplicity two, then \scrA j = \{ j - 1, j\} or \scrA j = \{ j, j + 1\} , etc. We also denote by \scrA cj = \scrA \setminus \scrA j its complement. (N )

We may thus write cj (171)

as (N )

cj

(N )

(N )

= cj,\| + cj,\bot ,

where (172)

(N )

cj,\| :=

\sum

a\alpha e\alpha \in \scrH j ,

\alpha \in \scrA j

and (173)

(N )

cj,\bot :=

\sum

a\alpha e\alpha \in \scrH j\bot .

\alpha \in \scrA cj

From (167), (171)--(173) it follows that

(174)

\Bigl( \Bigr) (N ) (N ) 3(2\pi )d/2 \epsilon | | F | | L2 (\BbbR 2d ) > | | (\BbbF - \mu j Id) cj,\| + cj,\bot | | l2 \sum (N ) = | | (\BbbF - \mu j Id)cj,\bot | | l2 = | | a\alpha (\beta \alpha - \mu j )e\alpha | | l2 , \alpha \in \scrA cj

where \beta \alpha = \lambda 1 , \lambda 2 , . . . for e\alpha = c1 , c2 , . . .; \beta \alpha = \lambda - 1 , \lambda - 2 , . . . for e\alpha = c - 1 , c - 2 , . . .; and \beta \alpha = 0 for e\alpha = d1 , d2 , . . . . From Pythagoras's theorem, \sqrt{} \sum \sqrt{} \sum | \beta \alpha - \mu j | 2 | a\alpha | 2 \geq mj | a\alpha | 2 . (175) 3(2\pi )d/2 \epsilon | | F | | L2 (\BbbR 2d ) > \alpha \in \scrA cj

\alpha \in \scrA cj

5184


Let \beta \alpha 0 be the eigenvalue of \BbbF such that | \beta \alpha 0 - \mu j | = mj .

(176)

In other words, \beta \alpha 0 is the eigenvalue closest to \mu j . Assuming that N is sufficiently large so that (163) and (164) hold and also \alpha 0 \in \{ - N - , . . . , - 1, 1, . . . , N+ \} , we have from (141) that (177)

(N )

| \beta \alpha 0 - \mu j | \geq | \beta \alpha (N0 ) - \mu j

| - 4(2\pi )d \epsilon | | F | | L2 (\BbbR 2d ) .

And thus (N )

mj \geq mj

(178)

(N )

- 4(2\pi )d \epsilon | | F | | L2 (\BbbR 2d ) = Mj

> 0.

Finally, from (175), (178) \Bigl( \Bigr) \sum (N ) (N ) a\alpha e\alpha | | l2 dist cj , \scrH j = | | cj,\bot | | l2 = | | \alpha \in \scrA cj

(179) =

d/2

\sqrt{} \sum

| a\alpha | 2
0 if and only if (191)

\omega I (k) = 0,

\nu (k) = \nu 1 k + \nu 0 ,

\nu 1 , \nu 0 \in \BbbR .

If (191) holds, then Wa \psi 0 (x, k, t) is representable for all t \in [ 0, +\infty ). Proof. Suppose that at some instant \tau > 0, the Wigner approximation Wa \psi (x, k, \tau ) is representable. Then, there exists \phi \tau \in L2 (\BbbR ) such that (192)

Wa \psi (x, k, \tau ) = W \phi \tau (x, k)

for all (x, k) \in \BbbR 2 , and where we use the fact that Wigner functions are uniformly \widetilde 0 = \scrF (\psi 0 ) and \phi \widetilde \tau = \scrF (\phi \tau ). continuous in \BbbR 2 [22]. Let \psi We may reexpress the Wigner functions (1) as \int 1 \widetilde 0 (k - \theta /2)\psi \widetilde 0 (k + \theta /2)e - ix\theta d\theta , (193) W \psi 0 (x, k) = \psi 2\pi \BbbR


5187

and (194)

W \phi \tau (x, k) =

1 2\pi

\int

\widetilde \tau (k - \theta /2)\phi \widetilde \tau (k + \theta /2)e - ix\theta d\theta . \phi

\BbbR

Plugging (190), (193), (194) into (192) and applying the Fourier inversion theorem, we obtain for almost all (\theta , k) \in \BbbR 2 (195)

\widetilde 0 (k - \theta /2)\psi \widetilde 0 (k + \theta /2)ei\theta \tau \nu (k) = \phi \widetilde \tau (k - \theta /2)\phi \widetilde \tau (k + \theta /2). e2\tau \omega I (k) \psi

Changing variables to \theta p = k - , 2

(196)

q=k+

\theta 2

we obtain (197)

e2\tau \omega I (

p+q 2

) \psi \widetilde 0 (q)ei\tau (q - p)\nu ( p+q \widetilde \tau (q) \widetilde 0 (p)\psi \widetilde \tau (p)\phi 2 ) = \phi

for a.e. (p, q) \in \BbbR 2 . Setting p = q, we have \widetilde 0 (p)| 2 = | \phi \widetilde \tau (p)| 2 e2\tau \omega I (p) | \psi

(198)

for almost all p \in \BbbR . \widetilde \tau (k0 ) \not = 0. We thus Let q = k0 \in \BbbR be such that (197) holds for a.e. p \in \BbbR and \phi have for a.e. p \in \BbbR (199)

e2\tau \omega I (

p+k0 2

0 )+i\tau (k0 - p)\nu ( p+k ) \psi \widetilde 0 (p)\psi \widetilde 0 (k0 ) = \phi \widetilde \tau (p)\phi \widetilde \tau (k0 ). 2

From (198) and (199), it follows that (200)

\widetilde \tau (p) = \psi \widetilde 0 (p)e2\tau \omega I (p)+i\tau (k0 - p)\nu ( \phi

p+k0 2

)+ic\tau

for almost all p \in \BbbR and c\tau \in \BbbR is some constant. Upon substitution of (200) in (195), we obtain for a.e. (\theta , k) \in \BbbR 2 \widetilde 0 (k - \theta /2)\psi \widetilde 0 (k + \theta /2) = \psi \widetilde 0 (k - \theta /2)\psi \widetilde 0 (k + \theta /2) e2\tau \omega I (k)+i\theta \tau \nu (k) \psi (201)

\biggl[ \biggl( \biggr) \biggr) \biggr] \biggl( \theta \theta \times exp 2\tau \omega I k - + 2\tau \omega I k + 2 2

\biggl[ \biggl( \biggr) \biggl( \biggr) \biggl( \biggr) \biggl( \biggr) \biggr] \theta k + k0 - \theta /2 \theta k + k0 + \theta /2 \times exp i\tau k0 - k + \nu - i\tau k0 - k - \nu . 2 2 2 2 This is possible if and only if (202)

\biggl( \biggr) \biggl( \biggr) \theta \theta \omega I (k) = \omega I k - + \omega I k + 2 2

and \biggl( \biggr) \biggl( \biggr) \theta k + k0 - \theta /2 \theta \nu (k) - k0 - k + \nu 2 2 (203) \biggl( \biggr) \biggl( \biggr) \theta k + k0 + \theta /2 2\pi n\tau ,k0 + k0 - k - \nu = 2 2 \tau for a.e. (\theta , k) \in \BbbR 2 and n\tau ,k0 \in \BbbZ . Since, by assumption, \omega I and \nu are continuous, (202), (203) must in fact hold for all (\theta , k) \in \BbbR 2 . Choosing k = k0 , \theta = 0, we conclude

5188


that n\tau ,k0 = 0. If we set \theta = 0 in (202), then \omega I (k) = 2\omega I (k), which is possible only if \omega I (k) vanishes identically. Let k = u + v - k0 ,

(204)

\theta = 2(v - u).

From (203), we obtain (v - u)\nu (u + v - k0 ) - (k0 - u)\nu (u) + (k0 - v)\nu (v) = 0.

(205) The function

f (x) = (k0 - x)\nu (x)

(206)

is obviously continuous and satisfies (207)

f (u) - f (v) f (u + v - k0 ) = . u - v u + v - 2k0

Taking the limit v \rightarrow u in the previous expression, we conclude that f is differentiable, except possibly at u = k0 . If we differentiate (208)

(u + v - 2k0 ) (f (u) - f (v)) = f (u + v - k0 )(u - v)

with respect to u and with respect to v, we obtain (209) \Biggl\{ f (u) - f (v) + (u + v - 2k0 )f \prime (u) = f \prime (u + v - k0 )(u - v) + f (u + v - k0 ), f (u) - f (v) - (u + v - 2k0 )f \prime (v) = f \prime (u + v - k0 )(u - v) - f (u + v - k0 ). If we subtract the two equations, we obtain (210)

(u + v - 2k0 ) (f \prime (u) + f \prime (v)) = 2f (u + v - k0 ),

and setting v = u, we have that (211)

2(u - k0 )f \prime (u) = f (2u - k0 )

for all u \in \BbbR \setminus \{ k0 \} .

This means that f and hence \nu is twice differentiable except possibly at k0 . If we go back to (205) and differentiate first with respect to u and then with respect to v, we conclude that (212)

(v - u)\nu \prime \prime (u + v - k0 ) = 0.

Thus for v = k0 and u \not = k0 , \nu \prime \prime (u) = 0, and (191) follows. Finally, if (191) holds, then the Wigner approximation amounts at all times to an affine linear symplectic transformation. It is well known that under these circumstances it must be a representable function [19, 22]. As a concrete example, we consider the standard centered Gaussian for d = 1 as the initial Wigner function (213)

W \psi 0 (z) =

1 - | z| 2 e , \pi

and choose the dispersion relation (214)

\omega (k) =

k3 . 3


5189

The Wigner approximation (8) is then given by \bigl( \bigr) 2 2 2 1 (215) Wa \psi (x, k, t) = W \psi 0 x - k 2 t, k = e - (x - k t) - k . \pi We thus want to obtain the Wigner function closest to F (x, k, t) = Wa \psi (x, k, t) (216)

\bigr) 1 - x2 - k2 \bigl( e 1 + 2xk 2 t + k 4 (2x2 - 1)t2 + \scrO (t3 ). \pi The expansion coefficients (41) are =

fn,m (t) = 2\pi \langle Wa \psi , W (en , em )\rangle L2 (\BbbR 2 ) \int \int 2 2 2 =2 e - (x - k t) - k W (en , em )(x, k)dxdk.

(217)

\BbbR

\BbbR

The integral in the previous expression is uniformly convergent for all t \in \BbbR \int \int (218) | fn,m (t)| \leq 2 | W (en , em )(x, k)| dxdk = 2| | W (en , em )| | L1 (\BbbR 2 ) < \infty . \BbbR

\BbbR

We conclude that fn,m \in C \infty (\BbbR ). We calculate some coefficients to order \scrO (t2 ): (219) \int \int \bigr) 2 2 \bigl( fn,m (t) = 2 e - x - k 1 + 2xk 2 t + k 4 (2x2 - 1)t2 W (en , em )(x, k)dxdk + \scrO (t3 ). \BbbR

\BbbR

Using the Hermite basis (72)--(75), we obtain 3 2 t + \scrO (t3 ), 32 \surd 2 f0,1 (t) = f1,0 (t) = t + \scrO (t3 ), 8 3 f1,1 (t) = - t2 + \scrO (t3 ). 32 f0,0 (t) = 1 -

(220)

Thus F (2) (x, k, t) = f0,0 (t)W (e0 , e0 )(x, k) + f0,1 (t)W (e0 , e1 )(x, k) (221)

+f1,0 (t)W (e1 , e0 )(x, k) + f1,1 (t)W (e1 , e1 )(x, k) \biggl( \biggr) 2 2 tx 3t2 2 1 = e - x - k 1 + - (x + k 2 ) + \scrO (t3 ) \pi 2 16

and \Biggl( (222)

(2)

\BbbF

=

The eigenvalues of \BbbF (2) are \Biggl( 1 (2) \lambda 1 = 1 - 2 (223) \Biggl( 1 (2) \lambda - 1 = 1 - 2

3 2 32 t \surd 2 8 t

1 -

\surd 2 8 t 3 2 - 32 t

\Biggr) + \scrO (t3 ).

\Biggr) t2 t2 1+ = 1 - + \scrO (t3 ), 8 16 \Biggr) \sqrt{} 3 2 t2 t2 t - 1 + = - + \scrO (t3 ). 16 8 4 3 2 t + 16

\sqrt{}

5190

BEN-BENJAMIN, COHEN, DIAS, LOUGHLIN, PRATA (2)

Notice that \lambda - 1 < 0 for t > 0. This is in agreement with the result of Theorem 16. (2) The eigenvectors associated with \lambda 1 read \Biggr) \Biggl( \surd \sqrt{} t2 2 t, 1 + - 1 , (224) c = Kt 4 8 where Kt is an arbitrary (nonzero) complex function of time only. The minimizing wave function is given by \Biggl( \surd \Biggl( \sqrt{} \Biggr) \Biggr) 2 t2 (2) 1 + - 1 e1 (x) . (225) \psi 1 (x) = Kt te0 (x) + 4 8 If we impose (52), we obtain \sqrt{} (2)

(2)

| | \psi 1 | | 2L2 (\BbbR ) = \lambda 1 \leftrightarrow Kt =

(226)

1 2

\sqrt{} 2 1 + t8 \sqrt{} . 2 1 + 81 t2 - 1 + t8

1 -

3 2 16 t

+

Consequently \biggl(

(2)

(227)

1 -

\psi 1 (x) =

t2 32

\biggr)

t e0 (x) + \surd e1 (x) + \scrO (t3 ). 4 2

Thus, to this order the Wigner function closest to the Wigner approximation F = Wa \psi (215) is (2)

W \psi (0,2) (x, k, t) = W \psi 1 (x, k, t) \biggl\{ \biggr\} 1 - | z| 2 tx t2 2 = e 1+ + (2| z| - 3) + \scrO (t3 ). \pi 2 32

(228)

Let us estimate the error of the truncation. From (59), (216), (221), we obtain \epsilon 2 | | F | | 2L2 (\BbbR 2 ) > | | F - F (2) | | 2L2 (\BbbR 2 ) (229)

\biggl( =

t 2\pi

\biggr) 2 \int \int \BbbR

e - 2x

2

- 2k2 2

x (4k 2 - 1)2 dxdk + \scrO (t3 ) =

\BbbR

t2 + \scrO (t3 ). 16\pi

We conclude that t (2) | \lambda 1 - \lambda 1 | < \surd + \scrO (t2 ). 2

(230) We also have (231)

(2)

M1

=

1 2

\biggl(

\surd 3 1 - 4 2t - t2 16

\biggr)

+ \scrO (t3 ). (2)

This approximation makes sense for sufficiently small t in order that M1 (165)) Moreover, from (166) we conclude that (232)

\Bigl( \Bigr) \surd 3 (2) dist c1 , \scrH 1 < \surd t(1 + 2 \pi t) + \scrO (t3 ). 2

> 0 (cf.


5191

Finally, estimate (183) yields (233)

| | W \psi 1 -

(2) W \psi 1 | | L2 (\BbbR 2d )

4t \leq \surd \pi

\biggl(

9t 1 + \surd 2

\biggr)

+ \scrO (t3 ).

(2)

Thus if we choose, for instance, t < 0.01, then M1 \simeq 0.478 is positive and we have \Bigl( \Bigr) (2) (2) (234) | \lambda 1 - \lambda 1 | \lesssim 0.008, dist c1 , \scrH 1 \lesssim 0.022, and (235)

(2)

| | W \psi 1 - W \psi 1 | | L2 (\BbbR 2d ) \lesssim 0.024.

7. Schatten-class operators. As another application of our results, we can estimate the eigenvalues and the norms of certain Schatten-class operators. Let us briefly recall the definition of Schatten-von Neumann operators [10]. Let p \in [ 1, \infty [ . \widehat acting on a separable Hilbert space \scrH , we denote by Given some operator A \widehat = (A \widehat \ast A) \widehat 1/2 | A|

(236)

\widehat \ast A, \widehat where A \widehat \ast is the adjoint of A. \widehat Its pth Schatten norm is given the positive root of A by \Bigl( \Bigr) 1/p \widehat S (\scrH ) = T r| A| \widehat p (237) | | A| | . p \widehat is given by The trace of an operator B (238)

\widehat = T r(B)

\sum

\widehat n , en \rangle \scrH \langle Be

n

for some orthonormal basis \{ en \} n . If it is finite, then the result is independent of the orthonormal basis chosen. \widehat belongs to the pth Schatten class Sp (\scrH ) if its pth Schatten norm An operator A (237) is finite. Schatten class operators are compact. Particular cases are the traceclass operators (p = 1) and the Hilbert--Schmidt operators (p = 2). \widehat \in Sp (\scrH ) is self-adjoint, then it admits a decomposition of the form (19)--(24). If A We can thus write its pth Schatten-norm as \left( \right) 1/p \Biggl( \Biggr) 1/p \sum p \sum \sum \widehat S (\scrH ) = (239) | | A| | \mu j + | \mu - j | p = | \mu \alpha | p . p j\in \BbbU +

j\in \BbbU -

\alpha \in \BbbU

We have the following continuous embedding: (240)

\widehat S (\scrH ) \leq | | A| | \widehat S (\scrH ) \Rightarrow Sq (\scrH ) \subset Sp (\scrH ), | | A| | p q

1 \leq q \leq p < \infty .

From our previous results we obtain the following two propositions. Let F\widehat \in Sp (L2 (\BbbR d )) for some p \in [1, 2] with F\widehat self-adjoint. From (240) F\widehat is a Hilbert--Schmidt operator with Weyl symbol F = \scrW (F\widehat ) and hence it admits the matrix representation \BbbF as before with respect to some orthonormal basis. With the assumption (59), we have a truncated matrix \BbbF (N ) with the associated eigenvalues (N ) \{ \mu j \} j . Proposition 17 follows from Theorem 12.

5192


\bigl( \bigr) Proposition 17. Let F\widehat \in Sp L2 (\BbbR d ) for some p \in [1, 2]. Under the assumption (59), we have \Bigl\{ \Bigr\} (N ) (N ) (241) \mu j \leq \mu j \leq max \mu j + (2\pi )d/2 \epsilon | | F | | L2 (\BbbR 2d ) , 2(2\pi )d/2 \epsilon | | F | | L2 (\BbbR 2d ) for j = 1, 2, . . . , N+ , and \Bigl\{ \Bigr\} (N ) (N ) (242) min - \mu - j - (2\pi )d/2 \epsilon | | F | | L2 (\BbbR 2d ) , - 2(2\pi )d/2 \epsilon | | F | | L2 (\BbbR 2d ) \leq \mu - j \leq \mu - j for j = 1, 2, . . . , N - . Consequently, (N )

| \mu j

(243)

- \mu j | < 2(2\pi )d/2 \epsilon | | F | | L2 (\BbbR 2d )

for all j = - N - , . . . , - 1, 1, . . . , N+ . Schatten norms are, in general, very difficult to compute. The exception is the Hilbert--Schmidt norm, because it can be evaluated through the L2 norm of the Weyl symbol (cf. (36), (43)): | | F\widehat | | S2 (L2 (\BbbR d )) = (2\pi )d/2 | | F | | L2 (\BbbR 2d ) .

(244)

But for all the other Schatten norms, there is no such simple formula and one is forced to determine the complete spectrum of F\widehat to compute (239). Our results permit to approximate some Schatten norms. We consider this problem again from another perspective elsewhere [8]. \bigl( \bigr) Proposition 18. Let F\widehat \in Sp L2 (\BbbR d ) for some p \in [1, 2] with F\widehat self-adjoint. Under the assumption (59), we have for any q \in [ 2, \infty [ \bigm| \right) 1/q \bigm| \bigm| \left( \bigm| N+ \Bigl( N - \bigm| \bigm| \bigm| \bigm| \Bigr) \sum (N ) q \sum \bigm| (N ) \bigm| q \bigm| \bigm| (245) \bigm| | | F\widehat | | Sq (L2 (\BbbR d )) - \mu j + \bigm| \leq (2\pi )d/2 \epsilon | | F | | L2 (\BbbR 2d ) . \bigm| \mu - j \bigm| \bigm| \bigm| j=1 j=1 \bigm| \bigm| Proof. Under the assumption (59), we have | | F\widehat | | Sq (L2 (\BbbR d )) = | | F\widehat - F\widehat (N ) + F\widehat (N ) | | Sq (L2 (\BbbR d )) \leq | | F\widehat (N ) | | Sq (L2 (\BbbR d )) + | | F\widehat - F\widehat (N ) | | Sq (L2 (\BbbR d )) (246) \leq | | F\widehat (N ) | | Sq (L2 (\BbbR d )) + | | F\widehat - F\widehat (N ) | | S2 (L2 (\BbbR d )) \leq | | F\widehat (N ) | | Sq (L2 (\BbbR d )) + (2\pi )d/2 \epsilon | | F | | L2 (\BbbR 2d ) , where we used (59), (240), (244). From (239), we have (247)

| | F\widehat (N ) | | Sq (L2 (\BbbR d )) =

\left( N+ \Bigl( \sum j=1

(N )

\mu j

\Bigr) q

N - \bigm| \bigm| \sum \bigm| (N ) \bigm| q + \bigm| \mu - j \bigm|

\right) 1/q .

j=1

Finally from the monotonicity of the eigenvalues (119), (120), we have that \bigm| \bigm| \bigm| \bigm| (248) \bigm| | | F\widehat | | S (L2 (\BbbR d )) - | | F\widehat (N ) | | S (L2 (\BbbR d )) \bigm| = | | F\widehat | | S (L2 (\BbbR d )) - | | F\widehat (N ) | | S (L2 (\BbbR d )) , q

and the result follows.

q

q

q


5193

8. Conclusions and outlook. Let us briefly recapitulate our results. Let F \in L2 (\BbbR 2d ) be some real nonrepresentable function. By this, we mean that there exists no \psi \in L2 (\BbbR d ) such that F = W \psi . We then look for the Wigner function W \psi 0 which is closest to F in the L2 norm. We solved this problem exactly in the case where F is a one-dimensional radial function. For the general case, we used a truncated version \BbbF (N ) = \{ Fn,m \} 1\leq n,m\leq N of the complete expansion coefficients \BbbF = \{ Fn,m \} n,m\in \BbbN of the function F in a given orthonormal basis of Wigner functions \{ W (en , em )\} n,m . By resorting to the Courant-Fischer min-max theorem, we obtained precise estimates for the errors of the approximate eigenvalues and eigenvectors. We proved that the function F = Wa \psi obtained by the Wigner approximation method developed in [15, 36, 37, 38] is never a Wigner function. We then used our methods to determine approximately the Wigner function closest to Wa \psi . Finally, we have shown that certain Schatten norms of self-adjoint Schatten class operators can be evaluated to any precision with our methods. In a future work, we wish to study other quasi-distributions. In the previous sections we have used only the Wigner distribution. However, there are an infinite number of other phase space distributions [12, 13, 14] and a number of questions arise when the considerations of the previous sections are applied to other distributions. We briefly discuss the general class of quasi-distributions. For simplicity we will consider the one-dimensional case. One can characterize the distributions by way of the kernel method. All bilinear distributions are given by \int \int \int 1 \psi ( x\prime - 21 \tau , t) \psi ( x\prime + 12 \tau , t), W \Phi \psi (x, k, t) = 2 4\pi (249) \prime \Phi (\theta , \tau )e - i\theta x - i\tau k+i\theta x d\theta d\tau d x\prime , where \Phi (\theta , \tau ) is called the kernel and characterizes the particular distribution. For the Wigner distribution, \Phi (\theta , \tau ) = 1. Here is how one can understand the previous expression. We assume that \psi \in \scrS (\BbbR ) and hence W \psi \in \scrS (\BbbR 2 ). By inverting the partial Fourier transform with respect to the second variable in (2), we obtain \int \prime \prime 1 1 \prime \psi ( x - 2 \tau , t) \psi ( x + 2 \tau , t) = W \psi (x\prime , k \prime , t)eik \tau dk \prime . (250) \BbbR

\widetilde be its Fourier transform, Let \Phi \in \scrS \prime (\BbbR 2 ) and let \Phi \int \int \widetilde k) = (\scrF \Phi ) (x, k) = 1 (251) \Phi (x, \Phi (\theta , \tau )e - i\theta x - i\tau k d\theta d\tau , 2\pi which should be understood in the usual distributional sense (252)

\langle \scrF \Phi , F \rangle = \langle \Phi , \scrF F \rangle

for all F \in \scrS (\BbbR 2 ) and where \langle \cdot , \cdot \rangle denotes the distributional bracket. If we use (251) and plug (250) into (249), we obtain (253)

W \Phi \psi (x, k, t) =

where \star denotes the convolution.

\Bigr) 1 \Bigl( \widetilde \Phi \star W \psi (x, k, t), 2\pi

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Some explicit relations between distributions are as follows. Two distributions W \Phi 1 \psi and W \Phi 2 \psi characterized by the kernels \Phi 1 and \Phi 2 are related by \int \int \int \int \Phi 2 (\theta , \tau ) \Phi 1 1 W \psi (x\prime , k \prime , t), W \Phi 2 \psi (x, k, t) = 2 4\pi \Phi 1 (\theta , \tau ) (254) \prime

\prime

ei\theta (x - x) + i\tau (k - k) d\theta d\tau dx\prime dk \prime . Equation (254) can be expressed in the form of a pseudodifferential operator, \biggl( \biggr) \biggl( \biggr) \partial \partial \partial \partial (255) W \Phi 2 \psi (x, k, t) = \Phi 2 i , i \Phi - 1 i , i W \Phi 1 \psi (x, k, t). 1 \partial x \partial k \partial x \partial k Just as for the Wigner distribution, one can calculate expectation values by re\widehat be some Weyl-operator with Weyl-symbol sorting to other quasi-distributions. Let A a \in \scrS (\BbbR 2 ). Given \psi \in \scrS (\BbbR ), we have \int \int \widehat \psi \rangle L2 (\BbbR ) = (256) \langle A\psi , a\Phi (x, k)W \Phi (x, k, t)dxdk. \BbbR

\BbbR

\widehat associated with the kernel \Phi . It is related with the Weyl Here a\Phi is the symbol of A symbol \Phi according to (257)

a=

1 \widetilde \Phi \star a\Phi . 2\pi

One can then ask whether one can apply the same ideas as described in the introduction to other quasi-distributions and obtain analogous approximations. This has been partially answered and we describe two such cases. The first is the Margenau-Hill distribution, the kernel for which is (258)

\Phi M H (\theta , \tau ) = e - i\theta \tau /2 ,

which results in the distribution (259)

1 W M H \psi (x, k, t) = \surd \psi (x, t)eikx (\scrF \psi ) (k, t), 2\pi

where \scrF \psi is the Fourier transform of \psi . Proceeding analogously as with the approximation for the Wigner distribution, we obtain the Margenau--Hill approximation, (260)

W M H \psi (x, k, t) \approx e2t\omega I (k) W M H \psi 0 (x - \nu (k)t, k).

For the spectrogram, the kernel is \int (261) \Phi SP (\theta , \tau ) = w(x + \tau 2 ) e - i\theta x w(x - \tau 2 ) dx. \BbbR

where w(x) \in \scrS (\BbbR ) is the window function. The distribution is (262)

\bigm| \bigm| 2 \int \bigm| 1 \bigm| \prime W SP \psi (x, k) = \bigm| \bigm| \surd e - ikx \psi (x\prime , t) w(x\prime - x)dx\prime \bigm| \bigm| . 2\pi \BbbR


5195

For a particular window w, this can be seen as the modulus squared of the Fourier-Bros--Iagolnitzer transform [19]. Analogous to the Wigner distribution, the approximation works out as (263)

W SP \psi (x, k, t) \approx e2t\omega I (k) W SP \psi 0 (x - \nu (k)t, k).

Comparing the Wigner approximation (8) with the MH approximation (260) and the spectrogram (263) we see that they are of the same functional form. This gives rise to various questions that we discuss and that are currently being studied. \bullet While the approximations are of the same functional form, the accuracy of the approximations is not necessarily equivalent. One can ask, which is closest to the exact corresponding distribution? Second, which produces a more accurate wave function by whatever method one can use to invert the distribution and obtain an approximate wave function? \bullet It is probably the case that none of the approximations are representable. Can one define approximate representability and see which distribution is most representable? \bullet What are the next (higher-order) approximations and are they the same for the different distributions? \bullet Related to the previous issues, if one can find a series approximation, will the successive approximations be more and more representable? \bullet Another method of approximation is the differential equation approach. Does that approach give the same approximations. \bullet The L2 norm seems natural in this setting because Wigner functions belong to L2 (\BbbR 2d ) and Moyal's identity leads to natural orthogonality relations. Moreover it tends to be pervasive in physical applications. However, in statistical estimation the fundamental measure is the L1 norm, since it allows us to control the MSE error in the estimations. (This is detailed in [3] for stationary signals.) In a future work we will try to investigate whether it is also possible to find the Wigner function closest to a given function in phase space with respect to the L1 norm. Acknowledgment. The authors would like to thank Franz Luef for drawing their attention to references [5, 6]. REFERENCES \" [1] L. D. Abreu and K. Grochenig, Banach Gabor frames with Hermite functions: Polyanalytic spaces from the Heisenberg group, Appl. Anal., 91 (2012), pp. 1981--1997. [2] L. D. Abreu and J. M. Pereira, Measures of localization and quantitative Nyquist densities, Appl. Comput. Harmon. Anal., 38 (2015), pp. 524--534. [3] L. D. Abreu and J. L. Romero, MSE bounds for multitaper spectral estimation and off-grid compressive sensing, IEEE Trans. Inform. Theory, 63 (2017), pp. 7770--7776. [4] L. D. Abreu and H. G. Feichtinger, Function spaces of polyanalytic functions. Harmonic and Complex Analysis and its Applications, Trends Math., Springer, New York, 2014, pp. 1--38. [5] P. Balazs, Basic definition and properties of Bessel multipliers, J. Math. Anal. Appl., 325 (2007), pp. 571--585. [6] P. Balazs, D. Bayer, and A. Rahimi, Multipliers for continuous frames in Hilbert spaces, J. Phys. A, 45 (2012), 244023. [7] M. Bayram and R. G. Baraniuk, Multiple window time-varying spectrum estimation, in Nonlinear and Nonstationary Signal Processing, Cambridge University Press, Cambridge, 2000, pp. 292--316. [8] J. S. Ben-Benjamin, L. Cohen, N. C. Dias, P. Loughlin, and J. N. Prata, On the topology of Wigner functions, submitted.

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WHAT IS THE WIGNER FUNCTION CLOSEST TO A GIVEN SQUARE

WHAT IS THE WIGNER FUNCTION CLOSEST TO A GIVEN SQUARE

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