integral transformations

7/21/2019 Integral Transformations

1/20

Publ

RIMS, Kyoto

Univ.

35 (1999),737-755

A n Integral

Transformation and its

Applications

to

Harmonic Analysis

on

the Space of Solutions ofHeat

Equation

By

Soon-Yeong

C H U N G *

and Yongjin

YEOM**

Abstract

W e

introduce an integral transformation

T

defined by

ixy

-W

2

f(y)dy,

in

order to do harmo nic analysis on the space of C solutions of heat equation.

First, the Paley-Wiener type theorem for the transform ation T will be given for the C

functions

and distributions with compact support.

Secondly, as an application of the transformation the solutions of heat equation given on

the torus J

will

be characterized.

Finally, we represent solutions of heat equation as an infinite series of Herm ite temp eratures,

which are to be defined as the images of H ermite polynomials under the transforma tion

T.

1.

Introduction

B y

a

temperature

we

mean

a C

solutions

of the

heat equation.

In

this

paper

we

will

do

harmonic analysis

on the

space

of

solutions

of

heat equation

(3

f

-A) i


2/20

73 8 SOON-YEONG

C HUNG

in

the usual harmonic analysis.

In

this paper

we first

introduce

an

integral transformation

T

as a

variant

of the

F ourier transfo rm, which

is defined by

We introduce several interesting properties of such an integral transfor-

mation

T

which

will

be

widely

used throughout this

paper.

As

the first main theorem we shall give a Palay-

W iener

type theorem of

this integral transforma tion

T.

Especially,

C functions

with compact support

and distributions with compact support

will

be characterized by entire

temperatures (see Definition 3.1) respectively, via their images of this integral

transformation T.

Next ,

we

shall characterize periodic temperatures

on J T x ( 0 ,

T)

or

temperatures on the w-dimensional torus J" without any condition on their

initial temperatures. The initial temp erature of any temperature on the torus

will

be

identified

with the help of the integral transformationTi n an easy way.

In the

last section,

we

introduce

the

Hermite temperature

J J ?

n

( x

9

t) as an

image of the H ermite polynomial under the integral transform ation

T.

Then

it will be shown that Hermite temperature has a similar orthogonal property

and can be

used

to

expand

the

temperatures with some growth condition into

infinite series

of

H ermite temperatures.

2. An Integral Transformation T

We start with a fundamental space on which we introduce the integral

transformation. For

l < / ? < o o

the

space

L

p

(d^) is the set of all functions

whose p-th

power

is

Lebesgue integrable with respect

to the

measure

\ J L on

IT. By

y*(fP)

or

simply &

p

we

denote

the set of all functions/

such that

for every

e

>0 ,/ belongs

L

p

(e~

E

^

2

dx) ,

where

dx

is the Lebesgue measure. Then

it

is easy to see that

for eachp> 1. The topolo gy is given on the space

<

p

( R

n

)

by the convergence

that

f j - + Q in

&

P

(R )

if and

only

if for

every e>0, /}-0

in L

p

(e~

E

^

2

dx) . In

fact,

the topology is the projective limit of

L

p

(e~

E

^

2

dx)

as e-0 . i.e.


3/20

AN

INTEGRAL TRANSFORMATION

73 9

N o w w e

introduce

the integral transformation as follows:

Definition

2.1. or

fe R )

and f>0 we difine

(2.1)

From this we can easily see that (Tf)

(x

9

t)

satisfies the heat equation

( d

t

- A)

( T f )

(x, 0 = 0 onR

n

x (0,

oo).

Definition

2.2.

By

3~(R

n

+

+

1

)

w edenote the set of all

infinitely differentiable

solutions u(x,f) of the heat equation

(d

t

A)u(x,

0 = 0 on 1?" x (0, oo) and we call

its element a temperature on

J? +

+ 1

.

W e give a topology on ^"(/?+

+1

) with

the

projective limit

of the

topology given

by the

semi-norms

for each t>Q. Then we have thefollowing:

Theorem 2.3. Th e integral transformation T\Q.

The transformation

T

plays a similar role as the Fourier transform in the

usual harmonic analysis, since it is actually the Fourier transformof /

after

multiplication by the Gaussian function e~^

x

\

2

. Thus th e transformation T

can be extended to the generalized function in anaturalway. For example,

for

a

tempered distribution

ue^ the

temperature

(Tu)(x, f) is defined by

by

considering

the function e~

ixy

~

f

^

2

as a

test function.

N ow we present some properties of the transformation T, which

will

be

used later. From now on * denotes the convolution product with respect to

x

variable.

(i) For feL

l

(R

n

)


4/20

740 SOON-YEONG

C HUNG

= $E(x-y

9

where

A

denotes

the

Fouriertransform

an d

E( x

9

1)denotes

a

fundamental

solution

of the

heat operator

d

t

A.

(ii)

where 8 is the Dirac measure and D

K

=^rd

x

. M ore generally, if

P(x)=

Y,

a

< *

x 0 and s> Q when

( v i i )

An

analogue

ofParseval's

identity

for the

transformation T

can be

derived as follows: For / andgin

( 7 7 ) (x, Wg) (x,t)dx=p-yj (3 0

=\E*fE

(271)

J

=

( 2 7 r ) \f(-x)3=

=(27c)"

(viii) ForftL

l

(R

n

) we can take a (pointwise) limit

,0

+ )= lim

U-^-

t - * o +

J

Thus(7/)(x,0 is a temperature whose initial value isjust/(x),so that

the integral transformation can be regarded as a generalization of the

Fourier t ransform.

3.

PaSey-Wiener Type Theorems

The Paley-

W iener

theorem characterizes distributions or functions by the

grow th of their Fourier-Laplace tran sform s. Then it

will

be quite interesting

to

consider

an

analogue

for the

integral transformation

T

introduced

in the

previous section.

Let

K

be a compact subset of

R

n

an d

ueS (K).

Then since the

function

e

-iy$-t\y\

2

j

s

(

rea

rj

analytic everywhere

in

R

n

the

t ransformation T


6/20

742 SOON-YEONG

C H U N G

is well defined.

M oreover,

it is

easy

to see the

followings:

(i) (Jw) ((,r) is an entire func tion of in C ".

(ii) (d

t

- A

c

)

(Tii)

(C ,0 = 0, on q x (0,

oo).

Definition 3.1. An infin itely differen tiable fun ctio n w ( ,f) on C " x (0, oo) is

called

an

entire temperature

if it

satisfies

(i) and

(ii) above.

N o w w e state th e Paley-W iener type theorem for the transformation T

of

the

distributions

and

functions with compact

support.

Theorem 3.2.

Let K be a

compact convex subset

of R

n

.

(i)

If

feC%(K) then (T / ) ( f0 is an entire temperature satisfying that fo r

an y

N> Q

there exists

a

constant

C >0

such that

(3.1) |(17)(C,

01

0such that

(3.2)

Conversely, if F(^ i) is an entire temperature satisfying (3.2) then there

exists

a

unique

u < = ( K )

such that (Tu)(,t)=F(

9

t)

in

C

c

x(0

5

oo).

Proof.

The uniqueness is easily

obtained

applying the well known

uniqueness theorems

fo r

temperatures (see [16]).

(i)First, we

note

that for a

constant

C >0

3.3)

/ y \

< C l

a

U l

a | / 2

a

1 / 2

e x p ( | j c |

2

J ,

Thisinequality

can be

easily obtained

by use of

C auchy's integral formula. Then

we have


7/20

AN IN T E G R A L T R A N S F O R M A T I O N 743

=

(V'G'We-'M

1

f ( y ) \

\

={ ^^Z(

J K /* Q

and

\a\Q. Then

it

follows that

, f >0

for

a

constant

C>0.

N ow we prove the converse.

Define

=

^p

3 . 4 )

The righthand side is independent of?>0,since

where

^^ is the inverse Fourier transform . Thiscan be seenby takingthe

Fourier

transform

on

(d

t

A^)F(^t )

= 0.

B y the

decay condition (3.1)

we can

see that /( jc)

is a Cfunction and

(Tf)g,i)=F(^t)

for

(eq

1

and t>Q. Now

it remainsto show that suppfaK. Fixing t = t

0

and shifting the path in(3.4)

we have

C t


8/20

744

SOON-YEONG C HUNG

Now let

xK.

Since

K

is compact and convex the H ahn-B anach theorem

gives a

hyperplane

{yeR

n

\(y,ri

o

y = c}

fo r

some

rj

0

ER

n

such

that

< j , f /

0

> < c

9

yeK

an d

Then it

follows

that

Taking ri=kr]

0

(k>0) in (3.5)we have

\ f x ) \

Then

the

righthand side goes

to 0 as A:

goes

to oo,

which implies that/(jc)

0 . Thus we have suppfcK.

(ii)

First, we choose a cut-off function x

d

eCo(K

d

) fo r 0 < < 5 < 1where

K

6

={x+y\xK

and \y\


9/20

AN INTEGRAL TRANSFORMATION 745

where C denotes a constant which is not essentially the same in each step.

Then

if we

take < 5

=

1 /1+ |C |

we

obtain

which is required.

N ow we prove the converse. Since Fl;

f)

is a tempered distribution fo r

each t>Q we can take the inverse Fourier transform ^T

1

with respect to

variable.

Put

Then

u

is also a tempered distribution and does not depend on the parameter

?>0 by the same argument

before.

M oreover, it is true that

(Tu)(,t)=F(,t)

for all C e C " and t>0.

O n the other hand, it follows from (3.2) that the usual P aley-W iener

theorem gives co

t

ES

>

'(K)

fo r

each t> Q such that

Therefore,

we

have

u =

e

t

^

2

a)

t

.

But

since

e ^O

everywhere

and

supp

a>

t

c:K we can see that supp uaK which completes the proof.

4.

Periodic

Temperatures on theTours T

n

For a multi-index aeN% by |a| we have denoted |a| = a

t

+ a

2

H h a

n

. But

for aeZ", ||a|| denotes the Euclidean length i.e. ||a||

2

= af + al+ + ; [

throughout this section where Z is the set of all integers.

As

an application of the transformation Twe shall give a characterization

of

periodic temperatures

on

R

n

x (0,

T).

Here

by a

periodic

temperature

we

mean the temperature

u(x,t)

such that

u(x+

O L , t )

= u(x,t) on R

n

x(Q,T)

for any aeZ". A periodic temp erature can be regarded as a tem perature on

the

^-dimensional torus

T

n

. In fact, it is

well know n that

ifw(x,f ) is a

periodic

temperature on R x(0,1) with the initial temperature f(x) =u(x,0 ) which is

integrable on (0,1), then

u(x,t)

can be represented by the Fourier series


10/20

746

SOON-YEONG C HUNG

(4.1) u(x,t)= a

n

e-

4n2n2t+2

*

inx

9

n= co

where

pi

(4.2) a

n

= f(x)e~

2ninx

dx.

Jo

Here,w eshall characterize arb itrary periodic temperature on R

n

x (0,T),without

any

condition on its initial temperature. M ore precisely, the initial tem perature

of

any periodic temperature will be given a meaning via the integral

transformation

T.

In

oreder

to do

this

we first

introduce some space

of

functions

of

certain

decay

rate.

Definition 4.1.

B y y^ we denote the set of all C

functions satisfying

the

following:

fo r every

aeN

n

there exists

a> Q

such that

( 4 . 3 ) | | ( | |

a > f l

= sup

\ d * < l > ( x } \ e

a

W

2

xeR

n

is

finite and by y\ we

denote

the

strong dual

space

ofy^

with respect

to

the

topology

defined by the

norm

(4.3).

Remark ,

(i)

Each element

u of the

space

y\

belongs

to the

space

of

Schwartz distributions, which is

defined

by the set of all continuous linear

functionals on the

space

C Q. In

fact,

it can be

represented

by u =d * f (x ) fo r

some

a and a

continuous

function/(*)onR

n

such that

for

some

a >0 and C> 0

This can be proved by the same method which was used for the structure

theorem

fo r

tempered distributions

(see [13]).

( i i ) It is

easy

to see that(7$)(

,

t)e^i for

each t

>0 and^e^.

Therefore,

the

transformation

T on 3 ~( is

also well

defined as, for

( i i i ) The space y^ is a little

different from

the space Si which was

introduced by G elfand-Shilov([6]). B ut using the same argu me nts in [6] we

can

see

that

the

Fourier transform

is an

isomorphism between

y^

and the

i


11/20

AN IN T E G R A L T R A N S F O R M A T I O N 747

space y given

by the set of all C

functions

$

satisfying that

for any

there

existh>Q

and C>0

such that

xe

n

i.e.

# =

&*

N ow w e are in a

position

to

state

the

main theorem

in

this section.

Theorem 4.2. L et u x,t) be aperiodic temperatureon R

n

x (0,T) .

Then

we can f ind a unique distribution v ( x )

=

Z

aeZn

c

a


12/20

748 SOON-YEONG

C HUNG

(4.9) the

Laplace operator

A and the

seminorm

in

(4.7)

can be

interchanged

in

their order of applications. B ut it may be nontrivial whether

8

t

and the

summation should

be

also interchanged.

B ut

since

d

t

u(x,

i)

is

also

a

periodic

temperature

if we use the decay condition and the uniquenessof the coefficients,

they can be also interchangeable. Then,

||a| fe

= 0

on R x (0, 7).

Thus from

the

uniqueness

of the

Fourier

coefficients we

obtain

an

ordinary differential equation

at

for each aeZ". Thu s wehave

which implies

that the function e

47c2 |

'

a| |2r

6(a,r)is

independentof

r > 0 , so that

we may write

c

p

L V

This

gives

the

expression

(4.6). M oreover, it follows from

(4.9)

that forevery

- J V *

2

n n

2

' ,

o < r < r .

Since

c

x

is

independent

of

r > 0 ,

we may

choose

t

arbitrarily,

so

that

fo r

every

>0 and some constant C>0 wehave

Now take v ( x )

=

S

aeZn

c

a

5(x+

27ta),

where c

a

's are the constants given

above.

For any 0 in ^ we

have


13/20

AN INTEGRAL TRANSFORMATION 749

aeZ"

for

some constant Q >0 and bytaking e>0 sothat s


14/20

750 SOON-YEONG C H U N G

property if

u(x,

t)= E(x-y,

t-s)u(y,

s)dy

J

n

for every t and s with

0 0

and

C(N,t) depending

on

N> Q

and

As an another application the backw ard C auchy problem for the heat

equation on the ^-dimensional torus

T

n

can be solved as

follows:

Corollary

4.5.

A

solution

of the

backward Cauchy problem

f(3

t

-A)i i ( jc ,0

=0 0 / ir

w

x(0,/

0

)

can be uniquely expressed as

u(x

9

t) = 0

a

where

2n i

*

x

d x,

aeZ

o o


15/20

AN

INTEGRAL T R A N S F O R M A T I O N 751

5. Hermite Temperatures

The Hermite polynomial H

n

(x) of degree

n

is defined by the Rodriguez

formula

Y

-) e~

x

, xeR

dx)

for n = 0,1,2,.... The orthogonality is defined by

H

n

(x)H

m

(x)e-**dx

O,

Then it is well known that if

e~^ f(x)

is a tempered distribution, then we

ca n write

the

distribution

/ as

( 5 . 1 ) /= Z

a

n

H

n

(x),

= 0

where

( 5 . 2 )

fln=

M oreover, the coefficients a

n

have an estimate that

for

some

Definition 5.1. Fo r

each neN

Q

we

call

the

temperature

(TH

n

)(x,t) the

H ermite temperatureoforder

n

and denoteit by

f fl

n

(x,t).

Infact,wehave

= H

n

(-D)E(x,t),

so that

J^f

n

(x

9

t){E(x,t)}~

1

is a

polynomial

of

degree n. Although

the

orthogonality of2?

n

(x,t)is notvalidanylonger they stillsatisfy the following:


16/20

75 2 SOON-YEONG C H U N G

(5.3)

: = f

*(*,t)JTJix

9

t)dx

JR

= H

n

(x)H

m

(x)e-

2t

^

2

d x

0,

when t=\.

N ow we

will

show that

if a

temperature

satisfies

some growth condition

at inifinity, it can be expanded via H ermite temperatures as

follows:

Theorem

5.2.

Let T>^ and u(x,f) be a temperatureon Rx (0,T) satisfying

that there

exist

M>0,

N>Q and C > 0 such that

(5.4) \u(x,t)\


17/20

AN

IN T E G R A L T R A N S F O R M A T I O N

753

and supp wcifT/4, T/2] such that

( 5 . 7 )

where d is the Dirac measure.

N ow take the integer m>N+2 where N is the constant in (5.4) an d

consider

functions

on

R

x (0

5

T/2)

[T/2

G(x,t) = u(x,t+s)v(s)ds

and

rn

M=

M)=f

Jo

H(x,

t)

= u(s, t+ s)w(s)ds.

J o

Then they are all temperatures on

Rx(Q,T/2)

an d satisfy

T

'2

and

T

M oreover,

G(x ,

t)

and /f

(x,t)

can be

continuously extended

to

R

x [0,T/2) and

Define

g(x) =G ( x ,Q ) and h ( x j = H(x,0). Then g and h are continuous and

This implies that g

and /z can be

regarded

as

tempered distributions.

B y the

uniquen ess theorem for temperatures (see [16]) we have

)=

)= E(x-y,t)h(y)d y = (


18/20

754 SOON-YEONG

C HUNG

If we define a

tempered distribution

M

O

by

then

2m

=u(x,t\

which

implies that

the

initial value

u(x,Q

+

) of u(x,t) is given by the

tempered

distribution

u

0

.

Since the inverse Fou rier tran sfor m ^~^ (U

Q

) of

u

0

is still a

tempered distribution we can write it as

(5.8)


19/20

AN INTEGRAL TRANSFORM ATION 755

References

[ 1 ] Aronszajn,N., Tracesofanalytic solutions

o f

heat equation ,

A sterisque,

2-3(1973), 5-68.

[ 2 ]

C hung, S.-Y., Positive

definite

temperature

functions and a

correspondence

to

positive

temperature functions,

Proc. Royal Soc. Edinburgh,

127A (1997),947-962.

[ 3 ] Chung , J.,C hung , S.-Y.,and Kim,D., Positive

definite

hyperfunctions,Nagoya Math.

J.,

140 (1995),139-149.

[ 4 ] C hung , S.-Y. and Kim, D., An example of nonuniqueness of the heat equation,

C o m m .

Part . D i f f . Equa.,

19

(1994),

1257-1261.

[ 5 ] , Distributions with exponential growth and Bochner-Schwartz theorem for

Fourier hyperfunction,P ubl RIMS , Kyoto

Univ.,

31 (1995),

829-845.

[ 6 ] G elfand,I. M. andShilov,G. E., GeneralizedunctionsII,Acad. Press,NewY ork , 1967.

[ 7 ] H ormander , L.,

The

analysis

of linear

partial differential operators

I,

Springer-

Verlag, B erlin-N ew Yo rk, 1983.

[ 8 ]

Haimo,

D. T.,

G eneralized temp erature

functions,

Duke MathJ. ,

33

(1966),305-322.

[ 9 ] , W idder tempera ture representations,J.

Math.

Andy.

App l ,

41(1973),170-178.

[10] Kim, K. W, C hung , S.-Y.,and Kim, D., Fourier hyperfunctionsas the boundary values

of

smooth solutions of heat equations,

Publ RIMS, Kyoto

Univ. ,

29 (1993),289-300.

[11] Lebedev, N . N ., Special

funct ions and

their applications, Dover Publications, 1972.

[12] M atsuzaw a, T., A calculus approach to hyperfunctionsII, Trans. Amer. Math. Soc. ,313

(1990), 619-654.

[13] Strichartz,

R., A guide to distribution

theory

an d

Fourier transforms,

C RC

Press, London,

1994.

[14] Stein, E . M . and Weiss, G .,

Introduc tion to

Fourier analysis

on Euclidean space,

Princeton

U niv., Prince ton, 1975.

[15] Widder,

D. V.,

Some analysis

from

classical analysis

in the

theory

of

heat equation,

Arch.

Rat ional Mech.

Anal,

21 (1966),108-119.

[16]

f

fj

eat

equat ion,Acad. Press,

N ew

York, 1975.


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integral transformations

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