Internat. J. Math. & Math. Sci.
VOL. 13 NO. 3 (1990) 431-442
431
FOURIER TRANFORMS IN GENERALIZED FOCK SPACES
JOHN SCHMEELK
Department of Mathematical Sciences
Box 2014, Oliver Hall, 1015 W. Main Street
Virginia Commonwealth University
Richmond, VA 23284-2014
P.,,cved June |6, ]q89)
ABSTRACT,
o
where
A classical Fock space consists of functions of the form,
C and
q L2
[R
3q
We will replace the
), q > I.
q
q >
with
q-symmetric rapid descent test functions within tempered distribution theory. This
space is a natural generalization of a classical Fock space as seen by expanding
The particular coefficients of such
functionals having generalized Taylor series.
series are multillnear functionals having tempered distributions as their domain.
A theorem will be
The Fourier transform will be introduced into this setting.
proven relating the convergence of the tranform to the parameter, s, which sweeps
out a scale of genralized Fock spaces.
Generalized Fock Sapces, tempered distlbutions, Fourier
transforms, and rapid descent test functions.
KEY WORDS AND PHRASES.
1980 AMS SUBJECT CLASSIFICATION CODE.
I.
Primary 46FXX
Secondary 44A15
INTRODUCTION.
Rapid
S’(Rq),
descent
test
functions,
S(Rq),
and
their
tempered distributions,
dual
are excellent spaces to do the analysis of the Fourier transform (Bogolubov
Logunov [I], Constantinescu [2], Freidman [3], Gelfand and Shilov [4], and
Lighthill [5]). The classical Fourier transform analysis examines spaces having test
functions defined on a finite number of independent variables.
By this we mean the
and
independent variables of a rapid descent test function,
dimensional Euclidean space.
@(tl,...tq),
belonging to a q-
This paper will indicate a method that will enjoy the
property that the number of independent variables becomes infinite, that is in some
sense
the
physics.
space.
dimension, q
.
The need for this analysis
is
essential
in advanced
An infinite number of particles are described by state vectors in a Fock
The classlcal results are developed in a Hilbert space.
Lebseque ntegrable functions,
LP(Rq),
Traditionally the
are implemented in the construction of a direct
432
J. SCHMEELK
However, when you want to describe
sum of these spaces.
Fourier
transform must be studied.
kernel, e
-2Itw
does not belong to any
LP(Rq)
space.
o
a frequency
This presents a significant
a parLice Lhe
problem since the
This kernel problem is solved
(Constantlnescu [2], Gelfand and Shilvo [4], Llghthill
[6]) but the infinite number of variables problem still remains.
in tempered distribution theory
[5], and
Zemanian
paper
This
functional
will
implement
tempered
theory
developed
in
distributions
Schmeelk
[7-10]
to
together
solve
with
the
a
holomorphlc
infinite
number
of
variables problem.
We briefly recal in S(Rq) and S’(Rq) the Fourier transforms are respectively
defined as
(F)(Wl...w)
q
A
R
q
exp-
2[tlW
+ t w
+
(t
q q
...tq)dt
...dt
q
and
F(wl...Wq) (tl...t q > A < F(Wl...Wq) F((tl...t q )) >
all (tl...tq) a e S(Rq) and all
F(Wl...Wq) e S’(Rq). The
<F
for
advantages of S(Rq)
and S’(Rq) are many but the fundamental result is that the Fourier transform exists a
homeomorphism and has
the appropriate derivative
multiplication property.
This
paper will not include a survey of the many Fourier transform properties which are
[2], Friedman [3],
[12], and Papoulis [13].
contained in Constantlnescu
Bracewell
[II], Gonzalez
We will extend the Fourier transform into generalized Fock spaces.
The principle
and Wintz
Zemanian
[6],
result will be the existence of the transform in the scale of Frechet spaces
U
pB
r
s
Fp’
sB
and its corresponding dual
(rPB)
these spaces are contained in Schmeelk [7-10].
A comprehensive examination of
We will only review these spaces in
sections 2 and 3.
2.
pB
THE SPACE, F
r
s I
For each s
p,sB
the space
rP’SB(p >
infinite dimensional Fock space.
{Bi} i=O Bi > Bj J >
I, B
Then p and
i
i), is called an
0 are all real numbers.
These
Bi,
spaces are topological spaces of complex valued functlonals on S’(R;
), the space of
complex valued distributions. The functionals which are members of rp’sB are all
C(S’(R);).
The complex or real valued functionals enjoy similar properties.
real valued functlonals which are members of
We also require if
O(x)
where a
space,
0
e
and
a x
q=O q
aq,
q
.
e
rp’sB,
a ix
q=O q
r
The
PB
are developed in Schmeelk [8].
then
x]
q )1 are q-multilinear symmetric continuous functlonals on the
p’sB
copies, q ) I) to C.
We identify for each
the
e r
S’(R)x...xS’(R), (q
433
FOURIER TRANSFORMS IN GENERALIZED FOCK SPACES
associated Fock state vector,
a0
a
(2.2)
++
aq
We equip our infinite dimensional Fock vector space with the following increasing
sequence of norms:
l/p
aq
x
q
(2.3)
0, I,...
m
(sB)q
m
where
x e S (R), m
(2.4)
0,
with
su
J<x,>j
I
0 e S(R), m
(2.5)
0,1...
and
Ii il
0 <
al
,tq)jDa(t l’" -tq)
M (t
sup
lllllm
m
m
tq)
(t
e R
(2.6)
q
where
,tq)
M (t
m
and
A
[(1+(2tl )2)
(1 +
(2Cq) 2)]m
(2.7)
+"’+%
D=
so
functions M (t ...,t
q
m I’
defined generate a sequence of norms equivalent to the sequence of norms Implementing
The
defined
norms
the functions,
M’m(t
expression
in
t
q
(2 6)
)--
using
the
q
It was proven in reference [I0] that each real valued functional,
a kernel
,sB,
representation which remains valid for complex valued functlonals.
has
This
representation is as follows,
(2.8)
lq(t
where
$^
’u
functions,
a0
t
and (t
.q
q
q) satisfying,
symmetric
complex
valued
rapid
descent
test
S+(R
III (R)IIIA
wher e
are
m
q! lip
II _qlim
(s B)
m
m
0, I,...
(2.9)
434
J. SCHMEELK
Da -(tl"’’t q )’I
t)l
q
M(t.
0
Cm
al
m
i Cq
(t
The representation for
t
c R
q
q
given in expression (2.8) enjoys the standard square summable
property often times postulated for Fock functlonals as seen by the following theorem.
p’sB its
THEOREM 2.10.
Given a
c r
kernel representation given in expression
(2.8) satisfies
II *1
PROOF
q
q=IRq
Clearly
2
the
constant,
problem of the result of the theorem.
does
not
Also since
contribute
e
rp’sB,
to
the
convergence
then by the requirement
given in expression (2 9) there must exist a sequence of positive constants,
such that
C
<
m
(sB)q
m
fr all q and
qO
q lRq
qo
q
m
q=l
M2m (t
t
q!
q=l
qo
"
q=l
q[
m
q!I/P(sBm )q
I/P(sBm)q
2/P(sB m )q
II qllmq, ’zp
(sB
dtl"’’dtq
q
)q
IImI/p
m
q!
"o
[
l,qll "q(SBm)q
m
m q=l
q!
qo
Cmq(sB m)2q
m q=1
q!
Since expression (2.11) converges for any
q0, the result follows.
{C
m
m=O’
435
FOURIER TRANSFORMS IN GENERALIZED FOCK SPACES
3
THE FOURIER TRANSFORM IN
rpB.
on
The Fourier transform
DEFINITION 3.1.
e
B
exp[-2
i
R
exp[-2
q
is defined as follows,
tlWl]l(tl)dtl
i(tlWl+...+t q wq )]q(t I’
tq)dt
dtq
R
.
LEMMA 3.2.
() is well defined for every
f
q=l
PROOF.
Rq
B
e
exp [-2
e
rp’sB
and moreover
tq)dt
i(tlWl+...+t q wq )]q(t l’
rp’sB
implies @ e
for some s
...dr
q
I.
We then have
f[exp
Rq
q-1
2w i
[tlw1/.../t q wq q(t
tq)dtl-..dtql
<
(R).
.,t
, ,o+lZ
/ ’M l(’t
I’
,tq)
Rq
q
I""’
q
f
Rq
M (t
..... tq)
.dt
dt
q
’ql lzq,1/p,z/p (SBm )q
"ql
q!
q-1
m
)q
q(sB )q
1,ol #1#(R)Ill;.m q-17. ql/pm
/
THEOREM 3.4.
PROOF
Since
e
<,
rpB
U
s I
rP,SB
we consider the Fourier transform on the space,
S
following,
IIsq
qq
R
m
to
Fourier transform is a linear continuous transfortion on
(s
’Bm)q
I’"
q
)dr
dtqllmq, tlp
B
m
J. SCHMEELK
436
{D
sup Mm(wl
w )](t
exp([-2,t(tlwl+...+t qq
Rq
Wq
(s ’B
q
Oai
m
,tq)dt I’" -dtqlq!
)q
m
I q
w )e R
q
(w
q
sup Mm(wl,...,w
q
q
0
i
m
q
(w
w
e R
q
q
IRq
...(-2it
qexp[-2i(tlWl+...+t qw q )](t l’’" .,tq)dt 1"" .dt q {q’ 1/p
q
(s ’B
M (w
m
w
I’
q
sup
R
q
)q
tq)l(t I"" .,t q )Idt I’’" dtq
Mm(t I’
(s ’B
m
i cq
m
)q
Ill
0=
(w
q
e R
l,***,wq)
f M2m(t I
tq){O(t
sup
q
(sB
q{
q
1/p
<q
i
f
sup
q
0at
i
Rq
0
M2m+l(,t
(t
,tq) {+(t
tq)
(s ’B
tq){dtl...dt q
q.’ 1/p
)q
m
q
sup
q
M2m+l(t
tq)
,tq){ +(t
(s ’B
ql m
i
(t
tq){dtl...dt q
q
R
q
m
)q
tq){
qq!l/p
1/p
FOURIER TRANSFORMS IN GENERALIZED FOCK SPACES
437
q
M2m+l (t
sup
q
tR)
(s
0(ai
(m
<i
<q
(t
tq)
e R
4.
i) lq! I/P(sB2m+l) q
(s ’B
m
)q
q
SB2m+1
B2m+l <
(3.5)
s’B
(SB2m+l)q
Noting that
t
B2m+l )q
1/p
sup
q
(t
B m and s’ > s implies expression (3.5) is finite.
THE FOURIER TRANSFORM ON
(rPB)
In a previous paper [9], it was shown that the dual of
the union of sets of the form,
rp,sB
-m
The
generallzed
{(F0’ F1
Fock
dual
,Fq
functionals
<
at
is denoted as
We
m.
< <
F,
also
note
> > A
0
p’sB)’
denoted (r
is
(4.1)
e C
described
Fq p’sB
considered as sequences where the
rank
): F
rp’sB
in
expression
(4.1) can also be
are symmetric tempered distributions all having
if @ e
I Fq, q
r
and F e
(rP’SB),
then
the
evaluation
>.
of
F
(4.2)
q--O
EXAMPLE.
4.3
All
p’sB
sets, (r
the
),
contain
the
generalized
Fock
Dirac
functional,
6
where 6 (R)6(R)
[3].
(4.3)
<=ffi>
66 is the tensor product of q copies of the Dirac delta functional
We immediately verify that
[I Fqll-m(sBm)q q!-I/p
II
"..."
(sB)q
m
ll-,,(SBm)q q!-l/P
q!-I/p <
-.
438
J. SCHMEELK
The Fourier transform on the spac
DEFINITION 4.4.
<
<
EXAHPLE 4.4.
. (rPB)
is
dFned
a’
> > =A << F,* > >"
F,
We compute the Fourier transform of
6(k) (tl-Zl)
(k)(t
T)
6(k) (t I_ Zl
<==>
(k) (t
qth
It suffices to consider the
<
6(k)(t
=A < 6(k)(t
(k)(t
<
(k)(t q
6
6
Zl
6
66(k)(t q
6...
6
(k)(t q
zl
component,
zl)
TI
(4.4)
(k) (t2_ z2
(k)(t q
q),(w1...Wq >
Zq)
Rq
exp[-2rl(Wlt1+...+wq t q )]@(Wl...wq)
dWl...dwq >
=(-l)qk<(tl-Tl
6...6(t
dk+k+.., k
q
-q) .----’-
dtl...dt ok
f exp[-2rl(Wltl+...+wq t q )](w ..Wq)
Rq
dWl...dwq >
q
(2rlw
)k
where
1)k e-2IwI’1
(2rlw)k
distribution
(2IWI)k
.(2rlw
)kexp[-2rl(Wlt +..+wq t q )]
(w
e
6
-2iw z
n n
(2,iw2)k e -2rlw2T2... 6(2rlwq )k e
Wq )dw
dWq
(1
n
< q)
is
being
-2iw T
considered
q q
as
@(Wl...Wq)>
a
regular
tempered
In summary we have
(2rlw
6(k
q
(2rlWq )kexp [-2i (w zl+...+Wq Zq @(Wl...Wq )dWl...dWq
Rq
<(2rlw
Rq
(4.5)
)k
-2 rlw
e
Zl
(4.6)
(t_ z) ++
2tw
)k (2 ,lw2)k
(2lw)k
e-2i[wl:l+*+q q
439
FOURIER TRANSFORMS IN GENERALIZED FOCK SPACES
It s clear that any q’th entry in expression 4.6) does not belong to
L2(Rq)
since
clearly
Ji(2iw )k
.(2lw
q
)k e -2i[wl I +’’’+wq q
(2w)]k
q
[(2 I)
q.
is not integrable over R
(pB),
However, the expression given in line (4.5) does belong to the
space since
-1/p
q=0
l
q=O
q
Jl(2Wl)k
I(sB
q=0
EXAMPLE 4.7.
m
)q
...(2Iiw )k e
q
q!-l/p <
-2I
[wl TI’" "Wq
q]l J-m(SSm )qq!-I/p
(R).
In a similar computation it can be shown that
and again the Fourier transform is a member of every set,
sB ).
It should be noted
that other spaces such as distributions of exponential growth [3] offer some technical
achievements that increase the space of Fourier transformable functions. However, we
wanted to relate our results to our specialized scales of Frechet spaces developed in
Schmeelk [7-10] and Schwartz [15].
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