Downloaded 14 Dec 2005 to 131.215.225.9. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp
Bargmann
transform,
Zak
transform,
and
coherent states
A.J.E.M. Janssen
California Institute
of
Technology, Pasadena, California 91125
(Received 7
November
1980;
accepted
for
publication
11
March
1981)
It
is well
known
that
completeness properties
of
sets
of
coherent
states
associated
with
lattices in
the
phase
plane
can
be
proved
by
using
the
Bargmann
representation
or
by
using
the
kq
representation
which
was
introduced
by
J.
Zak.
In
this
paper
both
methods
are
considered, in
particular,
in
connection
with
expansions
of
generalized functions in
what
are
called
Gabor
series.
The
setting consists
of
two
spaces
of
generalized functions (tempered
distributions
and
elements
of
the
classs·)
which
appear
in a
natural
way
in
the
context
of
the
Bargmann
transform.
Also, a
thorough
mathematical
investigation
of
the
Zak
transform
is given.
This
paper
contains
many
comments
and
complements
on
existing literature; in
particular,
connections
with
the
theory
of
interpolation
of
entire
functions
over
the
Gaussian
integers
are
given.
PACS
numbers:
02.30.Mv,
02.30.Lt
1.
INTRODUCTION
If
xER, yER,
then
G (x,y) denotes
the
function
(G (x,y))(t) =
2lexp(-
1T(t-
x)
2
+
21Tiyt-
1Tixy)
(tER);
G (x,y) is called a
coherent
state,
1
'
2
or
also a
Gabor
func-
tion.
3
.4
In
the
past
ten
years a
number
of
papers
1
•
2
•
4
-
7
ap-
peared
about
the
completeness
of
the
collection
! G (na,m(3 )ln,m integers l
where
a>
0,
(3
> 0.
These
papers
deal
with
the
following question:
if
/is
a (generalized) func-
tion
and
(f,G
(na,m(3
))
=
Oforall
integersn
andm,
then
does
it follow
that
f
_o?
As
early as 1932, von NeumannH noticed
(apparently
without
publishing a proof}
that
the
answer
is
"yes"
if
fEL
2
(R), a(3 =
1.
Two
proofs
of
this
fact were given
in 1970 by using
the
Bargmann
transform,
9
and
in 1975 a
proof
was given
by
using
the
kq representation.
The
most
complete
answer
to
the
above
question
was
probably
given in
1979.
It
is
shown
7
that
(f,G
(na,m(3)) = 0 for all integers n
and
m implies f 0 for a very large class
of
generalized func-
tions
/whenever
a(3 <
1.
Also, in case a(3 =
1,
a
character-
ization
of
all
tempered
distributions
/with
(J,G
(na,m(3
))
= 0 for all
nand
misgiven.
The
main
tools
are
a
Phragmen-
Lindelof
theorem
and
the
Bargmann
transform,
although
the
latter
is
not
explicitly mentioned.
A related
question
concerns
expansion
of
(generalized)
functions
fin
series
of
the
form
~n.m
cnm
G (na,m(3)
with
a(3 = 1
(Gabor
series).
In
1946
Gabor
10
suggested a
simulta-
neous
time-frequency analysis
of
signals based
on
these ex-
pansions.
In
1979 existence
and
uniqueness
theorems
about
Gabor
series
were
given
(cf.
Ref.
6,
where
expressions for
the
coefficients
cnm
are
given,
and
Ref. 4,
where
existence
of
Gabor's
expansions for
tempered
distributions
is proved; in
both
papers
the
kq
representation,
although
not
explicitly
mentioned,
plays
an
important
role).
We
give a survey
of
the
content
of
this
paper.
In
Sec. 2
we
consider
the
spaces S
of
smooth
functions
and
S •
of
gen-
eralized functions,
and
we
show
that,
in
connection
with
the
Bargmann
transform,
these spaces arise in a
natural
way.
In
Sec. 3
the
Zak
transform
T,
which
maps
functions f
of
one
real variable
onto
functions T
/defined
on
the
unit
square, is
introduced
and
studied
in
detail. A
peculiar
property
of
the
Zak
transform
11
is
the
following one:
if
fEL
2
(R)
and
T f is
continuous,
then
T
fhas
a
zero
in
the
unit
square.
In
Sec. 4 a
number
of
consequences
of
this
property
are
given.
One
of
the
consequences is
that
one
can
improve
the
convergence
of
Gabor
series (which, in general, converge
not
even in L
2
sense for elements
of
S)
by
shifting
the
lattice
over
a
distance
(a,b)
with
suitably
chosen
numbers
a
and
b.
Also,
the
results
about
completeness
after
deleting
one
or
more
coherent
states
2
•
5
•
7
are
completed
and
generalized,
and
a relation
with
classical results in
interpolation
theory
is indicated.
Al-
though
almost
all results deal
with
square
lattices
of
unit
area
with
axes parallel
to
the
x
andy
axis in
the
phase
plane,
some
indications
are
given
how
to
handle
general lattices.
Finally,
the
paper
shows existence
of
Gabor's
expansion for
elements
of
S
•.
2.
THE SPACES
SANDS
• AND THE BARGMANN
TRANSFORM
In
1961
Bargmann
12
constructed
a
unitary
mapping
of
L
2
(R)
onto
the
set F
of
all
entire
functions
/of
growth
<(2,!)
for
which
f
(:
I
f(zWe-
lzl'
dz <
oo.
On
the
space F
Fock's
so-
lution
5 =a
;a'T/
of
the
commutation
rule
[S,rJ]
= 1 is real-
ized.
In
1967
Bargmann
13
described several spaces
of
test
functions
and
generalized functions in
terms
of
certain
sub-
sets
ofF
and
duals
of
these.
In
particular
the
spaces
SandS'
(Schwartz's space
offunctions
of
rapid
decrease
and
oftem-
pered
distributions
respectively) were considered.
In
this
section we
shall
investigate
the
relation between
Bargmann
transform
and
the
spaces
SandS*
(of
smooth
and
general-
ized functions respectively)
which
were
introduced
in Ref.
14
and
studied
extensively in Refs. 3, IS,
and
16.
2.1.
The
spaceS
consists
of
all
entire
functions
/for
which
there
are
M > 0,
A>
0, B > 0
such
that
(*)1/(x + (Y)I<Mexp(
-1TAx
2
+
1TBy
2
)
(xER,yER).
s
A sequence (f,, ),
inS
converges
to
zero
inS
sense(/.,
---+0)
if
there
are M > 0, A > 0, B > 0
such
that
(
*)
holds
for all
/,
and
such
that
J,,---+0
pointwise.
The
spaceS*
consists
of
all con-
tinuous
antilinear
functionals defined
on
S.
The
action
of
FES •
on
/ES
is
denoted
by
(F,
f).
A sequence (F, ), in S •
720
J. Math. Phys. 23(5), May
1982
0022-2488/82/050720-12$02.50
©
1982
American Institute
of
Physics
720
Downloaded 14 Dec 2005 to 131.215.225.9. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp
s·
converges
to
zero
inS·
sense
(F"
~)
if(F"
,f)~
for all /ES.
Note
that
G (x,y) ES for all
xEIR,
yEIR.
2.2.
For
n = 0, 1,.··,¢" denotes
the
nth
Hermite func-
tion. We choose
our
normalizations such
that
17
exp(1rx
2
-
21T(X-
w)
2
)
= f cnw"rf'n(x)
(xEC,
WEC),
n=O
where
en
=
2-
114
(41T)"
12
/(n!)
112
for all n. We have
rf'nES.
There
is a one-to-one correspondence between
the
spaceS
and
the
spaceD
of
all complex sequences
(an
ln
with
an
= O(e-
"')for
some € >
0;
if
JES, then
((/.rf'n
))nED,
and
if
(an
)nED,
then
'!.nan
rfn
converges
inS
sense
to
an
element
of
S.
There
is a similar correspondence between the space S •
and
the
spaceD·
of
all complex sequences
(bn
ln
with
bn
=
O(en<)
for all € >
0.
It
follows
18
that
SC
Sand
that
S'cs·.
2.3. A different way
to
describe the
spaceS
is the follow-
ing one:
In
Ref.
14
the
spaceS
fi
(a > 0,
(3
>
0)
is
defined as the
set
of
all functions f
IR-C
for which there exist C > 0, A > 0,
B > 0 such that
(**)lxk Jlql(x)i
<;CA
kB
qk
kaqqf3,
for all
XEIR,
k = 0, 1,···,q =
0,
1,-···
Our
spaceS
can be identi-
fied with S
:~~
as follows:
If
JES, then the restriction
of
/to
lR
satisfies inequalities as in(**), and
if
we have an f
IR-C
satisfying inequalities as in(**), then
jean
be extended to an
entire function satisfying an inequality as in 2.1(*). Also,
the
notions
of
convergence in
2.1
for
Sand
in Ref.
14
for S
:~~
can be shown
to
be equivalent.
We note some topological properties
of
the
spaces
Sand
S
•.
If
we
considerS
and S
·with
the weak * topologies, i.e.,
with the linear topologies generated by all sets
of
the form
[ JES I(F,J)EOj (where
FES*,occ
open) and [FES*I
(F,/)EOJ (where JES, OCCopen), then
thedualofSisS*
and
the dual
of
S •
isS.
The
spaceS
(S
*)
is complete in
the
sense
that
if
In
ES
(F"
ES
*)
and
limn~=
(F,f") [limn
'"'
(F"
,f)]
ex-
ists for all
FES *(JES
),
then there is an
/ES
(FES
*)such
that
(F,f)
=
limn--oo
(F,fn)
((F,f)
=
limn~oo
(F"
,J)]
for all FES •
(JES).
More
information can be found in Ref.
15.
To
indicate how big the
spaceS
• is,
we
observe
that
any
measurable
F:IR-C
for which S':
oo
exp(-
Et
2
)IF(t
ll
dt
<
oo
for all € > 0 can be regarded as an element
of
S • by
putting (F,f): = S':
oo
F(t)
f(t
).
dt
for fES.
2.4. We give a list
of
operators
of
S:
if
fES,aEC,
bEC,
a>O,
then
(Ta
/)(t)
=
f(t
+a),
(Rb
f)(t)
=
e-
2rrtb'f(t
),
(NaJ)(t)
=
(-.-
1
-)
112
smha
xJOO
exp(
.-
1T
((t
2
+ z
2
)cosha-
2tz))f(z)
dz,
-
oo
smha
(.7
f)(t)
=
f_"'
=
e-
Zrrtrz
f(z)
dz,
(Y*
f)(t)
=
(Y
/)(
- t
),
(P
f)(t)
= (l/21Ti)j'(t
),
(Qf)(t)
=
tf(t
),
for
tEC.
1
"
These operators are continuous and have adjoints
that
mapS
into
S;
they can therefore be extended to continu-
721
J. Math. Phys
..
Vol. 23,
No.5,
May 1982
ous linear operators
of
S
•.
20
2.5. Definition:
For
FES*
the
Bargmann transform
BF
ofF
is defined by
(BF)(z) = el",(TzF,g)
(zEC).
Here g(t) = 2
114
exp( -
1Tt
2
)
for
tEC.
2.6.
The
following formula (for FES
')
is due
to
Barg-
mann
12
;
for
the
sake
of
completeness we give a proof.
Theorem: Let
FES
•.
Then
00
(F,rf'n)
(BF)(z) = ""
--(1r
112
z)"
(zEC).
n~o
(n!)
112
Proof
Put
hzlt) =
2
114
exp(~1Tr-
1r(t-
z)
2
)
for
zEC,
tEC.
Since (BF)(z) = if,h.) and
hz(t) = 2
114
exp(1Tt
2
-
21T(t-
~zl)
= f
rPn(t)
(1T
1
1
2
2
)n
- n
~
0
(n!)
1
12
'
with convergence
inS
sense for every
zEC,
we have
(BF)(z) = (Fh-) =
_f
(F,¢n)(1Tll2
2
)n.
' z n
~
0
(n!)
1/2
2.
7.
Let
3"
be the space
of
all entire functions
of
growth
<(2,1T/2), and let
11
be the measure on C defined by
dJ1(z)
=
e-
rrlzl' dz.
If
JY'
= 3"nL
2
(C,J1),
then
JY'
is a Hilbert
space for which
(~"z"
lv
n!)n
is
a complete orthonormal sys-
tem,
and
B maps L
2
(R)
isometrically onto
JY'.
12
Also,
13
B(S)
= [
/EWI/(z)exp(-
!1Tizi
2
)
=0((1
+ izi)-N)
forallN>OJ,
B(S)
=
[/EWI/(z)exp(-
!1Tizi
2
)
=0((1
+
izi)N)
forsomeN>OJ.
Theorem:
(i)
B (S) =
[jEW
I growth
off<
(2,1T/2
J,
(ii)
B(S*)=3".
Proof Let
FES.
There
is
an € > 0 such
that
(F,¢") = 0
(e-
"'). Hence, by Stirling's formula
[(F,rf'n
J/(n!)
1/2
]1Tn/2
= 0
(n-
I/4(1T/n)"l2e-
nl<+
ll).
It
follows from Ref. 21, Theorem 2.2.2
that
B
/has
growth
<
(2,1T/2).
Conversely, let jE3", growth
off<
(2,1T/2).
Writ-
ing f(z) = '!.nanz" we know from Ref. 21, 2.2.10
that
lim sup
n
ian
1
21
"
<
1Te.
Hence
bn:
=an
1T"
12
(n!)
112
= 0
(e-
nE)
for some
€>0.
SoifweputF=
'!.nbnrf'n,
thenFESandBF=
f
The
proof
of
(ii)
is
similar.
Remarks: (
1)
There are similar characterizations for the
elements
of
B (
'G')
and
B
(A')
(
'G'
is the convolution class and
(A')
is the multiplication class; cf. Ref.
16).
It
may be shown
that
B('G')=
[/E3"1V'P>l3q<l[f(x+iy)
= 0
(exp(1rqx
2
+
1rpy
2
))]
J,
B
(M)
= [ /E3" J
IV'P>
J3q<l
(f(x
+
1}1)
= 0 ( exp(
1rqy
2
+
1rpx
2
))]
J .
(2)
Theorem 2.7 shows
that
Theorem 2.8 in Ref. 7
is
in
some sense the best possible result
that
can be obtained by
using the Bargmann transform.
2.8.
In
the list below we have FES ·, aEC,
bEC,
a>
0,
zEC.
(1)
(BTaF)(z) =
e-
Jrra'-
rraz(BF)(z
+a),
(2)
(BR
6
F)(z) =
e-
Jrrb'-
,-tbz(BF)(z- ib
),
A. J. E. M. Janssen
721
Downloaded 14 Dec 2005 to 131.215.225.9. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp
(3)
(BNaF)(z) = e
-la(BF)(e-
az),
(4)
(B.YF)(z) = (BF)(iz),
(5)
(BPF)(z) = !iz(BF)(z) + (1/21Ti)(BF)'(z),
(6)
(BQF)(z) = ¥(BF)(z) + (l/21T)(BF)'(z),
(7)
(B(Q-
iP)F)(z) =z(BF)(z),
(8)
(B(Q + iP)F)(z) = (l/1r)(BF)'(z),
(9)
(B(Q
2
+ P
2
)F)(z)
= (1!21T)(BF)(z) + (z/1r)(BF)'(z).
The
proofs
of
these formulas
are
straightforward;
compare
also Ref.
12
where B U is calculated
with
U a canonical oper-
ator
associated
with
a symplectic
transformation
of
the
phase plane,
and
Ref.
3,
Sec. 27.3.
The
obvious
advantage
of
the
spaceS·
over
Y'
is
that
we
can
consider
in (
1)
and
(2)
complex values
of
a
and
b.
The
obvious disadvantage is
the
fact
that
S • is described in
terms
of
entire functions so
that
its elements
are
hard
to
localize.
Nevertheless, it
appears
that
one
can
say
at
least
something
22
about
the
carriers
of
the
elements
of
S • with
the
aid
of
the
Bargmann
transform
and
the
theory
of
analytic functionals.
Some
other
useful formulas
are
(BG
(x,y))(z)
= exp( -
~1T(x
2
+ y
2
)
+
1r(x
+
iy)z)
(zEC),
(Bo~
1
)(z)
=
(-
1)k(k
!)
112
(21T)k
12
¢dz/v2)
(zEC)
fork=
0,1,. ...
For
FES·, fES, aER, bER,
(F,G (a,b))
=
exp(-
!1T(a
2
+ b
2
))(BF)(a-
ib
),
(F,f)
=
Le-
7Tizi'(BF)(z)
(B/)(z) dz,
so
that
(integration
over
R
2
)
(F,f)
= J
f(F,G
(a,b
))(G
(a,b
),/)
da
db,
which agrees
with
the
formula
27 .12.1. 5 in Ref.
3.
3.
THE ZAK TRANSFORM
In
this
section we
study
the
Zak
transform
which was
introduced
in 1967 by
Zak
to
construct
a
quantum
mechani-
cal representation
(kq
representation) for
the
description
of
the
motion
of
a Bloch electron in
the
presence
of
a
magnetic
or
electric field.
23
-
25
This
representation
can
also be used for
the
quantum
mechanical description
of
angle
and
phase.
26
The
Zak
transform
T
maps
functions f defined
on
R
onto
functions T f
of
two
variables as follows:
(T
f)(z,w) = I
f(z-
n)e-
Z1Tinw.
n = - oo
Zak
denotes
the
first variable (quasiposition variable)
by
q
and
the
second variable
(quasimomentum
variable)
by
k.
We
consider
here T as a
mapping
of
L
2
(R)
into
L
2
([0, 1 ]
2
),
and
also as a
mapping
of
S •
into
S
2
. [and
of
5'
into
(5
2
)']. Al-
though
the
Zak
transform
looks,
at
first sight, less interest-
ing from
the
mathematical
point
of
view
than
does
the
Barg-
mann
transform,
it
pays (as we shall see in
the
next
section)
to
investigate its properties systematically. A striking
property
is
that
T
/has
a zero
in
[0,
1f,
provided
that
T
/is
continu-
ous.
We
further
give a
formula
for
the
product
T
F·
T f (in
case this
makes
sense)
which
is very convenient
when
prov-
722 J. Math. Phys., Vol. 23, No. 5, May 1982
ing completeness properties,
and
we
determine
T (S
),
T (S ·),
T(5),
and
T(5').
3.1. Definition:
Let
FES
·.We
define
TF:
= !
Ti~nR
~
1
(F®H),
n = - oc
where H = 1 [for
the
definition
of
the
tensor
product,
cf. Ref.
15,
Appendix
1,
1.17; we
have
(F
1
®F
2
, /
1
® /
2
)
= (F
1
,/
1
)(F
2
,/
2
)
for F;ES·,
J;ES(i
= 1,2)].
This
definition
makes sense, for
if
FES·, /
1
ES,
/
2
ES,
then
!
(T
1
~
nR
~
1
(F®H),
/1 ®
/2)
n = - oo
n = -
oo
converges absolutely by Ref. 16,
Theorem
5.5. By Ref. 15,
Appendix
1,
Theorem
3.7, the series
};n
T
1
~
nR
~
1
(F®H)
converges unconditionally
inS
2
. sense.
It
also follows from
Ref. 15,
Appendix
1,
4.14
that
Tis
a
continuous
linear
map-
ping from S • into S
2
.,
and
we have F = 0
if
TF = 0. Similar
things
hold
if
we consider T as a
mapping
from
5'
into
(5
2
)'.
In
the
case
/E5,
T f
can
be identified with the function
I
e-l1Tinwf(z-n)
[(z,w)ER
2
].
n-=-
-
oo
3.2.
Part
of
the
following
theorem
is
taken
from Ref.
4;
for
the
sake
of
completeness we include a proof.
We
also note
that
part
(i)
occurs in a
more
abstract
version in
the
proof
of
Ref.27,
Chap.
1,
Sec.
5,
Lemma
4.
Theorem:
(i)
TmapsL
2
(R)isometricallyontoL
2
([0,
1]
2
).
(ii)
Let
1
<p
<
2.
Then
T
maps
L
P(R)
into
L
P([O,
1f),
and
the
operator
norm
<
1;
Tis
injective
but
not
surjective.
Proof
(i)
Let
fEL
2
(R).
Since
the
functions
f(z-
n)e-
2
"inw
are
orthogonal
over
[0,
if
we see
that
.C.C
I(Tf)(z,wW
dzdw
n~~
oo.c.cl/(z-
n)e-
27Tinwl2
dz dw
=
f~"'
I
/(zW
dz.
Hence
Tis
well defined as a
mapping
of
L
2
(R)
into
L
2
([0,
1 )
2
),
and
it
is norm-prt-""·-·;ing.
Now
let gEL
2
([0,
1
JZ),
and
let
cnm:
= ililg(z,w)e21Timz +
21Tinw
dz dw
for integers n
and
m.
Putting
/(z
- n
):
=
};m
c
nm
e-
2
1Tinz
for
O<;z
< 1
and
integer n, we easily see
that
/EL
2
(R),
and
that
Tf=g.
(ii)
Let
/EL
1
(R).
Then
.C.CI(T/)(z,w)l
dzdw
<.C.C};n
1/(z-
n)l
dzdw
=
f~
"'1/(z)l dz.
Hence
T fEL
1
([0,
if)
and
II
T
/11
1
<II
/11
1
-It
follows from con-
vexity
theory
that
T
maps
L
P(R)
into
L
P([O,
1j2),
and
that
A.
J.
E.
M.
Janssen
722
Downloaded 14 Dec 2005 to 131.215.225.9. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp
IITJIIP<Il!IIP for
fELP(JR),
l<;p<.2.
Toshowinjectivity, let
/EL
P(JR),j
#0.
Let/,
EL
2
(lR)
be
such
that
1/,
1<1/,
+
1
1
and
fn-J,
Tfn-T
f a.e. By Fa-
tau's
lemma
and
part
(i),
ff1(T/)(z,w)l
2
dzdw
>li~~s~p
ff
I(Tf")(z,wW
dzdw
=
li~~s~pllfn
II~>
o.
It
is
trivial
that
Tis
not
surjective
if
I
<p
<
2;
otherwise
we would have
T(L
P(JR)):J
T(L
2
(lR)),
whence£
P(JR):JL
2
(lR).
Remarks: (I)
There
is
no
way to define
Tas
a mapping
of
L
P(JR)
into any L '((O,lf)
ifp
> 2
(cf.
Ref. 28, Chap.
XII,
2,
p.
I02).
(2)
As a mapping
of
L
P(JR)
into L
P([O,
1]
2
)
with
1
<.p
<
2,
Tis
not bounded below.
To
see this,
put
fc
: =
~nenXJn,n+
11
fore=
(en)nE[P.
IfTwerebounded
below
there would be
an
m > 0 such
that
liT
fc
ll>mllfc
liP=
m~n
len
IP)IIp
for
eE/P.
As
(T
fc)(z,w) =
~nene-
Zrrinw
for
eEIP,
this implies
that
[
~nene-
ZrrinwleEfP]
= L
P([O,
1]).
Contradiction.
3.3.
In
the
list below we have
FE.S
·,
aEC,
bEC,
a>
0,
(1)
T(TaF)
=
T~
11
(TF),
(2)
T(RbF)
= R
~
11
T~
1
(TF),
(3)
T\
1
'(TF) = R \
21
(TF),
(4)
n
21
(TF) =
TF,
(5)
T(NaF)
=N(II(
~
e-rrn'coshasinha(R
..
T
F)""H)
a
~
msmha
-
ncosha
'01
'
n = -
oo
(6)
TYF=e-lrrizwUTF,
(7)
T(PF) = P(
0
TF,
(8)
T(QF)
=
(Q(
11
+
P(2))TF.
Here U
is
(the extension
of)
the mapping
that
takes f(z,w)E.S
2
into
f(w,
-
z).
All formulas except
(6)
follow directly from a
computation.
To
prove
(6)
we
first take an
JE.S.
We have by
definition
(T.Y j)(z,w) = I
(.7
j)(z-
n)e-
2rrinw.
n=-
-:n
Observing
that
(.7
f)(z
- n
)e-
Zrrinw
= e -
Zrrizw(
.
.7
Rz
Tw
f)(-
n),
we
get by
the
Poisson summa-
tion formula
(T.Y j)(z,w) =
e-
27Tizw
I e2rrinzf(w-
n)
n-=-oo
=
e-
zrrizw(T
f)(w,-
z).
For
the
general case take a sequence (
fn
),
inS
that
converges
inS·
sense
to
F,
and
use continuity
ofT
(cf.
3.1
).
Remarks:
(1)
Formula
(6)
and
Theorem 3.2
(i)
give a
quick
proof
of
Plancherel's theorem since U maps L
2
([0,
1
fl
unitarily
ontoL
2
((0,1] X
[-
1,0]).
It
is
of
course
the
Poisson
summation formula
that
does
the
trick here.
(2)
If
T
fis
suffi-
ciently well behaved
we
can recover f
and
Y
fby
integra-
tion. We have
723
J. Math. Phys., Vol. 23, No.
5,
May 1982
j(z)
=
((Tf)(z,w)
dw, (.'7
f)(-
w)
Jo
= L ( T f)(z,w)e
2
1TIZI''
dz,
for
zElR,
WER
3.4. We calculate TG
(x,y)
and
Tl(;,
for
xElR,
yElR,
n =
0,
J,
.... We have by the formulas
of
3.3
(TG(x,y))(z,w) =
(T(e-
rrixyR
YT
xg))(z,w)
= e
rrixy
t
2rriyz(Tg)(z
_
X,W
_ y),
so
that
we
need
Tg.
In
general, we have by the generating
function
of
the Hermite functions
(cf.
2.2),
edTth){z,w)
=
c,,
[
e1TZ
2
-
2rr(z-
t)'~e
-1Tn
2
+
2rrin(w+
iz-
lit)]
=
C,,
[
e3(w
+
iz-
lit
)/~Oe,t
'if;,(x)
]·
Here
t9
3
(z)
=
~,exp(-
1rn
2
+
21rinz)
is
the 3rd theta func-
tion [in
the
notation
of
Ref. 29
we
have
t9
3
(z)
=
1J
(1rz,e-
1T)].
By
the
Taylor expansion
of
8
3
around the point w + iz we get
(Tl(;k)(z,w)
=,to
l(;~;:z)
e)
11
\
-1T-
112
i)
1
8~
1
(w
+
iz).
In particular,
(TG (x,y))(z,w) =
(G
(x,y))(z)8
3
(w
+
iz-
y-
ix),
and in case n and m are integers
we
get by 3.3,
(TG (n,m))(z,w) =
(-
l)"me2mmz-+-
2rrinwe-
rrz'e,(w
+
iz).
As another example, let
e"
(t
):
=
e-
Zrriar
whereaElR. We
have
00
T(ea) = L
ea
®D,_·a'
Hence,
if
a
is
an integer, T(ea) =
ea
®
~,8,,
and
if
JE.S
·is
periodic with period one, then T
f=
/®
L,D,.
Similarily, if
fis
a function
of
the form f =
~,e,t5,,
then T f =
e-
Zmzt"
(~,8,)
® .
.7
f
3.5.
It
is
easy to see
that
T
fhas
a zero in
[O,lf
if
T
fis
continuous
and
fis
real-valued,
or
even,
or
odd,
or
a
Gabor
function.
The
following theorem shows this
is
general.
Theorem:
Let fEL
2
(lR)
be such
that
T
fis
continuous.
Then T
fhas
a zero in
[0,1f.
ProoP
0
:
Assuming
(T
f)(z,w)#O for (z,w)E[O,lf we can
write
(T
j)(z,w) =
elr.-i<p(z,wl,
where
tp:lR
2
-c
is
continuous. Indeed, this follows
at
once
from Ref. 31,
Part
VI, Sec.
1,
Lemma
7.
We have
by
3.3
(Tf)(z
+
1,w)
= e
2
1TI"'(Tf)(z,w),
(T
f)(z,w +
1)
=
(T
f)(z,w),
for (z,w)E[O,lf, Hence, for some integers k
and
I
tp
(z
+
1,w)
=
tp
(z,w)
+ w + k,
tp (z,w +
1)
= tp
(z,w)
+ l,
for (z,w)E[O,lf. Calculating
q:>(l,
1)
in two different ways, we
A. J.
E.
M. Janssen
723
Downloaded 14 Dec 2005 to 131.215.225.9. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp
get
IP(O,O)
+ k
+I=
IP
(1,1)
=
IP
(0,0)
+ k
+I+
1.
Contradiction.
Remark:
Iff=
x
1
o,
1
1
,
then I(T f)(z,w)l = 1 for all
(z,w)eR
2
•
3.6.
In
the remainder
of
this section we determine
T(S
),T(s·), T(S),
and
T(S'). We first give a formula which
will also be used
in
Sec. 4 to answer questions about com-
pleteness (also cf. Ref.
6).
Theorem:
Let
(1)
FeS
·,
geS
or
(2)
FeS',
geS
or
(3)
FeL
2
(R),
geL
2
(R).
Then
I
(F,R
- m
T-
n
g)e21Tiriw
+
21Timz
=
TF·
Tg'
n,m
where the identity is to be interpreted
inS
2
•
for case (
1)
and
in (8
2
)'
sense for case
(2).
For
case
(3)
the identity must be
interpreted in the sense
that
the (mn)th Fourier coefficient
of
TF·
Tg
equals
(F,R
_ m T _"g).
Proof First take FeS, geS. Noting
that
(F,R_mT-ng)e
2
1Tirnz=
CrTz(F·
T
__
"g))(m)
for all n and m we get by the Poisson summation formula
(applied to the summation over
m)
I
(F,R
_ m T _ n
g)e21Tinw
+
21Timz
n,m
=
IF(m
+z)
g(-
n + m
+z)
e
2
1Tinw.
n,m
Now the formula easily follows by first summing over
nand
then over
m.
For
the general case (i.e.,
FeS
·)take
a sequence
(Fk
)k
in
S which converges to
FinS·
sense.
It
follows as in the proof
of
Ref.
16,
Lemma 5.2
that
for
every£>
0 there are positive
numbers
M
and
f3
such
that
I
(Fk
,R-
m T -"g) I
<,M
IIN(JFk
llzexp(1T£(n
2
+ m
2
)),
for all n,m, and k (note that
NfJFk
eS
for
f3
>
0).
Since
IINfJFk
11
2
is
bounded
ink
for every
f3
> 0 it is not
hard
to
complete the proof
of
the theorem for case ( 1
).
The proof for case
(2)
is similar to
that
for case
(1).
For
the
proof
of
case
(3)
we take
Fk
and
gk
inS
with
Fk----o.F,
gk-g
in L
2
(R)
sense. Now we note
that
- -
TFk·
Tgk
-TF·
Tg in L
1
([0,1f)
and
use the result already
proved with
Fk
and
gk
in the role
of
FeS, geS. Hence, the
(mn)th Fourier coefficient
of
TF·
Tg
is given by
(F,R
-m
T _"g).
3.7.
Theorem:
T(S)
equals the set
of
all entire functions
IP
of
two variables such
that
IP(z
+ 1
,w)
=
e-
l1TiwiP
(z,w),
IP
(z,w
+
1)
=
IP
(z,w)
for all
(z,w)e(:Z,
and such
that
there are
M > 0, A >
0,
B > 0 with
I<P
(x
+ iy,u +
iv)l
<,Mexp(21TXV
+
1rAy
2
+
1rBv
2
).
Furthermore, T
(S)
equals the set
of
all
q?EC
=(R
2
) such
that
<P
(z
+ 1
,w)
=
e-
2
1T;"'IP(z,w),
<P
(z,w
+
1)
=
<P
(z,w)
for all
(z,w)eC
2
•
Finally,
T(S•)
=
!FeS
2
.IT\'
1
F=
R \
21
F,T\
21
F=
F
l,
and
T(S') =
I
FE(S2)'1
T\llp
= R
1121p,
T\21p
= F ].
Proof LetfeS.
It
is
clear
that
Tfis
an entire function
of
two variables. Take K >
0,
C >
0,
and D > 0 such that
I
f(x
+
iy)
I <,Kexp( -
1rCx
2
+
1rDy
2
)
(xeR,
yElR).
724
J. Math. Phys., Vol. 23,
No.5,
May 1982
Then for real x,y,u,v,
I~,
f(x
+
iy-
n)e-
21Tin(u-+
i"'l
<,K
~"exp(
-1TC(x-
n)
2
+
1rDy
2
+
21Tnv)
=
Kexp(21TXV
+
1rv
2
/C
+
1rDy
2
)~,
Xexp(-
1rC(x
+viC-
n)
2
),
whence T (S)
is
contained in the set mentioned in the theo-
rem. Conversely, let
IP
be an entire function
of
two variables
such that
IP
(z
+
1,w)
=
e-
ZmwiP
(z,w),IP
(z,w
+
1)
=
<P
(z,w)
for all
(z,w)EC
2
,
and assume
that
M > 0, A > 0, B > 0 are such
that
liP
(x
+
iy,
u +
iv)l
<,Mexp(21Txv
+
1rAy
2
+
1rBv
2
).
Put
f/!(z)
=
fbiP
(z,w)
dw for zeC. Then
tP
is
an entire function
for which
lf/!(x
+
z:V)I
<,M
exp(1rAy
2
).
Also,
LIP
(x,w)
dw
=LIP
(x-
[x],w)e-
Z1Ti[xJw
dw.
Let
te[O,
1 ], ne'l. We have by analyticity
and
periodicity
of
IP
in its second variable and by the estimates on
IP
I
LIP
(t,w)e-
21Tinw
dw I
= I
{'
+
iy
IP(t,w)e-
21Tinw
dw I
Jo+
•Y
<,M
exp(21Tty
+
1rBf
+
21rny)
for all real y. Minimizing with respect to y gives
I
LIP
(t,w)e
2
1Tinw
dw I
<,M
exp(-
1TB
-
1
(t
+
nf).
Hence
f/!(x)
= 0
(exp(-
1rB
-•x
2
))
(xeR).
It
follows easily
from the Phragmen-Lindelof theorem
that
IPeS.
It
is trivial
that
T¢
=
IP·
The proof for the S case is similar and will be omitted.
To prove the assertion about
T(S
·)let
FeS
z•
satisfy
T\
1
'F = R \
2
'F,
T\
2
'F
=F.
For
any
f/!eS,
T¢
is
a multiplica-
tor
of
S
z·
and
it is easy to see
that
F·
T¢
is
an element
of
S
z·
which
is
periodic in its both variables. Hence
F·
T¢
has a
Fourier series
~n.m
c,m
(¢)e
2
1Timz
+
2
1Tinw
(cf.
Ref.
3,
27.24.3).
Define
G by
(G,f/1):
= c
00
{¢).
Then
GeS
·,and
by Theorem
3.6, the (nm)th Fourier coefficient
of
TG·
T¢
equals
(G,R _
m T
_,
¢) =
c,m
(¢).Hence
(F-
TG
)·
T¢
= 0 for all
f/!eS.
To
show
that
this implies F
1
:
=
F-
TG = 0, let
f/!eS,
f/!;:j:-0.
We see from the formula
T(R
_ b T
_a¢)=
e
2
"ibz(T¢)(z-
a,w-
b)
tha.t~--------
p1.
(Tf/!)(z-
a,w-
b)
exp(
-1r(z-
a)
2
-
1r(w-
b
f)=
0
for all a and
b.
Putting f(z,w) =
exp(-
1TZ
2
-
1TW
2
)
(
Tf/!)(z,W),
we have
feS
2
,
F
1
·T~
11
T~
1
f
= 0 for all aelR, hER.
So, if
heS
2
,
then
(cf.
Ref. 16, Sec.
S)
0 =
(F
1
·T~
11
T~
1
f,h)
= (R
1
~aR
~~by
f,
.Y(h·f';))
=
f~
,J~
=
e21Tiaz
+
2rribw
X(Y
f)(z,w) (Y(h·f';))(z,w) dz dw
for all real a
and
b.
As f
;:f-0,
this implies
that
Y(h·F';) =
0.
A.
J.
E.
M.
Janssen
724