Wavelet transforms in generalized Fock spaces

WAVELET TRANSFORMS

IN

GENERALIZED FOCK SPACES

JOHNSCHMEELK

Department

of Mathematical Sciences

Virginia Commonwealth University Richmond, Virginia 23284-201,t, U.S.A.

and ARPAD

TAKAtI

Institute of Mathematics University

o,f

NoviSad

TRG D.

OBRADOVICA

4,21000NoviSad Yugoslavia

(Received March 26, 1996 and in revised formMay 27, 1996)

ABSTP,CT.

A

generalized Fock space is introducedasit wasdevelopedby Schmeelk

[1-5],

also Schmeelk and Takai

[6-8].

The wavelet transform is then extended to thisgeneralized

Fock space. Since eachcomponent ofageneralizedFock functional isageneralized function, the wavelet transform acts upon the individual entry much the same as was developed by

Mikusinski and Mort

[9]

based upon earlier work ofMikusinski and Taylor

[10].

It is then shown that the generalized wavelet transform applied to a member ofour generalized Fock space producesa moreappropriate functional forcertainappfications.

KEY

WORKS AND PHRASES: Generalized Fockfunctional,wavelets, boundedintersection

property, generalized wavelet transforms.

1991AMS SUBJEC

CLASSIFICATION

CODES: 46F12,46F99

1.

INTRODUCTION

Wavelet transform techniques are rapidly becoming a primary mathematical theory being implementedin manyapplications. Thecurrentdevelopment by the FederalBureauof

Investigation

(FBI)

inestablishing aproper wavelet transform to store its30million criminal

fingerprints now stored in filing cabinets illustrates the wavelet applicational importance.

Theadvantagewill notonly betocompress the datafilesbuttodevelopafaster and superior method tofacilitatethematchingprocess. These techniquesaredeveloped and discussedina

work of

Strange

[11].

Wavelets are also shown to be of significant importance in the music industry. A

linear combinationof three waveletsareshown tobeanexcellentmodelfor the sound of the crash of cymbals in an orchestra shown byVon

Bseyer

[12].

The application of wavelets to seismic data developed by

Grossmann

and Morlet in reference

[13]

also illustrates the marvelous analysis of what the authors ofthis paper willterm,"roughdata". Perhapsin some situations theremovalof the "overshoot"knownasthe Gibbsphenomenonin Fourieranalysis giveswaveletanalysis akeyadvantagein approximatingour socalled "rough" datadiscussed

658 J. SCHMEELK AND A. TAKACI

Anatural question toaskishow can weapplywavelets to distributions? Inparticular

how to apply them to singtdar distributions or non-regular distributions such as the Dirac

deltafunctional whichagainin theauthorsopinion is avery "rough" object. Severalresearch

papersregarding theimplementation of wavelets todistributions canbe foundinthe research donebyWalter

[14-20]

andMikusinskiandMort

[9].

Wewill extendseveral of these results.

The application of waveletsto image processing has been developed as aconvolution

operatorinthework of Mallat

[21-24]

and furtherdevelopedin amathematical milieu inthe

workofMikusinskiandMort

[9].

Wewill first give ageneral overview ofwavelet transformsconsidered as convolution

operators for real valuedfunctionsbelongingtothespace,

L2(goq),

q

>

2. Wewill notrepeat here the marvelous multiresolution analysis conducted inthespace,

L2(9O2),

by Mallat

[21-24]

andMeyer

[25]

andgeneralizedtothespace,

L2(goq),

q

>

2 inthe work ofDaubechies

[26].

Wearealso interested in the space

L2(9O2)

sinceanimmediate applicationcanthen be made to image processing. The real valued functions in this application take on integer values rangingfrom 0 to255 depending ontheshadeofgray contained in ablack and whiteimage ,andits argumentis amember of9o2 designating thelocation within the image developedin

Lim

[27].

Moreover many signals in signal processing are bounded real valuedfunctions on 9oq and arealso members of the space,

LI(goq).

Thus we arenaturally lead tostudy the space,

L1B(%q),

namely thespaceof bounded functionswhich arealso members of

L2(goq).

Moreover

it is well known that the convolution of two functions belonging to

LI(go

q)

exists almost everywhere.

Wethenmove onandconsiderthe wavelet transform appliedto classical distributional

spaces. This then ctdminates by applying generalized wavelet transforms to our so-called generalized Fock spaces. The generalized Fock spaces are developed by Schmeelk

[1-5]

and

Takai

[6-8].

A

rapidreview of the key features for these spaces areincluded in section 3.

The threerepresentationsofourgeneralizedFock functionaJs will enableusto consideraform

of classical Fockspaces aswell asgeneralizedFock spaces. These so-cMledgeneralizedFock functionals contain functionals oftentimes non-linear which may

consist.of

aninfinite arrayof translatedDiracdeltafunctionals. Thisstatement will become clearasthe paperdevelopsthe surroundingmathematical considerations. Itis thewavelettransform generalizationstothese

spaces which willprovide uswiththeauthors so-called "somewhat smooth" or "verysmooth"

representations for the

"very

rough"functionals.

2.

SOME NOTIONS AND NOTATIONS

We recall some properties for our sequences of real numbers, p

(mq)qeNo

and sequences of functions,

Ah=(Mp(.

))peN0.

The sequences of real numbers,

p=(mq)qeNo

satisfy theKomatsuconditions

[28],

namely

(M.I)

(M.2)

(M.3)

m2q<_mq_l

mq+

1,

qeN;

thereexistrealnumbers,

A

and

H,

such that

mq

+

<

A

H

q

mq,

qeN_O oo

mq_

These conditions arecommonlycalledlogarithmicconvexity, stability under ultra differential

operators and strong non-quasi analyticity respectively. It is convenient to select m₀ 1.

Oneeasilyverifies, for instance,the sequence,

mq

qqs

s>1,

satisfiesthe threeconditions.

We next select a sequence of continuous functions,

.,

=_

_(MP(’))PeN0

on

q,

q

_>

1.

Werequirethetraditional conditions.

(P),(M)

and

(N)

hold

(see

Gelfand and Shilov

[29]

or

Pilipovic andTakai

[30]

aswellasthe inequalities,

1 A

MO

(tl

t)

_<

.--_<

Mp

(tl

tq)

_<---.

(2.1)

Thenaninfinitelydifferentiablefunction,

(t

t),

is in thetest space,

%(Jo),

ifand only if foreach

peN

₀thefollowingnorms arefinite;

where

Oq

t,

{1@

(’

)1

(h

tq) lMp

(t

tq)"

tR

q,

ji<p,

<i<q}

4

(jl

Jq)(t

tq)

Oil

+’"+jq

ojlt...Ojqt

q

(t

andt=(t

tq)

E

R

q

(2.2)

The family of norms,

(11"

),v0,

defines a locally convex topology on the vector space,

%(Mr,)

which in view of condition

(P)

turns it into a Fr6chet space. The classical test

functions,

@(q),

consisting of infinitely differentiable functions having compact support equippedwith itsusual topologyisthen densein

%(.hh).

Thisimplies that thespaceof hnear

continuous functionals referred to as generalized functions on the test space,

%(.Ah),

and

denoted

%’(.At,)

is in factacountablynormed spaceequippedwith norms,

II-p

sup

{I

(,)

p _< 1,

b

%(MI,),

p.e

NO}.

(2.3)

Moreoverweobserve that

o

I1.

>’--

>

I1.

>’"

for any z

/%’(.At,).

Duringour sequelwewilloccasionallyusethefunctions,

__(MP)PeNo

to bedefinedas

Mp

(t

tq)

[(I

+

tl

)

(i

+

tq)]

p

(2.4)

whereby the test space,

%(.A)

then becomes the test space of rapid descent,

Y(q)

and

%’(.,11,)

becomes the Schwartzspaceof tempereddistributions,

Y’(Rq).

660 J. SCHMEELK AND A. TAKACI

,-

%’()

x...x

’(.,r.,)

c

amultihneax continuous functionalonq-copies of

%’(.At,)

toC.

By

definitionwetake_a0 / C. Wedefine

aq

p

{laq

[,

11"

e

%’(),

II-

-<

P

}-

(3.2)

Then the infinitecolumnvector,

a

o

a

4-aq

(3.3)

is in the space,

F

s’p’,

s

>

1,if andonlyif thenorm

IIN,

III

<p) sup

aq

IIp

rnq

s,t,.Al q N_O

sq

(3.4)

isfiniteforevery p /N0.

Clearly thecanonicalinclusion,

F

s’/’Mb

F

s’’g’Mh

(3.5)

is continuousprovided

s’

>s>1.

DEFINITION 3.6.

A

generalized Fock space,

F/’’At’,

is the inductive hmit of the spaces,

F

s’/’lt’,

which isdenotedas

F

I’Mb

ind

F

s’t’’At’

It wasshown by Schmeelk

[2]

that eachfunctional, q

F/’’At’,

has akernel representation. Thisrepresentationwasshowntobe

b

0

bl(t

1)

.

(3.7)

dPq(tl

tq)

where

@0

ao

EC and

@q(gl

gq),

q

>

i are symmetric rapid descent test functions

wheneverthe functions, M1,

=(./p(.))pe.,Vo

aredefinedasinexpression

(2.4).

Moreoverour functional representationgivenin expression

(3.7)

will thensatisfy thenormconditions,

sup

<

c,

(3.8)

s,_I.t./&, q NO s

q

(3.3)

or

(3.7)

enjoyageneralized square summablepropertyas seen in thefollowingtheorem oftentimespostulatedforclassical Fock functionals.

THEOREM

3.9. Givena e

r

s’/z,tt,

s

>

ithenitskernelrepresentationgiveninexpression

(3.7)

satisfies

[l

I0l

2

/q-1

jq

ICq(t

tq)l

dt

dtq

<

oo.

PROOF:

SeeSchmeelk

[6],

Theorem 3.1.

sp= ,

’,

[3S-40]

e =h

F

E

(r/z’-),

wasshowntohave therepresentation,

F-[F

O,

F

1,

Fq,

...],

(3.10)

where

F

₀ECandeach

Fq,

q

>_

1 belongs tothe generalizedfunction space,

%(.),

andthey allhaveanorder _< p. Moreoveritwasshownthateach

F

satisfiesthe condition,

oo

sq

](p)

Fq

II-p

Wq [III

<

cx,

(3.11)

q 0

s,p,

forsomefixedorder,p.

The duality between

r/z’MI’

and

F

(/

(r/z,-A)

isthendefinedby

<(FI,))=

(Fq,

Cq)q.

(3.12)

The notation in expression

(3.12)

given as

(Fq,

dpq)q

denotes the duality between the test space,

%(),

anditscorresponding generalizedfunction space,

Wenowobservethatourdual space,

(r/z,

Mb)

canbeequippedwith thenorms,

ii

(’p)

q

IIIFI

-

Fq

II-p

(3.13)

s,/z,.A q 0

where p

_>

order of all

{Fq}q=

in the representation given in expression

(3.10).

Moreover whenever theseries in

(3.13)

converges,ourgeneraSzedFockfunctional isamember of

4.

WAVELEt TRANSFORMS

Forthis sectionand hereafterweconsiderthefollowingspaceinclusions,

(q)

C

Y(q)

C

LB(q)

C

L2(

q)

C

Y’(q)

C

(q).

(4.1)

Hereagain

LB(q

is thespaceof bounded functionson

q,

q

>_

whicharealso members of

L2(q).

The tempered distributions,

Y(q)

and distributions,

(q)

are acting on rapid descent and compact support test functions respectively having independent variables

belonging to

q.

Moreover wewant to preservethe mathematical analysis surrounding the

662 J. SCHMEELKAND A. TAKACI

transforms. This is very important in applications since many software packages contain

algorithms for Fast Fourier transforms, but do not support wavelet transform techniques. After we establish the wavelet transform to be a form of convolution, then these software packagesviatheexchangeprinciplecanbe extended to dowavelettransforms.

The wavelet analysis surrounding this section to include the mother wavelet, the admissibility condition and multiresolution analysis canbe found in Akansu

[31],

Benedetto and Frazier

[32],

Beylkin et al

[33],

Holschneider

[34],

Kaiser

[35],

Koornwinder

[36],

Ruskai

[37]

and Schumaker and Webb

[38].

Also theinversion of_{the wavelet transform stemming}

from the admissibilitycondition canalso befound withinthese references. Wewillbeginour analysis by considering the wavelet transform as a special type of convolution as in Mallat

[21-24]

and Milusinski and Mott

[9].

To this end we will adopt the following notation conventions.

Ifafunction, $(t) E

L2(q),

togetherwith two q-tuples of realnumbers, a

(ai)

a

#

O,

(1

< < q)andr

(ri)

aregiven, thenthe dilationand

tq-

rq

(4.2)

aq

t=l translationof $(t)is

a,(t

i

Moreoverwedenoteby

a(t)

thefunction definedas

ba(t) 1

tl

tq

)

(4.3)

Throughout our sequelwe will alwaysrequire the dilation sequence, a

(ai)=

satisfy the requirements,a

#

0, (1

< _<

q).

LEMMA

4.4 Givenafunction,$(t) E

L2(

q)

then

b(t)]I

_L2(tq)

a’r(t)

_L2Otq)

a(t)

_L2(q),where

isthestandardnormonthe equivalence classescontained in

L2(q).

PROOF.

Follows directly from the change of variables,

T

tiT-’iai

and

T

"-i

(1

_< <

q)respectively. This 1emma showsusthat the energyin areal valued signalremains

unchangedunderthe constructions,

ba’r(t)

and

cka(t).

Wenow selectamotherwavelet, b(t)

LB(q),

anddefine thewavelet transformofa

function,

f(t)

Ll(q),

tobe

-oo -oo

a

aq

_]

dtl...dq

f(t

,tq)

--q

_’\ -a

-aq

_]

dtl"’dtq

If(t)

where denotes the convolution product of the functions,

f(t)

and

a(t)

as defined by

Zemanian

[39].

Asanexampleofamotherwaveletweconsideraform of theMeyerwaveletpresented bySaitoh

[40],

q

-t-...-t

(t

,tq)=

F 7

|2r_//4

|

-t2)..-(

1-

t)e

(4.6)

L v,, _.J

which satisfiesthe aAmissibilitycondition,

andmoreover

Themotherwaveletgivenin expression

(4.6)

isarapiddescent test functiongivingus aclear

comeetion between wavelet analysis and tempered distributions. By applying the Schwtz inequality in the space, L2

(q),

we immediately have forour mother wavelet

(4.6)

and any

f(t)

E

L2(

q)

thefollowinginequality.

(4.8)

Thisinequalitywill remain validforanymotherwavelet,

Ca,

r(t),

whichhas beennormalized inthe

L2(

q)

norm.

5. WAVELET TRANSFORMS OF GENERALIZED FUNCTIONS

Webriefly recall thedefinitionof theconvolutionproduct fortwogeneralizedfunctions

developed byVladimirov

[41]

and Zemanian

[39].

DEFINITION

5.1. Giventwogeneralizedfunctions,

F,

G,

thentheirconvolutionproductis

(F G,)

<F(t

,tq),

(G(’r

,’rq)

(tl

+

"rl

,tq

+

rq))>

(5.2)

I I

F(tl

,tq) G(v

,Tq)

(t

+

r

,tq

+

rq)

dt

,dtq

dr

,drq

664 J. SCHMEELK AND A. TAKACI

Inherent in definition 5.1 istheproblemwiththeconstruction (t

+

r)

which destroys the compact support and polynomialgrowthconditionwhenthetestfunction,(t), belongsto

(Rq)

or

Y(q)

respectively. To overcomethis difficulty it is shown byVladimirov

[41]

and

Zemanian

[39]

thatif

F

andGboth havecompactsupport,orboth have boundedsupportson

the left orthe right, then the convolution productis well defined. Moreover it is shown by

VlaAimirov

[41]

that the convolution product remains well defined for F generalizedfunctionhaving compact support.

Since theconvolution of the translated generalized derivativeof the DiracfunctionaJ,

(Jl

Jq)(7.1

_.0

7.q-

rq),

and adistribution,

F,

is importantto oursequel,wecomputeit

q

implementingthe notationconvention,

Thisgivesus

/F

*

(Jl

Jq)

(7.1_

7.01

7.q_

7.q0),

(5.3)

<F(t,

,tq),

<5

(jl Jq)

(7.1-7.10

7-q-<F(,

,,q),

<(Jl

Jq)

(7.1,’",7.q),

/F(ti

,tq),

<(7.1,’;’,7.q),

(-1)

’j’g(jl

Jq)

(txq-7.1q’7.10

<F(Jl

Jq)

(t1_7.10

,,q_7.q0),

Since thisholds for alltestfunctions,

)(q),

wehave

F,

(J

Jq)

(’1-7.10

_{7.q_}

7.q0)= F(J

J)

(tx-

7.?

,tq_

7.q0).

(8.4)

We now consider a mother wavelet,

(t)

L2(

q)

or

Y(q)

such as the one in expression

(4.6).

Weconsider a distribution,

F(t),

where the support of

F(t)

intersects the

support of

=(t)

and

(t

+

7.)

in a bounded set, let us say fC

2q..

We then select a test

function,

A(t,7.)

(2q)

that is

equM

to one over some neighborhood of

lger region as shown in Zemanian

[39].

For such a distribution, F(t), satisfying what we termthe bonnddintcectionsupportpronertywiththe functions,

=(t)

and

(t

+ 7.),

wethen definethefollowingwavelet transform forF(t).

DEFINITION

5.5. The wavelet transform of a

distributior,

F(t)

satisfying the bounded

intersectionsupport propertywith

a(t)

and

(

+

7.)

isgiven by

(WcF)(=)

-

F(

,),

=(m

,)

As in Zemanian

[39],

this convolution product iswell defined makingourwavelet transform well defined.

We compute the wavelet transform of

(tl-b

tq-bq)

with a mother wavelet,

(7-1

,rq)

which satisfies the boundedintersection support property with

a(7-)

and

(t +

r)

whenever

(t)

E

().

Tothis endwehave

(t-b)

ea(t),)

(t-b),

lI

a

(81

_$q)

A(t,-as)(t

_-asl,

...,tq-aqSq)

dSl...ds

_q =1

-OO -OO

a (s

Sq)/(b,

as)(b

alSl,

...,bq

aqSq)

dSl...dsq

oo o

,/f

t-b

tq-bq

A(b,

b-t)(tl,..,tq)

_dt

1.

.dtq

I

I

(a(t-

b),(t))L2(q

).

Since thisholds for alltestfunctions,

(t)E

(8q),

wehave

(W(t-b))(a)

Ca(t-b).

Clearlythe"rough" deltafunctionalistransformedtoa "rathersmooth" or "smooth" regular distribution,

Ca(t-b)

dependingon the smoothness ofthe mother wavelet (t) selected for

theapphcation.

We now extend our wavelet transform to a tempered distribution,

F

e

Y’(q)

and a

motherwavelet,

(r)

e

Y(q),

following theconvolution construction Zemanian

[391.

DEFINITION 5.6. Given

F

Y’(q),

a mother wavelet,

(r)e

Y(q)

and a test function e

(q).

Wethen define the wavelet transform of

F

tobe

(WcF)(a)

&

F(t

,re),

Ca(r

,rq)

where

F(t

tq)

a(7-1

,rq),

(t

tq)>

-F(t

lq)

x

(7"1, ...,7"q),

)t(7"

,7"q)

ea(7"1-tl

,7-q-tq)>

666 J. SCHMEELK AND A. TAKACI

6. WAVELET TRANSFORMS OF GENERALIZED FOCK

SPACES

The technique of extending thewavelet transformtoourgeneralized Fockspaces is in

the spirit of extending the Fourier and Hankel transforms to generalized Fock spaces as

conducted bySchmeelk

[4.5]

respectively.

Forthissectionweconsider thefunctions,

(Mp(.))peNo

tobe

Mp(tl,

...,lq)

[(1

+

t12)...

(1

+

$q2).

Wenowdefinethe wavelet transformonfunctional,

q,

havingtherepresentative,

0

,

(.)

q(tl,

.,tq

and satisfying thenormconditions,

sup

sq

<

o,

(6.2)

forall

peN

_0.

DEFINITION 6.3 Given a normalized mother wavelet

(t) L2(q),

q

>_

i, the wavelet

transform of @ denoted

W(a,r)

is

Wv(a,

r):

0

l(tl)

Cq(t

tq)

0

[l(tl)

*

[q(t

tq)

*

d2al

aq(tl

tq)](r

(6.4)

LEMMA

6.5. The wavelet transform of

q, W@(a,r),

is well defined for every qe

r

g,,s,

having the representation given in expression

(6.1)

together with a normalized mother

PROOF: Let

F"

’’

forsomes_>1.Weconsider

-<

Iol

/

bq(tl,..

tq)

_al

q=l q -00 -00

rq-

tq

-aq

)

dr1, ...dtq

wheres ’s

2.

In

viewofLemma3.1 foundin Schmeelk and

Takai

[6],

page268, the last

sumisfinite.

Wenowconsiderthespace,

(Fg’")

defined in expression

(3.10)

and recall that an

F

(F

’

)’

has representation,

F[F

₀

Fq

(6.5)

where

F

₀ Cand

Fq

’(.At,),

q

>

1 and allhave order

<

p. Moreoverthe

F

then satisfies

thenormcondition

Fq

<

q=O -P

668 J. SCHMEELK AND A. TAKACI

(t -b

1)

$(t

-bl)

6(tq-bq)

x

sq

whereit isclear that

+

a(ta

-bl)

x...

(tq-bq)

il-

<

for everyp. q=l

Wethen define the wavelet transform ofourgeneralizedFock functional to be

(WcF)(a)

F

o

W(FI)(al)

W(Fq)(a

aq)

(6.7)

We compute the wavelet transform of the generalized Fock functional given in expression

(6.6)

withanormalizedmotherwaveletand obtain

F₀

Cal(t

-b

1)

ba

1,a2,..

.,aq(

bl

tq

bq)

We observe that each entry in expression

(6.3)

for q

>

satisfies the following norm condition.

I1,,,

aq(tl

-h,

...,tq-

)

_-p

sup

II,/,

,_<

-oo’"-

--:’-

]

(tl

,tq)

dtl...dt

q

<

sup

{

Thus we immediately have that our wavelet transform of our generalized Fock functionalgiveninexample

(6.8)

satisfiesthenormcondition,

o

sq

qsq

1/

e=

il,/,,,,,...,%

(h-b,...,e-bq)II

_

-<

1

-,e__z.q

<

.

(6.1o)

Weconcludewithconsideringthegeneralcasegivenin expression

(6.7).

Werequire thatour

generalized Fock functional has each entry,

Fq,

q

_>

1, and normalized mother wavelet satisfying the bounded pportintersectionproperty. Forthis situationwethen have

-P)

W(Fq)(,..,.,)II_p

q

IIIW()(a)lll_

,

q

o

.

(.)

IF012

+

Wb(Fq)(al,...,aq)

!1.

me

q=l

Nowfor eachq

>

1 in expression

(6.11)

wehave

W(Fq)(a

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