Hamiltonian Deformations of Gabor Frames: First Steps Maurice A. de Gosson University of Vienna, Faculty of Mathematics, NuHAG
Abstract Gabor frames can be advantageously be rede…ned using the Heisenberg operators familiar from quantum mechanics. Not only does this rede…nition allow us to recover in a very simple way known results of symplectic covariance, but it immediately leads to the consideration of a general deformation scheme by Hamiltonian isotopies (i.e. arbitrary paths of non-linear symplectic mappings). We will study in some detail a weak notion of Hamiltonian deformation of Gabor frame, using ideas from semiclassical physics involving coherent states and Gaussian approximations. This method can be practically and numerically implemented using symplectic algorithms. Key words: Gabor frame, symplectic group, metaplectic group, Hamiltonian isotopy, coherent states
1
Introduction
The theory of Gabor frames (or Weyl–Heisenberg frames as they are also called) is a rich and expanding topic of applied harmonic analysis. It has numerous applications in time-frequency analysis, signal theory, and mathematical physics. The aim of this article is to initiate a systematic study of the symplectic transformation properties of Gabor frames, both in the linear and nonlinear cases. Strangely enough, the use of symplectic techniques in the theory of Gabor frames is often ignored; one example (among many others) being Casazza’s seminal paper [9] on modern tools for Weyl–Heisenberg frame theory, where the word “symplectic”does not appear a single time in the 127 pages of this paper! This is of course very unfortunate: it is a thumb-rule in Corresponding author Email address:
[email protected],
[email protected] (Maurice A. de Gosson).
Preprint submitted to Elsevier Science
16 June 2013
mathematics and physics that when symmetries are present in a theory their use always leads to new insights in the mechanisms underlying that theory. To name just one single example, the study of fractional Fourier transforms belongs to the area of symplectic and metaplectic analysis and geometry (see section 3.3); remarking this would avoid to many authors unnecessary e¤orts and complicated calculations. On the positive side, there are however (a few) exceptions to this refusal to include symplectic techniques in applied harmonic analysis: for instance, in Gröchenig’s treatise [28] the metaplectic representation is used to study various symmetries in time frequency analysis, and the recent paper by Pfander et al. [52] elaborates on earlier work [30] by Han and Wang, where symplectic transformations are exploited to study various properties of Gabor frames. In this paper we consider deformations of Gabor systems using Hamiltonian isotopies. A Hamiltonian isotopy is a curve (ft )0 t 1 of di¤eomorphisms of phase space R2n starting at the identity, and such that there exists a (usually time-dependent) Hamiltonian function H such that the (generalized) phase ‡ow (ftH )t determined by the Hamilton equations x_ = @p H(x; p; t) , p_ =
@p H(x; p; t)
(1)
consists of the mappings ft for 0 t 1. In particular Hamiltonian isotopies consist of symplectomorphisms (or canonical transformations, as they are called in Physics). Given a Gabor system G( ; ) with window (or atom) and lattice we want to …nd a working de…nition of the deformation of G( ; ) by a Hamiltonian isotopy (ft )0 t 1 . While it is clear that the deformed lattice should be the image t = ft ( ) of the original lattice , it is less clear what the deformation t = ft ( ) of the window should be. A clue is however given by the linear case: assume that the mappings ft are linear, i.e. symplectic matrices St ; assume in addition that there exists an in…nitesimal symplectic transformation X such that St = etX for 0 t 1. Then (St )t is the ‡ow determined by the Hamiltonian function H(x; p) =
1 (x; p)T JX(x; p) 2
(2)
where J is the standard symplectic matrix. It is well-known that in this case there exists a one-parameter group of unitary operators (Sbt )t satisfying the operator Schrödinger equation i~
d b St = H(x; i~@x )Sbt dt
where the formally self-adjoint operator H(x; i~@x ) is obtained by replacing formally p with i~@x in (2); the matrices St and the operators Sbt correspond to each other via the metaplectic representation of the symplectic group. This suggests that we de…ne the deformation of the initial window by t = Sbt . It turns out that this de…nition is satisfactory, because it allows to recover, 2
setting t = 1, known results on the image of Gabor frames by linear symplectic transformations. This example is thus a good guideline; however one encounters di¢ culties as soon as one want to extend it to more general situations. While it is “reasonably”easy to see what one should do when the Hamiltonian isotopy consists of an arbitrary path of symplectic matrices (this will be done in section 4), it is not clear at all what a “good” de…nition should be in the general nonlinear case: this is discussed in section 6, where we suggest that a natural choice would be to extend the linear case by requiring that t should be the solution of the Schrödinger equation i~
d dt
t
c =H
t
associated with the Hamiltonian function H determined by the equality (ft )0 t 1 = c would then be associated with the (ftH )0 t 1 ; the Hamiltonian operator H function H by using, for instance, the Weyl correspondence. Since the method seems to be di¢ cult to study theoretically and to implement numerically, we propose what we call a notion of weak deformation, where the exact de…nition of the transformation 7 ! t of the window is replaced with a correspondence used in semiclassical mechanics, and which consists in propagating the “center”of a su¢ ciently sharply peaked initial window (for instance a coherent state, or a more general Gaussian) along the Hamiltonian trajectory. This de…nition coincides with the de…nition already given in the linear case, and has the advantage of being easily computable using the method of symplectic integrators (which we review in section 5.3) since all what is needed is the knowledge of the phase ‡ow determined by a certain Hamiltonian function. Finally we discuss possible extensions of our method.
Notation and terminology The generic point of the phase space R2n Rn Rn is denoted by z = (x; p) where we have set x = (x1 ; :::; xn ), p = (p1 ; :::; pn ). The scalar product of two vectors, say p and x, is denoted by p x or simply px. When matrix calculations are performed, z; x; p are viewed as column vectors. We will equip R2n with the standard symplectic structure (z; z 0 ) = p x0
p0 x; 0
1
B 0 IC
in matrix notation (z; z 0 ) = (z 0 )T Jz where J = @
A (0 and I are the n n I 0 zero and identity matrices). The symplectic group of R2n is denoted by Sp(n); it consists of all linear automorphisms of R2n such that (Sz; Sz 0 ) = (z; z 0 ) for all z; z 0 2 R2n . Working in the canonical basis Sp(n) is identi…ed with the group of all real 2n 2n matrices S such that S T JS = J (or, equivalently,
3
SJS T = J). We will write dz = dxdp where dx = dx1 scalar product on L2 (Rn ) is de…ned by ( j )=
Z
Rn
dxn and dp = dp1
dpn . The
(x) (x)dx
and the associated norm is denoted by jj jj. The Schwartz space of rapidly decreasing functions is denoted by S(Rn ) and its dual (the space of tempered distributions) by S 0 (Rn ).
2
Gabor Frames
Gabor frames are a generalization of the usual notion of basis; see for instance Gröchenig [28], Feichtinger and Gröchenig [17], Balan et al. [3], Heil [34], Casazza [9] for a detailed treatment of this topic. In what follows we give a slightly modi…ed version of the usual de…nition, better adapted to the study of symplectic symmetries.
2.1 De…nition
Let be a non-zero square integrable function (hereafter called window) on Rn , and a lattice in R2n , i.e. a discrete subset of R2n . The associated ~-Gabor system is the set of square-integrable functions G( ; ) = fTb ~ (z) : z 2 g
where Tb ~ (z) = e i (^z;z)=~ is the Heisenberg operator. The action of this operator is explicitly given by the formula Tb ~ (z0 ) (x) = ei(p0 x
p0 x0 =2)=~
(x
x0 )
(3)
(see e.g. [23,25,45]; it will be justi…ed in section 4.2). We will call the Gabor system G(g; ) a ~-frame for L2 (Rn ), if there exist constants a; b > 0 (the frame bounds) such that ajj jj2
X
z0 2
j( jTb ~ (z0 ) )j2
for every square integrable function is said to be tight.
bjj jj2
(4)
on Rn . When a = b the ~-frame G(g; )
4
Remark 1 The product ( jTb ~ (z0 ) ) is, up to the factor (2 ~) n , Woodward’s cross-ambiguity function [58]; it is up to a (symplectic Fourier transform) the cross-Wigner distribution W ( ; ) as was already observed by Klauder [43]; see [20,23,25].
2.2 Rescaling properties
For the choice ~ = 1=2 the notion of ~-Gabor frame coincides with the usual notion of Gabor frame as found in the literature. In fact, in this case, writing Tb (z) = Tb 1=2 (z) and p = !, we have j( jTb (z) )j = j( j (z) )j
where (z) is the modulation operator de…ned by (z0 ) (x) = e2
i! 0 x
(x
x0 )
for z0 = (x0 ; ! 0 ). The two following elementary results can be used to go from one de…nition to the other: 1
0
BI 0 C Proposition 2 Let D~ = @ A. The system G( ; ) is a Gabor frame 0 2 ~I if and only if G( ; D~ ) is a ~-Gabor frame.
PROOF. We have Tb ~ (x0 ; 2 ~p0 ) = Tb (x0 ; p0 ) where Tb (x0 ; p0 ) = Tb 1=2 (x0 ; p0 ). By de…nition G( ; ) is a Gabor frame if and only if X
ajj jj2 for every
2 L2 (Rn ) that is ajj jj2
z0 2
X
(x0 ;p0 )2
j( jTb (z0 ) )j2
bjj jj2
j( jTb (x0 ; p0 ) )j2
bjj jj2 ;
this inequality is equivalent to ajj jj2
X
j( jTb ~ (x0 ; 2 ~p0 ) )j2
(x0 ;p0 )2
bjj jj2
that is to ajj jj2
X
(x0 ;(2 ~)
1 p )2 0
j( jTb ~ (x0 ; p0 ) )j2 5
bjj jj2
hence the result since (x0 ; (2 ~) 1 p0 ) 2
means that (x0 ; p0 ) 2 D~ .
We can also rescale simultaneously the lattice and the window (“change of Planck’s constant”): Proposition 3 Let G( ; ) be a Gabor system, and set p ~ (x) = (2 ~) n=2 (x= 2 ~): p Then G( ; ) is a frame if and only if G( ~ ; 2 ~ ) is a ~-frame. PROOF. We have tion
~
c p =M 1= 2
M1=p2
~
~I;0
0
c p where M 1= 2 1=2
(2 ~) =B @ 0
I
~I;0
0
(2 ~)
1=2
1
(5)
2 Mp(n) has projec-
C A
I p on Sp(n) (see Appendix A). The Gabor system G( ~ ; 2 ~ ) is a ~-frame if and only X 2 c p bjj jj2 j( jTb (z0 )M ajj jj2 1= 2 ~I;0 )j p z0 2 2 ~
for every 2 L2 (Rn ), that is, taking the symplectic covariance formula (6) into account, if and only if ajj jj2
X p
z0 2 2 ~
cp j(M 2
~I;0
jTb ((2 ~)
1=2
x0 ; (2 ~)1=2 p0 ) )j2
bjj jj2 :
But this is inequality is equivalent to ajj jj2
X
z0 2D~
j( jTb (z0 ) )j2
bjj jj2
and one concludes using Proposition 2. Remark 4 In Appendix A we state a rescaling property for the covering projection ~ : Sb 7 ! S of metaplectic group Mp(n) onto Sp(n) (formula (61)). 3
Symplectic Covariance
The following formula will play a fundamental role in our study of symplectic covariance properties of frames. It relates Heisenberg–Weyl operators, linear symplectic transformations, and metaplectic operators (we refer to Appendix A for a concise review of the metaplectic group Mp(n) and its properties). 6
Let Sb 2 Mp(n) have projection
~
b = S 2 Sp(n). Then (S)
Tb ~ (z)Sb = SbTb ~ (S
1
(6)
z):
For a proof see e.g. [23,25,45]; one easy way is to prove this formula separately b M c b for each generator J, L;m , VP of the metaplectic group. 3.1 A …rst covariance result
Gabor frames behave well under symplectic transformations of the lattice (or, equivalently, under metaplectic transformations of the window). Let Sb 2 Mp(n) have projection S 2 Sp(n). The following result is well-known, and appears in many places in the literature (see e.g. Gröchenig [28], Pfander et al. [52]). Our proof is somewhat simpler since it exploits the symplectic covariance property of the Heisenberg–Weyl operators. Proposition 5 Let 2 L2 (Rn ) (or 2 S(Rn )). A Gabor system G( ; ) is a ~-frame if and only if G(Sb ; S ) is a ~-frame; when this is the case both frames have the same bounds. In particular, G( ; ) is a tight ~-frame if and only if G( ; ) is. PROOF. Using the symplectic covariance formula (6) we have X
z2S
j( jTb ~ (z)Sb )j2 =
=
X
z2S
X
z2
j( jSbTb ~ (S
j(Sb
1
1
z) )j2
jTb ~ (z) )j2
and hence, since G( ; ) is a ~-frame, ajjSb
1
jj2
X
z2S
j( jTb ~ (z)Sb )j2
bjjSb
1
jj2 :
The result follows since jjSb 1 jj = jj jj because metaplectic operators are unitary; the case 2 S(Rn ) is similar since metaplectic operators are linear automorphisms of S(Rn ). Remark 6 The result above still holds when one assumes that the window belongs to the Feichtinger algebra S0 (Rn ) (see Appendix B and the discussion at the end of the paper).
7
3.2 Gaussian frames
The problem of constructing Gabor frames G( ; ) in L2 (R) with an arbitrary window and lattice is di¢ cult and has been tackled by many authors (see for instance the comments in [29], also [52]). Very little is known about the existence of frames in the general case. We however have the following characterization of Gaussian frames which extends a classical result of Lyubarskii [48] and Seip and Wallstén [54]: 2
Proposition 7 Let ~0 (x) = ( ~) n=4 e jxj =2~ (the standard centered Gaussian) and = Zn Zn with = ( 1 ; :::; n ) and = ( 1 ; :::; n ). Then ~ ) is a frame if and only if j j < 2 ~ for 1 j n. G( 0 ;
PROOF. Bourouihiya [5] proves this for ~ = 1=2 ; the result for arbitrary ~ > 0 follows using Proposition 3.
It turns out that using the result above one can construct in…nitely many symplectic Gaussian frames using the theory of metaplectic operators: Proposition 8 Let ~0 be the standard Gaussian. The Gabor system G( ~0 ; ) is a frame if and only if G(Sb ~0 ; S ) is a frame (with same bounds) for every 0 1
A BC b Sb 2 Mp(n). Writing S in block-matrix form B @ A the window S C D Gaussian
where
Sb
~ 0 (x)
=
1 ~
n=4
(det X)1=4 e
1 (X+iY 2~
X = (CAT + DB T )(AAT + BB T ) Y = (AAT + BB T ) 1
~ 0
is the
)x x
(7)
1
(8) (9)
are symmetric matrices, and X > 0.
PROOF. That G( ~0 ; ) is a frame if and only if G(Sb ~0 ; S ) is a frame ~ b follows from Proposition 5. To calculate S 0 it su¢ ces to apply formulas (66) and (67) in Appendix A. 8
3.3 An example: fractional Fourier transforms
Let us choose ~ = 1=2 and consider the rotations 0
1
B cos t sin t C A
St = @
(10)
sin t cos t
(we assume n = 1). The matrices (St ) form a one-parameter subgroup of the symplectic group Sp(1). To (St ) corresponds a unique one-parameter subgroup (Sbt ) of the metaplectic group Mp(1) such that St = 1=2 (Sbt ). It follows from formula (A1) in Appendix A that Sbt is given for t 6= k (k integer) by the explicit formula Sbt
~ 0 (x)
= im(t)
1 2 ij sin tj
1=2
Z
1 1
e2
iW (x;x0 ;t) ~ 0 0 0 (x )dx
where m(t) is an integer (the “Maslov index”) and W (x; x0 ; t) =
1 (x2 + x02 ) cos t 2 sin t
2xx0 :
The metaplectic operators Sbt are the “fractional Fourier transforms”familiar from time-frequency analysis (see e.g. Almeida [1], Namias [50]). The argumentation above clearly shows that the study of these fractional Fourier transforms belong to the area of symplectic and metaplectic analysis and geometry. Applying Proposition 5 we recover without any calculation the results of Kaiser [40] (Theorem 1 and Corollary 2) about rotations of Gabor frames; in our notation: Corollary 9 Let G( ; ) be a frame; then G(Sbt ; St ) is a frame for every t 2 R. Notice that fractional Fourier transforms (and their higher-dimensional generalizations) are closely to the theory of the quantum mechanical harmonic oscillator: the metaplectic operators Sbt are solutions of the operator Schrödinger equation d 1 i~ Sbt = dt 2
!
d2 ~ + x2 Sbt . dx2 2
9
4
Symplectic Deformations of Gabor Frames
The symplectic covariance property of Gabor frames studied above can be interpreted as a …rst result on Hamiltonian deformations of frames because, as we will see, every symplectic matrix is the value of the ‡ow (at some time t) of a Hamiltonian function which is a homogeneous quadratic polynomial (with time-depending coe¢ cients) in the variables xj ; pk . We will in fact extend this result to deformations by a¢ ne ‡ows corresponding to the case where the Hamiltonian is an arbitrary quadratic function of these coordinates.
4.1 The linear case
The …rst example in subsection 3.3 (the fractional Fourier transform) can be interpreted as a statement about continuous deformations of Gabor frames. For instance, assume that St = etX , X in the Lie algebra sp(n) of the symplectic group Sp(n) (it is the algebra of all 2n 2n matrices X such that XJ + JX T = 0; when n = 1 it reduces to the condition Tr X = 0; see [20,23]). It is then easy to check that the family (St ) can be identi…ed with the ‡ow determined by the Hamilton equations z_ = J@z H for the function 1 T z (JX)z 2
H(z) =
(11)
which is a quadratic polynomial in in the variables xj ; pk . That ‡ow satis…es the matrix di¤erential equation d St = XSt : dt
(12)
We now make the following fundamental observation: to the path of symplectic matrices t 7 ! St , 0 t 1 corresponds a unique path t 7 ! Sbt , 0 t 1, of metaplectic operators such that Sb0 = I and Sb1 = Sb (see Appendix A). This path satis…es the operator Schrödinger equation i~
d b cS b St = H t dt
(13)
c is the Weyl quantization of the function H. Collecting these facts, one where H sees that G(Sb ~0 ; S ) is obtained from the initial Gabor frame G( ~0 ; ) by a smooth deformation
t 7 ! G(Sbt
~ 0 ; St
) , 0
t
1:
(14)
More generally, let S be an arbitrary element of the symplectic group Sp(n). The latter is connected so there exists a C 1 path (in fact in…nitely many) 10
t 7 ! St , 0 t 1, joining the identity to S in Sp(n). An essential result, generalizing the observations above, is then: Proposition 10 Let t 7 ! St , 0 t 1, be a path in Sp(n) such that S0 = I and S1 = S. There exists a Hamiltonian function H = H(z; t) such that St is the phase ‡ow determined by the Hamilton equations z_ = J@z H. Writing 1
0
BAt Bt C St = @ A Ct Dt
(15)
the Hamiltonian function is the quadratic form H(z; t) = 21 (D_ t CtT
C_ t DtT )x2 + (D_ t ATt
C_ t BtT )p x + 12 (B_ t ATt
A_ t BtT )p2 (16)
where A_ t = dAt =dt, etc.
PROOF. The matrices St being symplectic we have StT JSt = J. Di¤erentiating both side of this equality with respect to t and setting we get S_ tT JSt + StT J S_ t = 0 or, equivalently, J S_ t St
1
=
(StT ) 1 S_ tT J = (J S_ t St 1 )T :
This equality can be rewritten J S_ t St 1 = (J S_ t St 1 )T hence the matrix J S_ t St is symmetric. Set J S_ t St 1 = Mt = MtT ; then S_ t = Xt St , Xt =
JMt
1
(17)
(it reduces to formula (12) when Mt is time-independent). De…ne now H(z; t) =
1 T z (JXt )z; 2
using (17) one veri…es that the phase ‡ow determined by H consists precisely of the symplectic matrices St and that H is given by formula (16). Remark 11 Formula (16) also follows from the more general formula (31) about Hamiltonian isotopies in Proposition (14) below. Exactly as above, to the path of symplectic matrices t 7 ! St corresponds a path t 7 ! Sbt of metaplectic operators such that Sb0 = I and Sb1 = Sb satisfying the Schrödinger equation (13). Thus, it makes sense to consider smooth deformations (14) for arbitrary symplectic paths. This situation will be generalized to the nonlinear case in a moment. 11
4.2 Translations of Gabor systems
A particular simple example of transformation is that of the translations T (z0 ) : z 7 ! z + z0 in R2n . On the operator level they correspond to the Heisenberg–Weyl operators Tb ~ (z0 ). This correspondence is very easy to understand in terms of “quantization”: for …xed z0 consider the Hamiltonian function H(z) = (z; z0 ) = p x0
p0 x:
The associated Hamilton equations are just x_ = x0 , p_ = p0 whose solutions are x(t) = x(0) + tx0 and p(t) = p(0) + tp0 , that is z(t) = T (tz0 )z(0). Let now c = (zb; z ) = ( i~@ ) x H 0 x 0
p0 x:
be the “quantization”of H, and consider the Schrödinger equation i}@t = (zb; z0 ) :
Its solution is given by (x; t) = e
t (b z ;z0 )=~
(x; 0) = Tb ~ (tz0 ) (x; 0)
(the second equality can be veri…ed by a direct calculation, or using the Campbell–Hausdor¤ formula [20,23,25,45]). Translations act in a particularly simple way on Gabor frames; writing T (z1 ) = + z1 we have: Proposition 12 Let z0 ; z1 2 R2n . A Gabor system G( ; ) is a ~-frame if and only if G(Tb ~ (z0 ) ; T (z1 ) ) is a ~-frame; the frame bounds are in this case the same for all values of z0 ; z1 .
PROOF. We will need the following well-known [20,23,25,45] properties of the Heisenberg–Weyl operators:
Tb ~ (z)Tb ~ (z 0 ) = ei b~
0
T (z + z ) = e
(z;z 0 )=~
Tb ~ (z 0 )Tb ~ (z)
i (z;z 0 )=2~
b~
b~
(18) 0
T (z)T (z ):
(19)
Assume …rst z1 = 0 and let us prove that G(Tb ~ (z0 ) ; ) is a ~-frame if and only G( ; ) is. We have, using formula (18) and the unitarity of Tb ~ (z0 ), 12
X
z2
X
j( jTb ~ (z)Tb ~ (z0 ) )j2 =
z2
X
=
z2
X
=
z2
it follows that the inequality X
ajj jj2
(z;z0 )=~
Tb ~ (z0 )Tb ~ (z) )j
j( jTb ~ (z0 )Tb ~ (z) )j
j(Tb ~ ( z0 ) jTb ~ (z) )j;
j( jTb ~ (z)Tb ~ (z0 ) )j2
z2
is equivalent to
X
ajj jj2
j( jei
z2
j( jTb ~ (z) )j2
bjj jj2 bjj jj2
hence our claim in the case z1 = 0. We next assume that z0 = 0; we have, using this time formula (19), X
z2T (z1 )
j( jTb ~ (z) )j2 =
=
X
z2
X
z2
=
X
z2
j( jTb ~ (z + z1 ) )j2
j( jTb ~ (z1 )Tb ~ (z) )j2
j(Tb ~ ( z1 ) jTb ~ (z) )j2
and one concludes as in the case z1 = 0. The case of arbitrary z0 ; z1 immediately follows. Identifying the group of translations with R2n the inhomogeneous (or a¢ ne) symplectic group ISp(n) [8,20] is the semi-direct product Sp(n) n R2n ; the group law is given by (S; z)(S 0 ; z 0 ) = (SS 0 ; z + Sz 0 ): Using the conjugation relation S
1
T (z0 )S = T (S
1
z0 )
(20)
one checks that ISp(n) is isomorphic to the group of all a¢ ne transformations of R2n of the type ST (z0 ) (or T (z0 )S) where S 2 Sp(n). The group ISp(n) appears in a natural way when one considers Hamiltonians of the type (21) H(z; t) = 21 M (t)z z + m(t) z where M (t) is symmetric and m(t) is a vector. In fact, the phase ‡ow determined by the Hamilton equations for (21) consists of elements of ISp(n). 13
Assume for instance that the coe¢ cients M and m are time-independent; the solution of Hamilton’s equations z_ = JM z + Jm is zt = etJM z0 + (JM ) 1 (etJM
I)JM
(22)
provided that det M 6= 0. When det M = 0 the solution (22) is still formally valid and depends on the nilpotency degree of X = JM . Since X = JM 2 sp(n) we have St = etX 2 Sp(n); setting t = X 1 (etX I)u the ‡ow (ftH ) is thus given by ftH = T ( t )St 2 ISp(n): The metaplectic group Mp(n) is a unitary representation of the double cover Sp2 (n) of Sp(n) (see Appendix A). There is an analogue when Sp(n) is replaced with ISp(n): it is the Weyl-metaplectic group WMp(n), which consists of all b notice that formula (6), which we can rewrite products Tb (z0 )S; Sb 1 Tb ~ (z)Sb = Tb ~ (S
1
z)
(23)
is the operator version of formula (20) above.
5
The Group Ham(n)
In this section we review the basics of the modern theory of Hamiltonian mechanics from the symplectic point of view; for details we refer to [13,39,53]; we have also given elementary accounts in [23,25].
5.1 Hamiltonian ‡ows: properties
A Hamiltonian system (1) can be written in compact form as z_ = J@z H(z; t)
(24)
where J is the standard symplectic matrix. The Hamiltonian function H is assumed to be twice continuously di¤erentiable in z, and continuous in t. We denote by ftH the mapping R2n ! R2n which to an initial condition z0 associates the value z = ftH (z0 ) of the solution to (24) at time t. The family (ftH )t of all these mappings is called the phase ‡ow determined by the Hamiltonian system (24). It is often useful to replace the notion of ‡ow as de…ned as above by that of H H H 0 time-dependent ‡ow (ft;t 0 ): ft;t0 is the function such that ft;t0 (z ) is the solution 14
of Hamilton’s equations with z(t0 ) = z 0 . Obviously 1
H H H ft;t 0 = ft;0 ft0 ;0
= ftH ftH0
1
(25)
H and the ft;t 0 satisfy the groupoid property H H H H ft;t 0 ft0 ;t0 = ft;t00 , ft;t = Id
(26)
H for all t, t0 and t00 . Notice that it follows in particular that (ft;t 0)
1
= ftH0 ;t .
An essential property which links Hamiltonian dynamics to symplectic geometry is that each mapping ftH is a di¤eomorphism such that h
iT
DftH (z)
h
iT
JDftH (z) = DftH (z)J DftH (z)
= J:
(27)
Here DftH (z) is the Jacobian matrix of the di¤eomorphism ftH calculated at the point z = (x; p): @zt @(xt ; pt ) DftH (z) = = @z @(x; p) if zt = ftH (z). The equality (27) means that the matrix DftH (z) is symplectic: DftH (z) 2 Sp(n) for every z and t. Any di¤eomorphism f of phase space R2n satisfying the condition Df (z)T JDf (z) = Df (z)JDf (z)T = J
(28)
is called a symplectomorphism. Formula (27) thus says that Hamiltonian ‡ows consist of symplectomorphisms, which is a well-known property from classical mechanics [2,39,53]. A remarkable fact is that composition and inversion of Hamiltonian ‡ows also yield Hamiltonian ‡ows: Proposition 13 Let (ftH ) and (ftK ) be the phase ‡ows determined by two Hamiltonian functions H = H(z; t) and K = K(z; t). We have ftH ftK = ftH#K (ftH )
1
= ftH
with
with
H#K(z; t) = H(z; t) + K((ftH ) 1 (z); t):
H(z; t) =
H(ftH (z); t):
(29) (30)
PROOF. It is based on the transformation properties of the Hamiltonian …elds XH = J@z H under di¤eomorphisms; see [23,39,53] for detailed proofs.
We notice that even if H and K are time-independent Hamiltonians, then H#K and H are time-dependent. 15
5.2 Hamiltonian isotopies
Formula (28) above shows, using the chain rule, that the symplectomorphisms of R2n form a group, which we will denote by Symp(n). Let us now focus on the Hamiltonian case. We will call a symplectomorphism f such that f = ftH for some Hamiltonian function H and time t = 1 a Hamiltonian symplectomorphism. The choice of time t = 1 in this de…nition is of course arbitrary, and can be replaced with any other choice t = a: we have f = faHa where Ha (z; t) = aH(z; at): Hamiltonian symplectomorphisms form a subgroup Ham(n) of the group Symp(n) of all symplectomorphisms; it is in fact a normal subgroup of Symp(n) as follows from the conjugation formula g 1 ftH g = ftH
g
valid for every symplectomorphism g of R2n [39,23,25]. That Ham(n) is a group follows from the two formulas (29) and (30) in Proposition 13 above. The following result is, in spite of its simplicity, a deep statement about the structure of the group Ham(n). It says that every continuous path of Hamiltonian transformations passing through the identity is itself the phase ‡ow determined by a certain Hamiltonian function. Proposition 14 Let (ft )t be a smooth one-parameter family of Hamiltonian transformations such that f0 = Id . There exists a Hamiltonian function H = H(z; t) such that ft = ftH . More precisely, (ft )t is the phase ‡ow determined by the Hamiltonian function H(z; t) =
Z
0
1
z T J f_t ft
1
( z)d
(31)
where f_t =dft =dt.
PROOF. See Banyaga [4]; Wang [57] gives an elementary proof of formula (31). Remark 15 The idea is already present in Arnold [2] (p.269) who uses the apparatus of generating functions. We will a call smooth path (ft )t in Ham(n) joining the identity to some element f 2 Ham(n) a Hamiltonian isotopy. Proposition 14 above says that every Hamiltonian isotopy is a Hamiltonian ‡ow restricted to some time interval. 16
5.3 Symplectic algorithms
Symplectic integrators are designed for the numerical solution of Hamilton’s equations; they are algorithms which preserve the symplectic character of Hamiltonian ‡ows. The literature on the topic is immense; a well-cited paper is Channel and Scovel [60]. Among many recent contributions, a highlight is the recent treatise [41] by Kang Feng and Mengzhao Qin; also see the comprehensive paper by Xue-Shen Liu et al. [59], and Marsden’s online lecture notes [49] (Chapter 9). Let (ftH ) be a Hamiltonian ‡ow; let us …rst assume that H is time-independent H so that we have the one-parameter group property ftH ftH0 = ft+t 0 . Choose an 2n initial value z0 at time t = 0. A mapping f t on R is an algorithm with time step-size t for (ftH ) if we have f Ht (z) = f t (z) + O( tk ); the number k (usually an integer 1) is called the order of the algorithm. In the theory of Hamiltonian systems one requires that f t be a symplectomorphism; f t is then called a symplectic integrator. One of the basic properties one is interested in is convergence: setting t = t=N (N an integer) when do we have limN !1 (ft=N )N (z) = ftH (z)? One important requirement is stability, i.e. (ft=N )N (z) must remain close to z for small t (Chorin et al. [10]). Here are two elementary examples of symplectic integrators. We assume that the Hamiltonian H has the physical form H(x; p) = U (p) + V (x): First order algorithm. One de…nes (xk+1 ; pk+1 ) = f t (xk ; pk ) by xk+1 = xk + @p U (pk @x V (xk ) t) t pk+1 = pk @x V (xk ) t: Second order algorithm. Setting x0k = xk + 12 @p U (pk ) we take xk+1 = x0k + 21 @p U (pk ) pk+1 = pk @x V (x0k ) t: One can show, using Proposition 14 that both schemes are not only symplectic, but also Hamiltonian (Wang [57]). For instance, for the …rst order algorithm 17
described above, we have f t = f Kt where K is the now time-dependent Hamiltonian K(x; p; t) = U (p) + V (x @p U (p)t): (32) When the Hamiltonian H is itself time-dependent its ‡ow does no longer H enjoy the group property ftH ftH0 = ft+t 0 , so one has to rede…ne the notion of algorithm in some way. This can be done by considering the time-dependent 1 H H H . One then uses the following ftH0 ‡ow (ft;t 0 ) de…ned by (25): ft;t0 = ft H trick: de…ne the suspended ‡ow (fg t ) by the formula
H 0 0 H 0 0 fg t (z ; t ) = (ft+t0 ;t0 (z ); t + t );
(33)
H : R2n R ! R2n R (the “extended one veri…es that the mappings fg t g H H Hg phase space”) satisfy the one-parameter group law fg t ft0 = ft+t0 and one H may then de…ne a notion of algorithm approximating fg t (see Struckmeier [55] for a detailed study of the extended phase space approach to Hamiltonian dynamics). For details and related topics see the paper [10] by Chorin et al.
6
Weak Hamiltonian Deformations of Gabor Systems
We now turn to the central topic of this paper, which is to propose and study “reasonable”de…nitions of the notion of Hamiltonian deformation of a Gabor frame. 6.1 Description of the problem Let f 2 Ham(n) and (ft )0 t 1 a Hamiltonian isotopy joining the identity to f ; in view of Proposition 14 there exists a Hamiltonian function H such that ft = ftH for 0 t 1. We want to study the deformation of a ~-Gabor frame G( ; ) by (ft )0 t 1 ; that is we want to de…ne a deformation ft
G( ; ) ! G(Ubt ; ft );
(34)
here Ubt is an (unknown) operator associated in some (yet unknown) way with ft . We will proceed by analogy with the case ft = St 2 Sp(n) where we de…ned the deformation by St G( ; ) ! G(Sbt ; St ); (35) where Sbt 2 Mp(n), St =
}
(Sbt ). This suggests that we require that:
The operators Ubt should be unitary in L2 (Rn ); 18
The deformation (34) should reduce to (35) when the isotopy (ft )0 in Sp(n).
t 1
lies
The following property of the metaplectic representation gives us a clue. Let (St ) be a Hamiltonian isotopy in Sp(n) Ham(n). We have seen in Proposition 10 that there exists a Hamiltonian function H(z; t) = 21 M (t)z z with associated phase ‡ow precisely (St ). Consider now the Schrödinger equation @ c =H , ( ; 0) = 0 i~ @t c is the Weyl quantization of H (it is a formally self-adjoint operator where H since H is real). It is well-known [23,25,20] that = Sbt 0 where (Sbt )t is the unique path in Mp(n) passing through the identity and covering (St ). This suggests that we should choose (Ubt )t in the following way: let H be the Hamiltonian function determined by the Hamiltonian isotopy (ft ): ft = ftH . c using the Weyl correspondence, and let Then quantize H into a operator H b (Ut ) be the solution of Schrödinger’s equation i~
d b cU b , F =I : Ut = H t 0 d dt
The operators Ubt are unitary. Let in fact u(t) = (Ubt jUbt ) where c (we assume it contains S(Rn )); we have domain of H cU b jU b ) i~u(t) _ = (H t t
is in the
cU b )=0 (Ubt jH t
c is (formally) self-adjoint; it follows that (U b jU b ) = ( j ). since H t t
While de…nition (35) of a Hamiltonian deformation of a Gabor system is “reasonable”, its practical implementation is di¢ cult because it requires the solution of a Schrödinger equation. We will try to …nd a weaker, more tractable de…nition of the correspondence (34), which is easier to implement numerically.
6.2 The “Thawed Gaussian Approximation” The “weak Hamiltonian deformation” scheme method we are going to de…ne in nest subsection is so-called Gaussian wavepacket method which comes from semiclassical mechanics and is widely used in chemistry; it is due to Heller and his collaborators (Heller [37], Davis and Heller [11]) and Littlejohn [45]. (For a rather up to date discussion of various Gaussian wavepacket methods see Heller [37]). 19
For …xed z0 we set zt = ftH (z0 ) and de…ne the new Hamilton function zt ) + 12 Dz2 H(zt ; t)(z
Hz0 (z; t) = (@z H)(zt ; t)(z
zt )2 ;
(36)
it is the Taylor series of H at zt with terms of order 0 and > 2 suppressed. The corresponding Hamilton equations are z_ = J@z H(zt ; t) + JDz2 H(zt ; t)(z
zt ):
(37)
We make the following obvious but essential observation: in view of the uniqueness theorem for the solutions of Hamilton’s equations, the solution of (37) with initial value z0 is the same as that of the Hamiltonian system z(t) _ = J@z H(z(t); t)
(38)
H
with z(0) = z0 . Denoting by (ft z0 ) the Hamiltonian ‡ow determined by Hz0 H H we thus have ftH (z0 ) = ft z0 (z0 ). More generally, the ‡ows (ft z0 ) and (ftH ) are related by a simple formula involving the ‚linearized ‡ow”(St ): Proposition 16 The solutions of Hamilton’s equations (37) and (38) are related by the formula z(t) = zt + St (z(0) z0 ) (39) where zt = ftH (z0 ), z(t) = ftH (z) and (St ) is the phase ‡ow determined by the quadratic time-dependent Hamiltonian H 0 (z; t) = 12 Dz2 H(zt ; t)z z:
(40)
Equivalently, ftH (z) = T [zt
St (z0 )]St (z(0))
where T ( ) is the translation operator.
PROOF. Let us set u = z
zt . We have, taking (37) into account,
u_ + z_t = J@z H(z(t); t) + JDz2 H(zt ; t)u that is, since z_t = J@z H(zt ; t), u_ = JDz2 H(zt ; t)u: It follows that u(t) = St (u(0)) and hence z(t) = ftH (z0 ) + St u(0) = zt which is precisely (39).
20
St (z0 ) + St (z(0))
(41)
Remark 17 The function t 7 ! St (z) = DftH (z) satis…es the “variational equation” d St (z) = JDz2 H(ftH (z); t)St (z) , S0 (z) = I (42) dt (this relation can be used to show that St (z) is symplectic [23,25]; it thus gives a simple proof of the fact that Hamiltonian phase ‡ows consist of symplectomorphisms (ibid.)). The thawed Gaussian approximation (TGA) (also sometimes called the nearby orbit method) consists in making the following Ansatz: The approximate solution to Schrödinger’s equation i} where
@ c =H @t ~ z0
,
( ; 0) =
= Tb ~ (z0 )
~ z0
~ 0
(43)
is the standard coherent state centered at z0 is given by the formula e (x; t) = e ~i
(t;z0 )
Tb ~ (zt )Sbt (z0 )Tb ~ (z0 )
1 ~ z0
(44)
H(zt0 ; t0 ) dt0
(45)
where the phase (t; z0 ) is the symmetrized action (t; z0 ) =
Z
0
t
1 2
(zt0 ; z_t0 )
calculated along the Hamiltonian trajectory leading from z0 at time t0 = 0 to zt at time t. One shows that under suitable conditions on the Hamiltonian H the approximate solution satis…es, for jtj T , an estimate of the type jj ( ; t)
e ( ; t)jj
p C(z0 ; T ) ~jtj
(46)
where C(z0 ; T ) is a positive constant depending only on the initial point z0 and the time interval [ T; T ] (Hagedorn [32,33], Nazaikiinskii et al. [51].). Remark 18 Formula (44) shows that the solution of Schrödinger’s equation with initial datum ~0 is approximately the Gaussian obtained by propagating ~ 0 along the Hamiltonian trajectory starting from z = 0 while deforming it using the metaplectic lift of the linearized ‡ow around this point.
6.3 Application to Gabor frames
Let us state and prove the main results of this paper. 21
In what follows we consider a Gaussian Gabor system G( nearby orbit method to ~0 yields the approximation ~ t
i
= e~
(t;0)
Tb ~ (zt )Sbt
~ 0;
); applying the
~ 0
(47)
where we have set Sbt = Sbt (0). Let us consider the Gabor system G( where t = ftH ( ).
~ t;
t)
Proposition 19 The Gabor system G( ~t ; t ) is a Gabor ~-frame if and only if G( ~0 ; ) is a a Gabor ~-frame; when this is the case both frames have the same bounds. PROOF. Writing It ( ) =
X
z2
t
j( jTb ~ (z)
~ 2 t )j
(48)
we set out to show that the inequality ajj jj2
It ( )
bjj jj2
(49)
(for all 2 L2 (Rn )) holds for every t if and only if it holds for t = 0 (for all 2 L2 (Rn )). In view of de…nition (47) we have X
It ( ) =
z2
t
j( jTb ~ (z)Tb ~ (zt )Sbt
~ 2 0 )j ;
the commutation formula (18) yields Tb ~ (z)Tb ~ (zt ) = ei
and hence It ( ) =
X
z2
=
X
z2
t
j( jTb ~ (zt ))Tb ~ (z)Sbt
(z;zt )=~
Tb ~ (zt )Tb ~ (z)
~ 2 0 )j
j( jTb ~ (zt )Tb ~ (ftH (z))Sbt
~ 2 0 )j :
Since Tb ~ (zt ) is unitary the inequality (49) is thus equivalent to ajj jj2
X
z2
j( jTb ~ (ftH (z))Sbt
~ 2 0 )j
bjj jj2 :
(50)
In view of formula (39) we have, since St z0 = 0 because z0 = 0, ftH (z) = St z + ftH (0) = St z + zt hence the inequality (50) can be written ajj jj2
X
z2
j( jTb ~ (St z + zt )Sbt 22
~ 2 0 )j
bjj jj2 :
(51)
In view of the product formula (19) for Heisenberg–Weyl operators we have Tb ~ (St z + zt ) = ei
(St z;zt )=2~
so that (51) becomes
ajj jj2
X
z2
Tb ~ (zt )Tb ~ (St z)
j( jTb ~ (zt )Tb ~ (St z)Sbt
~ 2 0 )j
bjj jj2 ;
the unitarity of Tb ~ (zt ) implies that (52) is equivalent to ajj jj2
X
z2
j( jTb ~ (St z)Sbt
~ 2 0 )j
bjj jj2 :
(52)
(53)
Using the symplectic covariance formula (6) we have Tb ~ (St z)Sbt = Sbt Tb ~ (z)
so that the inequality (53) can be written ajj jj2
X
z2
j( jSbt Tb ~ (z)
since Sbt is unitary, this is equivalent to ajj jj2
X
z2
j( jTb ~ (z)
~ 2 0 )j
~ 2 0 )j
bjj jj2 ;
bjj jj2 :
The Proposition follows. The fact that we assumed that the window is the centered coherent state ~0 is not essential. For instance, Proposition 12 shows that the result remains valid if we replace ~0 with a coherent state having arbitrary center, for instance ~ ~ b~ z0 = T (z0 ) 0 . More generally: Corollary 20 Let G( ; ) be a Gabor system where the window Gaussian 1=4 i M x x ~ det Im M e 2~ n M (x) = ( ~) where M = M T , Im M > 0. Then G( it is the case for G( ; ).
~ t;
t)
is the (54)
is a Gabor ~-frame if and only if
PROOF. It follows from the properties of the action of the metaplectic group on Gaussians (see Appendix A) that there exists Sb 2 Mp(n) such that ~M = b be the projection on Sp(n) of S; b the Gabor system Sb ~0 . Let S = ~ (S) ~ ~ G( M ; ) is a ~-frame if and only if G(Sb 1 M ; S 1 ) = G( ~0 ; S 1 ) is a ~frame in view of Proposition 5. The result now follows from Proposition 19. 23
6.4 Possible extensions; the Feichtinger algebra
We …nally remark that the fact that we have been using Gaussian windows (coherent states and their generalizations) is a matter of pure convenience. In fact, the de…nition of weak Hamiltonian deformations of a Gabor frame as given above is valid for arbitrary windows 2 S(Rn ) (or 2 L2 (Rn )). It su¢ ces for this to replace the de…ning formula (47) with i
t
= e~
(t;0)
Tb ~ (zt )Sbt :
(55)
One can prove that if is su¢ ciently concentrated around the origin, then t is again a good semiclassical approximation to the true solution of Schrödinger’s equation. However, when one wants to impose to the initial window to belong to more sophisticated functional spaces than S(Rn ) or L2 (Rn ) one might be confronted to technical di¢ culties if one wants to prove that the deformed window (55) belongs to the same space. However, there is a very important case where this di¢ culty does not appear, namely if we assume that the initial window belongs to Feichtinger’s algebra S0 (Rn ) (reviewed in Appendix B). Since our de…nition of weak transformations of Gabor frames only makes use of phase space translations Tb ~ (z) and of metaplectic operators it follows that n 2 S0 (Rn ) (see de Gosson [24]). This is due to the t 2 S0 (R ) if and only if fact that the Feichtinger algebra is the smallest Banach algebra invariant under these operations, and is thus preserved under the semiclassical propagation scheme used here. It is unknown whether this property is conserved under passage to the general de…nition (35), that is ft
G( ; ) ! G(Ubt ; ft )
(56)
where Ubt is the solution of the Schrödinger equation associated with the Hamiltonian operator corresponding to the Hamiltonian isotopy (ft )0 t 1 : one does not know at the time of writing if the solution to Schrödinger equations with initial data in S0 (Rn ) also is in S0 (Rn ) for arbitrary Hamiltonians. The same di¢ culty appears when one considers other more general functions spaces (e.g. modulation spaces).
7
Discussion and Additional Remarks
Since our de…nition of weak deformations was motivated by semiclassical considerations one could perhaps consider re…nements of this method using the asymptotic expansions of Hagedorn [32,33] and his followers; this could then lead to “higher order”weak deformations, depending on the number of terms that are retained. 24
The scheme we have been exposing is a standard and robust method; its advantage is its simplicity. In future work we will discuss other interesting possibilities. For instance, in [35,36] Heller proposes a particular simple semiclassical approach which he calls the “frozen Gaussian approximation” (FGA). It is obtained by surrounding the Hamiltonian trajectories by a …xed (“frozen”) Gaussian function (for instance ~0 ) and neglecting its “squeezing” by metaplectic operators used in the TGA. Although this method seems to be rather crude, it yields astoundingly accurate numerical results applied to superpositions of in…nitely many Gaussians; thus it inherently has a clear relationship with frame expansions. A more sophisticated procedure would be the use of the Kluk–Herman (HK) approximate propagator, which has been widely discussed in the chemical literature (Herman [38] shows that the evolution associated with the HK propagator is unitary, and Swart and Rousse [56] put the method on a …rm mathematical footing by relating it with the theory of Fourier integral operators; in [31] Grossmann and Herman discuss questions of terminology relating to the FGA and the HK propagator). Also see the review papers by Heller [37] and Kay [42] where the respective merits of various semiclassical approximation methods are discussed. Still, there remains the question of the general de…nition (56) where the exact quantum propagator is used. It would indeed be more intellectually (and also probably practically!) satisfying to study this de…nition in detail. As we said, we preferred in this …rst approach to consider a weaker version because it is relatively easy to implement numerically using symplectic integrators. The general case (56) is challenging, but probably not out of reach. From a theoretical point of view, it amounts to construct an extension of the metaplectic representation in the non-linear case; that such a representation indeed exists has been shown in our paper with Hiley [26] (a caveat: one sometimes …nds in the physical literature a claim following which such an extension could not be constructed; a famous theorem of Groenewold and Van Hove being invoked. This is merely a misunderstanding of this theorem, which only claims that there is no way to extend the metaplectic representation so that the Dirac correspondence between Poisson brackets and commutators is preserved). There remains the problem of how one could prove that the deformation scheme (56) preserves the frame property; a possible approach could consisting in using a time-slicing (as one does for symplectic integrators); this would possibly also lead to some insight on whether the Feichtinger algebra is preserved by general quantum evolution.
APPENDIX A: METAPLECTIC GROUP AND GAUSSIANS
Let Mp(n) be the metaplectic representation of the symplectic group Sp(n) (see [20,23,25]); it is a unitary representation of the double cover Sp2 (n) of 25
Sp(n): we have a short exact sequence ~
0 ! Z2 ! Mp(n) ! Sp(n) ! 0 where ~ : Sb 7 ! S is the covering projection; we explain the appearance of the subscript } below. The metaplectic group is generated by the following elementary operators: Fourier transform: Jb
1 2 i~
=
n=2
Z
Rn
e
ixx0 =~
(x0 )dx0
(57)
(notice the presence of the imaginary unit i in the prefactor); Unitary dilations: c M
L;m
=i
m
q
j det Lj (Lx) (det L 6= 0)
(58)
where m is an integer depending on the sign of det L: m 2 f0; 2g if det L > 0 and m 2 f1; 3g if det L < 0; “Chirps”: 2 VbP = e iP x =2~ (x) (P = P T ): (59) The projections on Sp(n) of these operators are given by ~ c (ML;m ) = ML , and ~ (VbP ) = VP with 0
1
1
0
1
L 0C B I 0C ML = B @ A , VP = @ A 0 LT P I
~
b = J, (J)
(60)
(the matrices VP are sometimes called “symplectic shears”). The projection of a covering group onto its base group is de…ned only up to conjugation; our choice –and notation– is here dictated by the fact that to the ~-dependent operators (57) and (58) should correspond the symplectic matrices (60). For instance, in time-frequency analysis it is customary to make the choice ~ = 1=2 . Following formula relates the projections ~ and = 1=2 : b cp b = ~ (M c p S (61) (S) 1= 2 ~ M 2 ~ )
cp where M 2
~
cp =M 2
~I;0 .
Metaplectic operators are not only unitary operators on L2 (Rn ) but also linear automorphisms of S(Rn ) which extend by duality to automorphisms of S 0 (Rn ). There is an alternative way to describe the metaplectic group Mp(n). Let W (x; x0 ) = 21 P x2 26
Lx x0 + 12 Qx2
(62)
where P; L; Q are real n n matrices, P and Q symmetric and L invertible (we are writing P x2 for P x x, etc.). Let m be a choice of arg det L as in formula (58); each Sb 2 Mp(n) is the product to two operators of the type Sb
W;m
(x) =
1 2 i~
n=2 m q
j det Lj
i
Z
Rn
iW (x;x0 )=~
e
(x0 )dx0
(see [44,23,25]). The operators SbW;m can be factorized as SbW;m = Vb
c
b Vb
P ML;m J
(x; p) = SW (x0 ; p0 ) ()
~
(SbW;m ) is characterized
8 > < p = @x W (x; x0 ) > : p0 =
(A2)
Q
and hence belong to Mp(n). The projection SW = by the condition
(A1)
@x W (x; x0 )
;
this condition identi…es W with the generating function of …rst type, familiar from Hamiltonian mechanics [2,23,25,44]. A straightforward calculation using the expression (62) of W yields the symplectic matrix 0
1
1
1
L Q L C B SW = @ A: P L 1 Q LT P L 1 The metaplectic group acts on Gaussian functions in a particularly simple way. Let M be a complex n n matrix; we assume in fact that M belongs to the Siegel half-space + n
= fM : M = M T ; Im M > 0g:
We call generalized centered coherent state a Gaussian function of the type ~ M (x)
and for z0 2 R2n we set
=
det Im M ( ~)n
1=4
i
e 2~ M x x
(63)
= Tb ~ (z0 ) ~M (64) (it is a Gaussian centered at the point z0 ). The symplectic group Sp(n) acts transitively on the Siegel half-space via the law [20] ~ M;z0
(S; M ) 7 ! (S)M = (C + DM )(A + BM ) 1 :
(65)
One can show [25] that if M = X + iY then X = (CAT + DB T )(AAT + BB T ) Y = (AAT + BB T ) 1 :
27
1
(66) (67)
This action induces in turn a transitive action b (S;
~ M;z0 )
7 !
~
(S)M;Sz0
of the metaplectic group Mp(n) on the set Gn of Gaussians of the type (64). These actions make the following diagram Gn ! Gn
Mp(n) #
# + n
Sp(n)
+ n
!
b commutative (the vertical arrows being the mappings (S; and ~M;z0 7 ! M , respectively).
~ M;z0 )
7 ! (S; M )
The formulas above can be proven by using either the properties of the Wigner transform, or by a calculation of Gaussian integrals using the operators SbW;m de…ned by formula (A1).
APPENDIX B: FEICHTINGER’S ALGEBRA The Feichtinger algebra S0 (Rn ) was introduced in [14–16]; it is an important particular case of the modulation spaces de…ned by the same author; we refer to Gröchenig’s treatise [28] for a complete study of these important functional spaces. Also see Feichtinger and Luef [19] for an up to date concise review. The Feichtinger algebra is usually de…ned in terms of short-time Fourier transform Z 0 V (z) = e 2 ip x (x0 ) (x0 x)dx0 ; (68) Rn
which is related to the cross-Wigner transform by the formula W ( ; )(z) = where
p
2
2 ~
p (x) = (x 2 ~) and ~ V
(z) =
2 ~
n=2
e
n=2
_
2i
e ~ p xV
_ p
2 ~
p
2
~ (z
q
2 ) ~
(69)
(x) = ( x); equivalently
i px
W(
p
1= 2
~;
_p 1= 2 ~ )(z
q
~ ): 2
(70)
The Feichtinger algebra S0 (Rn ) consists of all 2 S 0 (Rn ) such that V 2 L1 (R2n ) for every window . In view of the relations (69), (70) this condition is equivalent to W ( ; ) 2 L1 (R2n ). A function 2 L2 (Rn ) belongs to S0 (Rn ) if 28
2 L1 (R2n ); here W
and only if W The number
= W ( ; ) is the usual Wigner function.
jj jj~ ;S0 = jjW ( ; )jjL1 (R2n ) = is the norm of
Z
R2n
jW ( ; )(z)jdz
(71)
relative to the window . We have the inclusions
S(Rn )
S0 (Rn )
C 0 (Rn ) \ L1 (Rn ) \ F (L1 (Rn ))
(72)
where F (L1 (Rn )) is the image of L1 (Rn ) by the Fourier transform. One proves that S0 (Rn ) is an algebra, both for pointwise multiplication and for convolution. An essential property of the Feichtinger algebra is that it is closed under the action of the Weyl-metaplectic group WMp(n): if 2 S0 (Rn ), Sb 2 Mp(n), and z0 2 Rn we have both Sb 2 S0 (Rn ) and Tb ~ (z0 ) 2 S0 (Rn ). In particular 2 S0 (Rn ) if and only if F 2 S0 (Rn ). One proves that S0 (Rn ) is the smallest Banach space containing S(Rn ) and having this property. Acknowledgements
This work has been supported by a research grant from the Austrian Research Agency FWF (Projektnummer P23902-N13).
Acknowledgements
The author wishes to thank Hans Feichtinger and Karlheinz Gröchenig for valuable suggestions, criticism, and discussions about Gabor frames.
References
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