,
~;::-""'I'
Ann, Inst. Henri Poincare,
Vol. 45, n° 3,1986,
Physique
p, ~0'-309,
thearique
«"c\
1
Transforms associated
to square integrable group representations II:
examples
by
A. GROSSMANN,
e.
J. MORLET
(*) and T. PAUL
Centre de Physique Theorique C**), Section II,
N. R. S" Luminy, Case 907, 13288 Marseille, Cedex 9, France
ABSTRACT. We give examples of integral transforms defined through
square integrable group representations, described in the first paper of
this series, We focus on the Weyl and « ax + b » groups and study their
relevance in physics and applied mathematics.
RESUME, On donne des exemples de transformations integrales definies au moyens de representations de carre integrable de groupes, decrites
dans Ie premier article de cette serie. On se concentre sur Ie groupe de Weyl
et Ie groupe « ax + b » et on examine leur pertinence en Physique et en
Mathematiques appliquees.
I. INTRODUCTION
In a preceding paper [1], we discussed orthogonality relations for
square integrable group representations, and claimed that they allowed
a unified discussion of many situations of interest in quantum mechanics
C*) ELF Aquitaine
Company,
Ruei1- Malmaison,
France,
C**)
Laboratoire
propre du
e.
ORIC
Lab"
370 bis, avenue
Napoleon
N, R, S,
Annales de l'Inslitul Henri Poincare - Physique theorique - 0246-0211
VoL 45/86/03/293117)$ 3,70/'£J Gauthier-Villars
Bonaparte,
<:;
294
A. GROSSMANN,
and in signal analysis. The main purpose
claim.
of this paper is to illustrate
We first recall the statement of orthogonality
identical to the one given in [1].
Let G be a locally compact
right Haar measure dflR(X),
Let U:
"I
TRANSFORMS
1. MORLET AND T. PAUL
U(~')
TO SQUARE INTEGRABLE
GROUP
In this series of papers, we shall be mainly
<PI = <P2' If we define then the number
REPRESENTATIONS
concerned
295
with the case
relations, in a form almost
C
group, with left Haar measure
-->
this
ASSOCIATED
df.l(x)
'"
and
=(<P,<P)a
(<p, <p)
we obtain the following statement:
For fixed admissible <P, the correspondence
and the function
L'EG)
between the vector
l/J E
£(U)
1
be a continuous unitary representation
in a Hilbert space £(U).
In the present paper, we assume U to be irreducible. We shaII see however,
in the comments and remarks at the end of this paper, that the irreducibility
assumption
is unnaturalIy restrictive. It wiII be relaxed in work in preparation.
We say that U is square integrable if there exists in yt(U) at least one
non-zero vector <P such that the matrix element (U(y)<p, <,o),-a complexvalued function on G-is
square integrable with respect to the measure
df.l(Y). It follows then that the function (U(y)<p, <,0) is also square integrable
with respect to the right-invariant
measure df.lR(Y)' Any such vector will
be called an admissih/e
(/}w/I'::inr;
\I·are/et.
If U is square integrable, then the set A(U) of admissible analyzing
wavelets is dense in Jlf(U).
Now choose any <PE A(U), and an arbitrary ifi E £(U), which need not
be admissible.
Then,
The matrix element (U(y)<,o, l/J) is square integrable with respect to
i. e. it belongs to U(G, df.l(Y)). (It does not necessarily belong to
L 2(G, df.lR(Y)) ; there is no difficulty in deriving the appropriate L 2(G, dflR(Y))statemen ts).
i)
(L",ifi)(y)
= v'~C,p (U(y)<,o,
in U(G, df.l(Y)), is an isometry from £(U) into U(G, df.l(Y)).
The function L",ifi is called the left transform of l/J with respect to G, U
and <,0.
It is sometimes also convenient to introduce the right transform
(R,pifi)(y)
ii)
There exists
in A( U) a positive quadratic
(f(ip
1· (P 2)
=
(<,0 "
(P2)"
(<,0 [, «2)
form
=
(L",ifi)(y-1)
.
Examples will be found in aII the sections that follow.
The range of L", is in general a proper closed subspace of U(G, df.l(Y)).
The usefulness of the transforms L", is largely a consequence of the fact
that there exists a simple general characterization
of this range.
Consider on G the function
1
P",(y)
Consider
= C(U(y)<p,
'"
on L Z(G,
df.l(Y)
(<,oEA(U),
<,0)
the integral
dfl(Y),
We next consider two admissible analyzing wavelets <PI' <,02, and two
arbitrary vectors in £(U), namely l/J to and ifi 2' The orthogonality relations
are a statement about the inner product of (U((')<,ol>l/J I) and of (U(y)<,o2' 1/12)'
in L2(G, dlt(Y)), namely:
l/J)
P,p(~',/)
operator
YEG).
P", given by the kernel
= P,h'-ly)
(a convolution operator on the group). Then P", is the orthogonal projection
operator on the range of L",. It acts as the identity on any function l/J(y)
in the range of L"" and sends to zero any function orthogonal to that
range. The range of L", is a Hilbert space with reproducing kernel P q>(Y, y').
The above results can be written very transparently
with the help of
Dirac notation, in which < I denotes the linear functional corresponding
to a vector <P = 1<,0> in a Hilbert space (To avoid possible misunderstandings, we mention that we deal here in bona fide Hilbert space vectors,
and that a label on a vector, say I y > does not imply that I y > is the eigenvector of some operator). If we choose any admissible analyzing wavelet
and write I y;
> (or I y > for shortness) to denote
1
<,0
E A(U)
such that A(U) is a Hilbert space with respect to the scalar product ( ,
and such that
)a,
<,0
<,0
I
(V(y)<,ol'
ifil)(U(y)<,oz,
ifiz)dfl(Y)
Annalesde
=
(<pz, <Pl)a(l/JI,
l/lz)·
rInstitIIl Henri Poincare - Physique theorique
Iy>
Vol. 45, n° 3;1986.
= ylC",
~
U(y) I <P >,
12
296
J. MORLET AND T. PAUL
A. GROSSMANN,
TRANSFORMS
ASSL .ATED TO SQUARE INTEGRABLE
(identity operator in J'f(U)). The transform Lrpassociates, to the vector
ljJ E J't'(U), the function < y IljJ > on G.
The reproducing equation becomes evident: it states that
=f
< y
I y'
> d.u(y') < y'
1
< y I y' >
=
IljJ >
=
U(y')cp)
(P
C(U(y'-ly)cp,
rp
TX + x'
(x, x'
E IR)
Eb+b'
(b, b'
E IR)
DYDY'
oyy'
(y, y' E IR, y
EbTX
=
eibxTxEb
shift by x
ii)
Eb:
-
x')
DYTx
FDY
=
eibx'ljJ(x')
=
iv)
TXDY
=
F-1DY
(parity operator)
F2
= D(-l).
=f.
0).
D1/YF-1
a) The group.
The Weyl-Heisenberg group appears naturally whenever one considers
simultaneously displacements (shifts) in a space IRn and in the space of
Fourier conjugate variables. In order to motivate the definition below,
consider, in the Hilbert space L 2(lRn, dnx) the unitary displacement operators
Tb
(TbljJ)(x)
=
ljJ(x -
b)
(b E
IRn) .
If we define F (the Fourier transform) as the unitary map from
to L2(lRn,dnp)
defined by
bE IR.
=
l{;(p)
U(lRn,
dnx)
= (2n)-n/2 f e-ipxljJ(x)dnx
we notice that
YEIR,
y
=f.
O.
F-1TbF
where
(FljJ)(w)
TXYDY
= ~
Fourier transform:
1 f
=.jhr
0, y'
= DYEbY
= DYTx/y
EbDY
F4
(FljJ)(p)
= IYI-1/2ljJC)
F-1Eb
= D1/YF
Unitary dilation operator:
(DYljJ)(t)
FEb
(identity)
XE IR.
multiplication by exponential
(EbljJ)(x')
iii) DY:
= ljJ(x
=f.
2. THE WEYL-HEISENBERG GROUP
Many calculations in this paper can be made trivially with the help of
straightforward operator identities which we give here for the reader's
convenience.
Consider first a space L 2(1R) (line with Lebesgue measure). Introduce
the following operators:
(rljJ)(x')
.
= e - ibxEbTx
= TbF
= T-bF-1
TXEb
FTX = E-xF
F- ITx = EXF-1
DYEb = Eb/YDY
cp).
Notations: some families of operators.
r:
eLWtX(w)dw
Furthermore
There exists a vast literature dealing with square integrable representations of various classes of groups, and we shall not attempt here even a
superficial survey. Our main purpose in this paper is the study of some
very particular cases which fall under the general scheme just described,
and which are of independent interest.
We shall start with the Weyl-Heisenberg group and some closely related
objects, and then discuss transforms associated with the « ax + b »-group
(affine group in one dimension).
i)
1 f.
.jhr
=
=
=
TXTX'
1
C(U(y)cp,
297
All these operators are unitary in L 2(1R). They satisfy
EbEb'
IljJ >
= --
(F-1X)(t)
L I y > d.u(y) < y I = ~
<y
REPRESENTATIONS
Inverse Fourier transform:
then the basic isometry relation takes the familiar form
where
GROUP
e is the (unitary)
.
e-lO)tljJ(t)dt.
Annales de I'lnstitut Henri Poincare - Physique theorique
Eb
operator of multiplication by an exponential
(EbljJ)(x)
Vol. 45, n° 3-1986.
=
= eibxljJ(x).
298
A. GROSSMANN,
J. MORLET
Consider now the set of operators generated
(multiplication) by the T' (s E IRn) and the Eb
.sb
The operators
W(s,
W(s,
in the sense of operator
One has
(but not essential) to introduce
W(s, b)
b)
-A.TED TO SQUARE INTEGRABLE
REPRESENTATIONS
299
<po
For the sake of simplicity, we consider now the case n = 1; the generalization to arbitrary n is trivial in the case of the Weyl-Heisenberg
group.
The results of the introduction
give now the following statements.
Let <p, (the analyzing wavelet) be an arbitrary vector in L2(1R). For every
bE IR, s E IR, consider the vector <p(s,b)E L 2(1R) defined as
the two-parameter
. sb
= e'2TsEb = e -12EbTs.
')
=
b')
!-(bs' -sb')
e2
<p(s.b) =
W(s
+
S', b
+
1
S,
-b ,1: -I}
~W(s,
b)<p
,le,p
b') .
where
We are now ready to define the Weyl-Heisenberg
group:
Consider the set G¥I<V of triplets y = {s, b, 1:} with s E IRn,
I 1:I = 1. Define
!-(bs' - sb')
yy' = {s + S', b + b', H' e2
}
-1_f_
-
GROUP
b) The transform L<p: arbitrary
satisfy
b)W(S',
Y
AS~
(b E IRn).
= e-ibsEbTs
TSEb
and it is convenient,
family of operators
TRANSFORMS
AND T. PAUL
=
C,p
bE
IRn,
1:E C,
To every if; E L 2(1R) associate
2rc
II
(P
112
.
the function
(L(pif;)(s,b)
=
(<p(s,b),
If;).
.
With these operations, G\1<V becomes a group, the Weyl-Heisenberg group.
The discussion preceding this definition gives immediately a unitary
representation of G¥I<V in L 2(lRn, dnx):
Then the transformation
L,p is isometric from U(IR, dx) into U(1R2, dsdb).
The range of Lq>is characterized as follows: consider the function
=
<1>(s,bl(S',b/)
(<p(S',b'l, cp(s,b»)
= (Fcp(S',b'l, F <p(s.b»)
U(s, b, 1:) = 1:W(s, b) .
1 e2!-(b'S'-bslf'
= C<p
It is not difficult to see that U is irreducible.
The group
measure IS
G¥I<V
is unimodular.
A suitable
Then a function
dJ.l.(Y)
with 1: =
-
S/)<p(X -
s)dx.
f(s.
h)
belongs to the range of L,p if and only if it satisfies
= dnsdnbd<p
ei<P.
We shall see that the representation
{ a, y]
f(s,
U is square integrable.
Remark, - The Weyl-Heisenberg
group with dilations.
Let, as above, s E IRn, bE IRn, 1:E C, I 1:I = 1, y = { s, b, 1:}. If
non-zero real number, a E IR, a =f. 0, consider the pair { a, y } :
= { a, s, h, 1: }
b)
If F denotes the Fourier
a IS
=
transform
S', b',
1:/}
= { aa',
(-1
{ a, s, b,
1: J
s+as',
b+
~bl,
If D(a) is the unitary
, -
H' expG(
abs' -
~Sbl))
-[
-;; s, -
(This is a semidirect product construction).
This group will be exploited in another
ab,
dilation operator,
s) .
one has
1:
•
= (LD(Qlq>D(a)if;)(
If the displacement
operator
are defined as in p. 8, then
paper.
Henri Poincare - Physique theorique
as, ~ b).
}
(L<pT'°if;)(s,b)
Annales de l'lnstitat
as in (1), one has
by defining
-1
= {I}
a
b')ds'db'.
= (LF<pFlf;)(b,-
(L,plj;)(s,h)
(Lq>if;)(s,b)
s, b, 1:}{ai,
f (j5(S,b)(S', b')f(s',
any
.
The set of such pairs can be made into a group
{a,
el(b-b ,_)xcp(x
(right and left) invariant
(L<pEboljJ)(s, b)
Vol. 45, n° 3-1986.
T'0 and the multiplication
=
=
e-ibso/2(L<pif;)(s
-
so, b)
eibso/2(Lq>if;)(s,
b -
bo).
operator
Ebo
300
J. MORLET AND T. PAUL
A. GROSSMANN,
TRANSFORMS
AS~
'The adjoint of Lq>(which is the inverse of Lq>on the range of Lq>and zero
on the orthogonal complement) is given by
tj;(x)
= ff
TO SQUARE INTEGRABLE
+
3. THE «ax
The «ax
cp(S,b)(x)Lq>(tj;)(s, b)dsdb
'ATED
+
Assume that tj; is such that the support of the Fourier transform of tj;
is contained in a finite interval ~. Then in addition to the reproducing
equation, Lq>i/tsatisfies a more stringent reproducing equation, determined
by the kernel <1l~s,b)(S', b') = (Fcp(S',b'l, Fcp(s,b»)d where the subscript ~ means
that the inner product is defined by integration over the interval ~,
a fixed rr
{ a, b } { a', b' }
The « ax
J2
211:
II
= V ;::.;
211: for all rr.
conveniently in terms
fi
1 -
can be explicitly
.!.lzIL.!.lz'12
calculated.
It is
_,
= e 2 2 eZ
1 - .!.lz-z'12
= -e
2
ei1m(z'zl
211:
and the right Haar measure
U(b, a)
is
b)
= dadb
Ia I .
=
TbDa
(b E
(U(a,
b)(p,
tj;) =
(TbDacp,
cpo
+
b »-group
and where
IR, a =F-0)
(ThD"cp,
admissihilitr
Ij;)
<p E
U(IR,
dt))
= (E -hD1"<p.I[;).
condition.
The expression for the left Haarmeasure
that a function Ip EU(R dt) is admissible
f(z)
where
is entire analytic.
This is the canonical
coherent states representation.
belongs to Bargmann's
space of analytic functions.
dt),
can also be written as an expression involving the Fourier transforms
(p = Fcp and I[; = Fi/t. The commutation
relations of Sec. 1 give
a) The
Annales de l'lnstitut
1R1,
a E
(tj; E U(IR,
tj;)
-l.lzI2
2
is
that hold for an arbitrary (admissible) analyzing wavelet, and then compute
it more explicitly for some special analyzing wavelets.
It is important to notice right from the start that the matrix element
The reproducing
equation shows then that the range of Lq>~consists
exactly of functions
that are i) square integrable
with' respect to
d(Re z)d(Im z) and ii) of the form
e
} .
= -2a
We now consider the case where G is the « ax
U is the irreducible representation
Z
211:
f
=F-0,
acting in the Hilbert space L 2(1R1,
dt). The left transform defined by U(b, a)
has arisen independently
in mathematical
analysis and in the study of
signals. Following Y. Meyer, we call it a wavelet transform.
In this section we first discuss some features of the wavelet transform
z=----brr.
(cp~.bl, cp~·.b·l)
1R1,bE 1R1,a
3.1. Wavelet transforms on the line:
i
firr
E
Its left Haar measure
general admissible
1 s
P(z, z')
+ ab'
b
3/4 e-x2/2,,2
The normalization
has been chosen so that II Cp"
The transform (Lq>~tj;)(s,b) can then be written
of the complex variable
kernel
with a
b},
dadb
dJlR(a,
1
The reproducing
= { aa',
dJI(a, b)
rr-1/2
=
of the real axis
ax+b.
+ h »-group is not unimodular.
> 0, and define the gaussian analyzing wavelet
cp,,(x)
GROUP
b»
It can be defined as the set of pairs {a,
with the group law
c) The transform Lq>: gaussian analyzing wavelet.
Choose
--+
301
REPRESENTATIONS
h» group is the group of affine transformations
xEIR1
with appropriate definition of convergence.
We mention finaly that the Lq>-transform allows an easy characterization
of « band-limited»
(or « Paley-Wiener ») functions.
GROUP
The function
If
f
Henri Poincare - Physique theorique
yl (Tb D a <p,
Vol. 45, n° 3-1986.
Ip)
of the « ax + h »-group shows
if and only if one has
7
I2 dadb '< 00.
302
TRANSFORMS
J. MORLET AND T. PAUL
A. GROSSMANN,
With the help of Sect. 2, one can calculate
II
I
=
(TbDa<p, <p) 12 d:~b
This means that a function
<p
L 2(1R,
=
2nI
[ cp(w)
Iw[
I
~~~
dw.
12
is admissible if and only if its
dt)
U(IR, IdW);
w [
Fourier
is
now transform belongs to
Cep
E
<p 1122n I
II
one sees that the number
12
Cep
= 0,
we see that it implies I <p(t)dt = 0; an integrable admissible
analyzing wavelet has zero mean. Consequently if the wavelet transform
is extended from L 2(1R, dt) to a larger space containing functions that are
constant over fR, the wavelet transform of all such functions vanishes.
Stronger statements hold if cp(w) has zeroes of higher order at w = O.
The general definition of LepljJ can now be written as
(LepljJ)(b, a)
b) Explicit expression
1
= vCep
;r:;- (TbDa<p,
ljJ)
GROUP
REPRESENTATIONS
<p
-
oj
=
(L'r~J)(h,
oj
and one can restrict one's attention to the half-plane a > O.
At this stage, we do not discuss decompositions involving symmetrization
or support properties, since such decompositions may affect the smoothness
of the functions concerned.
d) Isometry and inversion.
The general isometry equation becomes now
[ t/J(t) 12dt
I
=
If
I (L,pljJ)(b,
a)
127
dadb
which gives also
~ ~
= vCep
;r:;- (E -bD1/a<p,
ljJ).
=
I- ljJl(t)ljJ2(t)dt
for the transform.
If LepljJl(b,
a)LepljJ2(b,
a)7'dadb
The transform Lepis inverted, on its range, by its adjoint, i. e.
If the operators and the inner products are written out explicitly, one
finds
(L",ljJ)(b,
a)
I (b t)
= -JC: I .- ~
1- 1/2
= I aJC:
Ia
7p -a-
t/J(t)dt
e'bw cp(aw) ljJ(w )dw.
[1/2
These equations can of course be the starting point of a discussion of
the wavelet transform, shortcircuiting the generalized nonsense.
c) Reality
and symmetry
=
ljJ(t)
(A)
~
1
If
(Lept/J)(b, a) [ a
1, 1/2<p (t
-a b)
with suitable definition of integral.
e) Characterization
The function
properties.
of the range of Lep.
P,p()')
of Sec. 1 is now
P,p(b, a) =Cep1 [ a 1-1/2
In contrast to Fourier transform and to transforms based on the WeylHeisenberg group, the wavelet transform need not involve integrals over
Annales de I'Institut Henri Poincare - Physique tMorique
303
rapidly oscillating functions. This can be seen from the first equation (A),
and is one of the main features that singles out wavelet transforms among
more general left transforms, and makes them computationally and theoretically appealing.
If and t/J are real, then Lept/Jis real. There is no loss in generality in
assuming that t/J is real. There is also no loss of generality in assuming
that cp is real, since the admissibility condition is valid, separately, for the
real and imaginary part of cp.
Under the two assumptions above one has
(L,pljJ)(h.
<p
1
IATED TO SQUARE INTEGRABLE
dw.
If t is interpreted as time, then the admissibility condition says essentially
that the analyzing wavelet <p(t) has no zero-frequency component. Indeed,
regularity conditions on together with the assumption cp(O) =1= 0, imply
the divergence of the integral C",.
If, with slight loss of precision, we write the admissibility condition as
ijJ(O)
A
= ~C,p [a
Vol. 45, n° 3-1986,
f 7p (b-a- -
t) <p(t)dt
11/2f7P(wa)eibWcp(W)dW
dadb
a2
304
J. MORLET AND T. PAUL
A. GROSSMANN,
and the characterization
L~is
ljJ(a, b)
TRANSFORMS
of functions
ljJ(a, b) that belong
= velr r Pcp(b
-n-'
- b'
\..,..
n'
a \}ljJ(b', a')~.db'da'
..,."/
to the range of
Lio
This characterization
is important in practice, since it underlies
polation procedures adapted to the range of Lq>'
f) Wavelet
transform
of shifted
inter-
functions.
Consider the shifted « signal»
(L,prljJ)(b,
TXljJ. The definition
=
a)
undergoes
REPRESENTATIONS
305
= (M)2(~W)2
I]
the same displace-
fp(w) that effectively influence
to values of
(M)2
to
locali::ation.
1
at a given point b, a, correspond
GROUP
where
» representation;
components
INTEGRABLE
corresponds of course to musical intervals; for this reason wavelet transforms were called « cycle octave transforms ».)
If a is large, then (LcpljJ)(a, b) catches low frequencies of ljJ, in concordance
with the picture in t-space.
The above discussion has been voluntarily kept imprecise. The compact
sets I(<p)and J(iP) cannot be true supports of <p and iP and the definition of
effective support will depend on the problem on hand.
A natural rough measure of the (lack of) concentration
of is the « uncertainty parameter»
shows then that
The transform (Lcp) is non-local, in the sense that the integral (LcpljJ)
involves all values of ljJ. If however the analyzing wavelet is suitably « concentrated », then the value of LcpljJ at a given point of the cut (b, a)-plane
depends effectively only on the values of if; and $ in appropriate domains.
Assume that <p is in some sense negligible outside a compact set I(cp),
(<<effective support of cp »), and thatcp is in some sense negligible outside
a compact set J(cp); (<< effective support of rp »). Then the integral (LcpljJ)
can be effectively replaced by an integral over the set b - aI(cp). The arguments t of ljJ that effectively influence a given value (LcpljJ)(b, a) lie in the
« neighbourhood
» b - t E aI(cp). This neighbourhood
increases linearly
with a. If a is large, then (LcpljJ)(b, a) is a very smoothed-out
picture of ljJ,
showing only large-scale features of ljJ that lie in the set b - aI( cp).
The integral in w can be effectively replaced by an integral over
-J(ip).
The Fourier
a
TO SQUARE
(L,pif;)(b - x, a) .
If the function if; is displaced, its transform
ment along the b-axis of the cut b, a-plane.
g) L,plj; as (/ « t ill/('~fi'e(l(/ency
ASSOCIATED
(I)
(Lq>ljJ)(b,a)
that satisfy
1
UJE-J(fp).
(~W)2
= ~II
<p
II
= ~II
<p
II ft
f(t
-
to)21 <p(t) 12dt
I cp(t) 12dt
=~
II fp II
f(W
= --21
II fp II
f wi <p(w)
~
Wo
-
fp(w)
WO)2j
I
2
12dw
dw.
It is well known that
I]?-for every
1
4
<p.
h) Wavelet
transform
on the line:
special
analyzing
wavelets.
A gaussian is not admissible,
i) Linear combinations
of gaussians.
even after shifts and multiplication
by exponentials,
since its Fourier
transform does not vanish at zero. It is however easy to construct linear
combinations of gaussians that are admissible. The corresponding
reproducing kernels can be explicitly calculated. Some details can be found
in [3].
a
ii) Derivative
It is telling to use a logarithmic
scale for wand
to write
Schwartz
space
The derivative
is an admissible analyzing
of gaussians.
Y'(IR)
of any function in the
wavelet, as can be seen
d
by a look at the Fourier
log(w)
-
10gG)ElOgI(fp).
Consider
The relevant Fourier
bourhood
» of log (~)
transform
components
of ljJ lie in a fixed « logarithmic
(The logarithmic
distance
between
in particular
of -dt g.
the case where g is a gaussian:
neighg(t)
frequencies
Annates de I'lnslilut Henri Poincare - Physique·theorique
Vol. 45, n° 3-1986.
= exp
( _ t~)
30t>
TRANSFORMS
J. MORLET AND T. PAUL
A. GROSSMANN,
£12
= - -2
= (1 -
get)
£It
The wavelet transform with respect to this
t2)g(t)
0
=
cP
has a very simple intuitive
=-
-=
.jC",
1
(T
b
a 2
Dog,
I/J)
a2
= - --
(TbDag,
02I/J)
~
f g (t-a--
= - ~a3!2
b) I/J"(t)dt
iii)
Logarithmic
gaussian.
-
Let
ex
In this section we restrict our attention to the subgroup of affine transformations with positive dilation parameter a > O. The group law and
the two Haar measures are the same as for the fuIl group described
at the beginning of this chapter. It is weIl known [6] [7], that this group
has two faithful unequivalent irreducible unitary representations (up to
unitary equivalence). Both are square integrable in the sense of (I) and
we can apply the method of (I).
We will often use in this section the two foIlowing realisations of the
two representations of the « ax + b » group (a > 0) U ± defined on
U(IR +, x"- Idx) for n ?o 1, by
(D ±(a,
For fixed a, the function (L",I/J)(b, w) is a gaussian average of the negative
second derivative of I/J, and is thus reaIly a picture of the negative second
derivative of I/J, on a scale given by a. (Compare the less specific discussion
of section g).
If the function I/J has an isolated discontinuity at, say t = to, then the
function L,pI/J(b, a) (a constant and sufficiently smaIl) will try to imitate
the derivative of a Dirac b-function in a neighbourhood of h = to. The
imitation will be closer, and the variation of LtpI/J(b, a) more rapid, as a
tends to zero.
rhese facts have been used, (without the formalism of wavelet transforms)
in models of vision (edge detection) (We are grateful to R. Balian who
brought this to our attention.)
rhe methods described here aIlow the construction of appropriate
extrapolation procedures in the parameter a.
defrned as
> O. Consider the function
= exp( - ~ln2(W))
= 0
w
w
(V ±(a,
The function ga(w) belongs to 9'(IR). It has a zero of infinite order at
w= O.
Consider now ga = F- Iga and choose it as analyzing wavelet. The
a)
=
inen(n+ [l!Zaa-n(Lal/J)(b,
b)I/J)(k)
__
= an!zI/J (k~ + b) .
= X-1!2I/J(X)
(O/J)(x)
defined on the domain
f0
= { I/J E U(IR+,
This means that, for I/J1 and
we have
xn-1dx),
I/JZ in
f II/J(x)
12xn-Zdx
f0 and every
r
cp1
and
< +
cPZ in
CfJ }.
U( IR + , xn -
~~
7=
(CPl, U ±(a, b)I/J 1)(U ±(a, b)I/J z, CPz)
(L;oI/J)(a,
b)
= y'~C"'~ an!2. Jor+oo
1dx),
.
(cpt> CPZ)(CI/J2'
CI/Jr)
(R;oI/J)(a,
b)
=
e+ibXcpo(ax)I/J(x)xn-ldx
and
d
resultmg L",-transform La has the following basic property: If 0 = -dt then
(LaonI/J)(b,
an!2e±ibxI/J(ax)
and aIlows to define
< O.
.
=
An easy computation gives the fact that the (unbounded) operator defined
in [1] by (3.1) associated to U ± is given by
ga(w)
> 0
b)I/J)(x)
and their Fourier transforms defined by
J
ga(w)
307
REPRESENTATIONS
i) The group.
£It
(L",I/J)(b, a)
GROUP
.
d
interpretation. One has, with
TO SQUARE INTEGRABLE
3.2. Wavelet transform on the half-line:
affine coherent states.
and choose as analyzing wavelet the negative second derivu.[ive of g
cp(t)
ASSOCIATED
J C"'~
1 an!Z Jor+oo
cpo(x)e±ibxI/J(ax)xn-1dx
en/aa)
as isometric transform between L 2(1R+,xn-
which wiIl be exploited in a forthcoming paper.
Annales de /'InstitUi Henri Poincare - Physique theorique
Vol. 45, n° 3-1986.
Idx)
and L 2(IR +
x 1R'-2a
dadb)
.
309
A. GROSSMANN, J. MORLET AND T. PAUL
3CB
TRANSFORMSASSOCIATEDTO SQUARE INTEGRABLE GROUP REPRESENTATIONS
The reproducing kernels of the range of the transfon
given by the function
.
Pio(a, b)
(see 1(V)) IS
= c~~an!2 f e+ibXlpo(ax)lpo(x)xn-ldx.
_ they provide " natural setting for large N (or I) limits [5]
_ they allow us to introduce Wigner functions for the radial Schrodinger
equation [5].
llie following paragraph is devoted to a family of functions
which
gives a characterization of the range or L';o in terms of analyticity in the
nriable z = b + ia (affine coherent states).
CPo
_
b»
o
In this section we consider the representation U _ and the family of
analyzing wavelets» given by the functions
=
lpl(X)
we can consider the following realisation of the two representations of
« ax + b » W +, W _, obtained by restricting the metaplectic representation
1.
l2:
xl-1e-x
This family of analyzing wavelets gives rise to the following transforms
+ib~
(Vl{i)(a, b)
CljJ
=
f
=.
2 2
J2nr(2l
a
+ 2n)
2
1+!!.-li+CO
0
with the following choice of
x1-1e'bX-aXl{i(x)xn-1dx,
1+!!.-1
2
•
f 1 f(a,
analyzing wavelets»
l>
O.
b)
12
7
dadb
=
11l{i
112
.
V is exactly the set of func-
I+!!. -1
tii>ns which are the product of
a
They give rise to affine coherent states very well adapted to the radial
harmonic oscillator ([5 D·
REFERENCES
It turns out [5] that the range of the transform
2
by an analytic function, square
.
dadb
Integrable with respect to the two-dimensional measure
analogy with Bargmann space.)
Let us call HI the range of V i. e.
-2-
J. MORLET and T. PAUL, Journ.
[2] T. PAUL, J. Math. Phys., t. 25, 1984, p. 3252.
[11 A. GROSSMANN,
canonical
(Note the
a
f(a, b)
=
a2
!!.+1-1
g(z),
=
z
b
+ ia,
a
g analytic with
r
I
f(a, b)
transformations.
L. S. Ferreira
In : Resonances.
and L. Streit,
editors.
Models
Springer,
~dadb
U
t. 26, 1985, p. 2473.
t. 23, 1984-1985, p. 85.
of the group of affine
and Phenonema.
S. Albeverio,
Lecture
[5] T. PAUL, Affine coherent states and the radial Schrodinger
prints,
12
Phys.,
Notes
in Physics,
Vol. 211,
1984, p. 128.
> 0,
J
Math.
[3] P. GOUPILLAUD, A. GROSSMANNand J. MORLET, Geoexploration,
[4] A. GROSSMANN and T. PAUL, Wave functions on subgroups
oscillator
<+
ctJ 1
(.
)
and hydrogen
Luminy.
mi>ving on the half-line, or of a particle with radial hamiltonian.
The family of spaces HI has been used in [5] and has many properties
of the Bargmann space described in section (2):
they solve the time-independent Schr6dinger equation for the hydro-
gel atom in an explicit way [5]
Annales de I'lllslilu! Henri Poincare - Physique theorique
atom,
Submitted
II Large
to Annals oj
N limit and
I. H.
equation 1. Radial harmonic
Wigner Junctions, pre-
III Affine
P.
[6] 1. M. GELFAND and'M. A. NAIMARK, Dokl-Akad-Navk
SSSR, t. 55, 1954, p. 570.
[7] E. W. ASLASKENand J. R. KLAUDER, J. Math. Phys., t. 9, 1968, p. 206.
(Aianuscrit
re,'u Ie 6 mars 1986)
HI can be considered as the state space of quantum mechanics of a particle
-
2lj1(a1!2x)
= x1e-X2
ljIl(X)
The general theory exposed in (I) and recalled in
tte introduction gives that
{
«
•
an!4e -
is clearly the product of an analytic function of the variable
= b + ia by a
H=
=
(W ±ljJ)(X)
l+!'. -1
Z
group as a subgroup of SL(2, R)
fi
ill Affine coherent states.
«
+
If we consider the « ax
via the parametrization
Remark.
Vol. 45, n° 3-1986.