Some theorems on Gabor operators

J. DE GRAAF Dept. of Mathematics, Technological University, Eindhoven, The Netherlands Abstract. The object of this study is the class of closable Gabor operators. That is the set of operators which map a Gabor function (or 'note') into a multiple of a Gabor function. By using the Bargman space (sometimes called Bargman representation) some general properties of these operators are derived. It is shown that the set of Gabor operators whose adjoint is also a Gabor operator establishes a six-dimensional complex manifold with a partial Lie-group structure and with an involution. The corresponding Lie-algebra and the infinitesimal generators are calculated. Further it turns out that, at least locally, a Gabor operator with a Gabor adjoint results from an evolution process. The proofs of the theorems are a hybridization of Hilbert space techniques and classical complex analysis (theorems of Osgood, Montel, etc.). The proofs will be published elsewhere in a wider context. 1. Introduction In physics, signal theory and music the so-called Gabor function

ga, ~, 7 ( 0 = exp (

(t--a)2+iwt} 27

with a, w, 7 E N and 7 > 0, plays an important role. This function describes an impure tone of width 2 V ~ at time a and angular frequency co. In music it could be the performance of a note of length 2 x / ~ . Gabor functions are eigenfunctions of the eigenvalue problem du

7 -~ + tu = Xu. Each X E C is an eigenvalue and X = a + iTco. For 7 = 1 our operator is just the annihilation operator and the corresponding Gabor functions are the so-called coherent states. I am interested in a description of all linear signal processors which turn each Gabor function into (a multiple of) a Gabor function. Mathematically this problem can be cast in the following form: Try to find or characterize all closable, densely defined, possibly unbounded operators in L 2 ( ~ ) which turn a Gabor function into a multiple of a Gabor function. Operators having these properties will be called Gabor operators. 2. Preliminaries I will use the Bargman space ~e~. This is a Hilbert space of entire analytic functions. The inner product is

(f,g) = j f(z)g(z) d/~(z). 45

Applied Scientific Research 37:45-52 {1981) 0003-6994/81/0371-0045 $01.20. © Martinus Ni]hoffPubIishers, The Hague. Printed in the Netherlands.

46 Here d/~(z) = 7r-1 exp (-- Iz [2) dx dy. If not mentioned otherwise integrations are carried out over the whole of C considered as N2. In ,~ßone has the principal vectors ew(Z) = e x p ~ z with the property ( e w , f ) = f ( w ) for all w E C. Next, following Bargman, I introduce the mappingA : ~ ½ f f r o m L2(N) into ?S defined by B(z) = (A~)(z) =

S ~ A(z,q)~(q)dq

with AQ, q) = rr1/« exp {-~(z 2 + q2) + x/'2zq}. This mappingA is a bijective isometry. It maps the orthonormal Hermite basis of L2(R) into the orthonormal basis {(n!)-l/2zn} of ~ . The expansion of an element u in ~ ' w i t h respect to the last mentioned basis is easily obtained from the Taylor expansion of u. Further,-the mapping A establishes a unitary isomorphism between the linear operators in . ~ a n d those in L2(N), namely M = A-1LA. Here L is an operator in .Y and M the corresponding operator in L2(N). The domains B ( L ) and .~(M) are related by . ~ ( M ) = A - a . ~ ( L ) . Some examples of operators in L2(N) together with their counterpart in ~ a r e : the annihilation operator M = 2-1/2(x + d/dx), L = d/dz. The creation operator M = 2-1/2(x -- d/dx), L = z. The Fourier operator M = ~ , (Lu)(z) = u (iz). Any bounded operator L in ~ c a n be represented as an integral transform. For any f E g/~we have ( L f ) . (w) = (ew, Icl)= (L* ew, f), so that

(Lf)(w) = f L(w, z)f(z)dts(z), where

L(w,z) = L*ew(z ) = (ez,L*ew) = (Lez, ew). Or

L(w, z) = (ew, Le~) = (Le~y(w). For the results of this section I refer to [1]. A short and very readable account of the Bargman space theory can be found in [3], pp. 215-218. Finally calculation of the Bargman representation of a Gabor function yields

(Aga, to,-r)(z) = ~o(a,w, 3') exp {½(3' -- 1)(3' + 1)-lz 2 + + 21/23'(3' + 1)-103' -1 + ko)z}, where the 'phase factor' is ~0(a, co, 7) = rr-1/4(2rr3')l/2(7 + 1) -1/2 exp {--~3'-1a2 + ~(3'-1a + ia~)2}. This is an eigenfunction with eigenvalue a + ko3' of the operator 2 -'/2 {(1 -- 3')z

47 + (1 + 3,)d/dz} defined in ~ . It is clear that the eigenfunction belongs to,~ß whenever 1(7 -- 1)(3, + 1) -11 < 1. That means Re 3, > 0. Next we throw away the phase factors and define a Gabor function in U ' t o be the function eo, w(z) = exp {½az 2 + ff~z} with Io[ < 1. Obviously ea, ~ is an eigenfunction with eigenvalue ~ of the operator - - o z + d/dz. In case o = 0 one retains the annhilation operator in the Bargman representation and {eo, ~}, w E C , are the coherent states. Now our mathematical problem is the following: Characterize the set of densely defined and closable operators in ~ w h i c h map a Gabor function in a multiple of a Gabor function. From now on these operators will be called Gabor operators. 3. Some results on general Gabor operators As mentioned in the preceding section each element in . ~ c a n be written as a continuous superposition of the coherent states e w = eo, w u(w) = f eü~--~u(z)dp(z). Intuitively one suspects that the action of an operator G is determined by its action on the elements e w . Heuristically

(Gu)(w) = ~ (Gez)(w) u(z) du(z). Necessary conditions for this expression to have a meaning is that ez E ~ ( G ) for all z E C~and some sort of convergence of the integral. Definition 1 Ler f, g, h be mappings from C into itself, f, g, h are not necessarily continuous or even measurable. Suppose [h(z) l < 1 for all z E C . On the linear span (ew), i.e. the set of finite linear combinations of the functions ew, we define the Gabor operator Gf# h by (Gtghew)(Z) = f ( ~ ) exp {g(ff0z + ½h(ff0z 2 } and linear algebraic extension. Algebraically speaking the functions ew, w EC:, are independent but toplogically they are not. So it is not surprising that imposing a topological condition on Gmh implies some smoothness of the functions f, g, h. Theorem 2 A necessary condition for the operator Gfgh to be closable is that f, g and h are analytic functions on an open and dense subset of C.

48

A question that immediately comes up is: Does it follow from the closability of Gfg h that f, g and h are entire functions? I don't know whether this follows from the closability condition alone. However, with a gentle additional condition it tbllows that f, g and h are entire analytic functions.

Theorem 3 Suppose that Gfg h is closable and suppose that f and g map bounded sets into bounded sets. Then h must be constant and f and g must be entire functions. The condition on f and g in this theorem is satisfied i f f and g a r e supposed to be continuous or if Gfg h is supposed to be a bounded operatdr in ~ . The operator Gfg h is closable iff the domain.~(G~gh) of its adjoint G~gh is dense. If we area little specific on-~(G~gh) then it also follows that f, g and h are entire. This is done in the next theorem.

Theorem 4 (A)

{zn}n=o C ..~r(G~gh), i.e. the domain of the adjoint contains all polynomials, if and only i f f and g a r e entire, h is constant andfg n E ~/~for n=0,1,2 .....

(B)

{e«}«~ c C-~(G~gh),i.e.the domain of the adjoint contains all coherent states, if and only i f f ( ~ ) exp {g(~)z + ½h(~)z 2 } for each fixed z E , as a function of w, belongs to ~e-.

(C)

The conditions on f, g, h as mentioned in (A) and (B) are sufficient for Gfg h to be closable.

In the sequel I consider closable Gabor operators with f and g entire functions and h a complex constant, [hl < 1. The next theorem contains some information on the domaln.~(Gfgh) of the closure Gfgh of Gfg h and also on

Theorem 5 (A)

Let g2 c C be a compact subset. Let ~ EL 1(g2). Then fa ~ (w) e ü~zd#(w), as a function of z, belongs t o ß ( G f g n ) .

(B)

Let u E ~'. Suppose 2ißn~ flwl
(C)

Let v E-~(G~eh). Then (Gfgnv)(a) = i f ( ~ exp { g ( ~ ~ + ~h~~} v(w) d#(w).

49 Any bounded operator in 7 can be represented by a kernel, see [1]. In the text theorem a class of unbounded closable operators is introduced which can be represented by a kernel. The result is then apptied to Gabor operators.

Theorem 6 Let A be a densely defined operator in 7" and suppose . ~ ( A ) D {es} , B ( A * ) D {%}. Then there exists an entire function a(u, v) o n C 2 such that for each ~0E ~ ( A )

(A'~o)(z) = I a(z, ~)~o(w)dia(w). For each u E C a (u, -) E Y and for e ach v E C a ( ' , v) E ~ i

Theorem 7 Suppose G is an injective Gabor operator such that .~(G*) D {es}. Then G* is also a Gabor operator and the kernel of G is given by

a(z, V0 = exp {.40 + ~AaW2 + A2ff~ + A3Wz + A 4 z +

½Asz2}

where A i, 0 ~< i < 5, are complex cônstants, lA a I < 1, lA s I < 1.

Remark 8 If both G and G* are Gabor operators, the injectiveness of G and G* follows. 4. On the composition of Gabor operators whose adjoint is also a Gabor operator Let D C C denote the open unit disc. Consider the manifold I~ =C/27ri x D × C x C\{0} × C x D. Denote A = ( A o , A a , . . . ,As) E U. With each A E P we associate a Gabor operator G A by

(GAU)(Z) = ~ g(A, z, ffO u(w) dia (w) with

g(A,z,~) = exp {A0 + ~A1 ~2 + A 2 w +A3wz +A4z +

+ ½Asz~}. From the preceding section it follows that G~, = GÄ with Ä = (Ao, A s, A4,

A3,A2,A1). All functions e«, aEC, are in the domain of each G A but in general functions of type exp {äz + ~~z 2 } are only in _,~(GA) if [~[ is small enough. Since applica-tion of G A to some es may leäd to the exponential of a second degree polynomial, the composition GA o GB is not always an operator with each es in its domain. If for given B the constants A 1 , .. ,As are chosen

5O sufficiently close to 0, 0, 1, 0, 0, composition is possible such that each % is in the domain of G A ° GB. Definition 9

P is endowed with a partial Lie-group structure and with an involution in the foUowing way. The composition law is A*

B =

(Ao,A1,

. . , A s ) * (Bo,B1 . . . . ,Bs) =

(Bo + A o + ~(A1B4 1 2 + BsA22 + 2B4A2)(1 - - A 1 B s ) - 1 - - ~ !og {1 - - A 1 B s } , B1 + A 1 B ~ ( 1 - - A 1 B s ) - I , B 2 + A 1 B 3 B 4 ( 1 - - A 1 B s ) -1 + + BaA2(1 - - A 1 B s ) -1 ,B3A3(1 - - A 1 B s ) -1, Ag + B s A 2 A a ( 1 - - A 1 B s ) -1 + A a B « ( 1 - - A I B s ) -1 , As + B s A ~ ( 1 - - A I B 5 ) - l ) .

(t)

The involution is Ä = (Ao,AI ..... As)R emarks 10

1.

2. 3. 4.

= (Ao,A5,A4,Aa,A2,Ax).

J

The expression (t) makes sense for each A, B E F. The result of the composition however is not necessarily an element of F. For given A composition with an element B sufficiently near to (0, O, O, l, O, O) leads to an dement in F. ( A * B)~ = ~ * Ä. (0, 0, 0, 1, 0, 0) acts as a unit element. The left inverse of B is given by (Ao, B1 (B1Bs -- B i ) - 1 , (B3B2 -- B , B « ) ( B 1 B s -- B i ) - 1 , --B3 (B, Bs -- B~) -1 , (B3B4 -- BzBs )(BIBs -- B i ) - 1 , B5 (BI B s --B32)-1).

5.

A0 can be calculated by putting the first component in (t) equal to zero and substitution of the known variables. Obviously the inverse exists. On a sufficiently small open neighbourhood of the unit element the inverse is again an element of P. With the composition rule as introduced above the elements A E F for which A 1 = A s = 0 establish a Lie-group.

Theorem 11

Suppose that the product operator GA o GB contains all coherent states es, a E C , in its domain, then G A o GB = G A • B.

51 The variable Ao only plays a subordinate role. It has to do with phase factors. Therefore I introduce the manifold E = DxCxC\{0}xC

xD.

The notation of the variables is x = (Xm. . . . . xs). With the composition (t) where the first component is left out, N is again a partial Lie-group with involution. The tangent bundle TE is trivializable. A right invariant vector field v(x) is a vector field with the property 0x (x * y ) ' v ( y ) = v(x * y). For each y there is an open neighbourhood ~2y of unity such that this expression makes sense for each x E ~y. Denote the tangent space at the unit element e by Te 2. Let « = ( c q , . . , as) T«Z. Then a right invariant vector field is given by "-1

x~

v(x) = O ( x ) ' « =

0

0

0

0

a~ [

x4xs

x3

0

0

0

a2 !

xsx3

0

x3

0

0

c~3

xsx4

xs

x3

1

0

a4 ]

x~

0

2xs

0

1

_as A]

Obviously v(0, 0, 1,0, 0) - (al . . . . , as). One-parameter partial sub-Lie-groups of 2 are obtained by solving the differential equation dx dt

O

«

with

x(0) = e.

The tangent space T«Z at the unit element is made into a Lie-algebra in the usual way with the aid of the local formula for the Lie-product [u(x), v(x)] = Du(x)" v(x) --Dv(x) • u(x). Evaluation of this Lie-product leads to the expression [a,[3 ] = column (2(al/33 -- 131aa), (a2f13 -- ~2(~3) "~

+ («~ ô» - & e s ) , 2 (a3~~ - ~3 «»))-

Theorem 12 Consider the set ~ of Gabor operators whose adjoint is also a Gabor operator. Ler c~ be provided with the operations of composition (whenever defined

52 on the whole of span (e~)) and taking the adjoint. Then ~ is a bijective representation of the partial Lie-group with involution F. By this I mean: (i) (il) (iii) (iv)

The mapping F 9 A ½ GA is bijective. GA o GB = G A , B. The left-hand side is defined iff the right-hand side is defined. G~. = GÆ. If P(t), -- e < t < e, e > 0, P(0) = I, is a one-pararneter partial subgroup of ~', then P(t) has an infinitesimal generator of the form d2

I«1 ~ (v)

(vi)

d +

+ «~ dz

d

a3z ~ + «,z + ½«,z~

with al . . . . . as E C . The set of differential operators of the form mentioned in (iv) can be made into a Lie-algebra in the usual way by defining the Lie-product equal to the commutator. This Lie-algebra is isomorphic to the Liealgebra TeE. There is a neighbourhood t2 of e in E such that each element in 12 is an element of a one-parameter partial subgroup of E. In the set of Gabor operators ~corresponding to fl each Gabor operator is the result of an evolution process.

Final remarks

(a)

(b)

(c)

I don't know whether (vi) of the last theorem is globally true. That is whether each Gabor operator that has a Gabor adjoint results from one evolution process. The translation from . ~ b a c k to L 2 ( ~ ) can be made via (iv) of the last theorem. The set of infmitesimal generators has exactly the same form there. Th~s can be seen by using the translation mies for creation and annihilation operators as mentioned in the Preliminaries. The class of operators mentioned by De Bruijn [2], Section 27.3, consists of Gabor operators whose adjoint is also a Gabor operator. It is a proper subset of the class of operators I discussed in the last section.

This paper is dedicated to L.J.F. Broer on the occasion of his retirement.

References 1. 2. 3.

Bargmann V (1961) On a Hilbert Space of analytic functions and an associated integral transform. Comm Pure Appl Math 14:187-214 Bruijn NG De (1973) A theory of generalized funcfions with applications to Wigner distribution and Weyl correspondence. Nieuw Archief voor Wiskunde 21: 205-280 Jauch JM (1968) Foundations of Quantum Mechanics. Reading, Massachusetts: Addison-Wesley