A characterization of the spaces $S%5E{k/k+1}_{1/k+1}$ by means of holomorphic semigroups

A characterization of the spaces $S%5E{k/k+1}_{1/k+1}$ by

means of holomorphic semigroups

Citation for published version (APA):

Eijndhoven, van, S. J. L., Graaf, de, J., & Pathak, R. S. (1982). A characterization of the spaces

$S%5E{k/k+1}_{1/k+1}$ by means of holomorphic semigroups. (Eindhoven University of Technology : Dept of Mathematics : memorandum; Vol. 8202). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1982 Document Version:

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EINDHOVEN UNIVERSITY OF TECHNOLOGY

Department of Mathematics and Computing Science

Memorandum 1982-02 january 1982

A h c aracter1zat10n 0 • • f t e spaces h Sk/k+ 1 l/k+l

by means of holomorphic semigroups by

S.J.L. Van Eijndhoven, J. De Graaf, R.S. Pathak

Technological University Department of Mathematics PO Box 513, Eindhoven The Netherlands

A CHARACTERIZATION OF THE SPACES

S~~:::

BY MEANS .OF HOLOMORPHIC SEMIGROUPS

S.J.L. Van Eijndhoven, J. De Graaf, R.S. Pathak

Department of Mathematics, Technological University Eindhoven, The Netherlands

Abstract.

The Gelfand-Shilov spaces

S~,

a

=

k~1

' 8 =

k:l

,are'specU.l castm' of·

a

general

type of test function spaces introduced by De Graaf. We give a

self-ad-joint operator so that the test functions in those Sa spaces can be

ex-a

panded in terms of the eigenfunctions of that .elf-adjoint operator.

A.M.S. Subject Classifications 46F05, 35KJ5.

One of the authors (SJLVE) was supported by a grant from the Netherlands

2

-1. Introduction

De Bruijn's theory of generalized functions based on a specific

one-para-meter semigroup of smoothing operators [IJ was generalized considerably

by De Graaf [4]. In brief this extended theory can be described

as

follows:

In a Hilbert space X consider the evolution equation

(I. J)

~

_dt

= -

Au

where A is a positive, self-adjoint operator, which is unbounded in order

-tA

that the semigroup (e )t~O is smoothing. A solution u of (1.1) is called

a trajectory if u satisfies

(1. 2. i)

(1.2.ii) u(t) €

X.

The limit lim u(t) does not necessarily exists in

X

!

NO

The complex vector space of all trajectories is denoted by

TX

_,

AO The elements

of

TX,A

are called generalized functions.

The test function space SX,A is the dense linear subspace of

X

consisting

-tA

of smooth elements of the form e h, where h € X and t > 0; we have

Sx A

=

u e-tA(X). The densely defined inverse of e-tA is denoted by etA

, t>O

.A

For each , € SX,A there exists • > 0 such that e ,makes sense. The

pairing between SX,A and

TX,A

is defined by

(1.3)

Here (0,0) denotes the inner product in X. Definition (1.3) makes sense

3

-it does not depend on the'specific choice of T. For further results

con-cerning this theory we refer to [4].

The aim of the present paper is to show that for certain Gelfand-Shilov

spaces SB, [2], there exists an operator _~

A

such that SB • Sx _{a ,}

A.

This

leads to the result that the elements of the dtlal of SB can be interpreted

a

as trajectories. Furthermore, we find that a function in the studied

SB-spaces can be developed in a series of certain orthonormal functions. _a

2. Eigenfunction expansions of test functions in S!

Let us consider the following eigenvalue problem in i2(~)

(2.1) d

2 _2k

- Y + (A - x )y

=

0

dx2

where A is a real number and k a positive integer. It is well-known that

2

h d 2k h . • d h f ' 1

t e operator - ---2 + x as a p01nt spectrum an t e set 0 e1genva ues

dx

(An) is real, positive and unbounded. In the sequel we shall regard it

as ordered with An+1 ~ An' n

=

0,1,2, •••• The corresponding normalized

eigenfunctions {~n} form a complete orthonormal basis in L2(~)' SO by

the Riesz-Fischer theorem every f € L2(~) can be represented by

(2.2)

First of all we gather some of the estimates for the eigenvalues An and the eigenfunctions .n of the problem (2.1), and then characterize {.n} as elements

of certain SB-spaces. We take _a

~

_n (x) > 0 for large positive values of x, cf. Titchmarsh [5, Ch. VIII].

From Titcbmarsh (5, p. ]44] we have

(2.3)

2k.

A

=

d(nk+1)

n

According to Titcbmarsh we have the following estimates for the normalized eigenfunctions

(2.4) for all x € :R, n € :N, [5, p. 168].

x

(2 • .')

l~n(x)l::> ~n(xO)

exp{ -

I

(u2k -

An)~dU}

for

x~ xO~A!k

, [5, p. 165].

Xo

We take

Xo ..

<4

An)2k~

From a straightforward calculation it follows

:3

2 1 + 4k

{ I l l

1 k+ I}

l~n(x)l::>

3

An exp -

4

k+T

x

1

for Ixl ;::

2A~k

•

For any number a, 0 < a < _4(k+l)' 1 _{we have}

(2.6) where

The eigenfunction ~n(x) can be extended to an entire function ~t1(z). We want

to estimate ~ _n (z) in the complex plane. First we produce an estimate for

l~~

(0)1· Let

~

> 0 denote a point at which

~2

reaches its absolute maximum.

5

-I

We have 0 s

~

s _{. n}

~2k.

Integrate the equality

from 0 to ~. A crude estimate yields

1+1..+2-I

tV I (0)

I

s;

1

It

+ 2k' A 2 4k

n 3 n

Next, following the technique of Titchmarsh [5, p. 172] it can be shown that

00

tVn(z) == y(O)(z) +

I

{y(m)(z) - y(m-l)(z)}, z E C •

m=1

Here y(O)(z) ==

~

(0) + z

~'(O)

and y(m)(z). m

~

1, can be obtained from

n n

+ ( (s 2k - A _n }y (m-I) (s}(w - s}ds •

o

With

we get the estimate

I~

(z)i S K (lzi)exp(lzlk+1 + Ai

Izl) .

n n n

3 4

1+-Here Kn<lzl> ==

'3

An 4k (1 + (1 + 2k)i,,!

Izl)

2:

I/O)(z)l.

Now let d > O.

Then

1

whenever

Izi

~

dA2k and

k+l

expO ..

!

I

z

I)

s exp (dA;k )

whenever Izi

~dA!k.

Thus we have

k+l 6

-(2.7)

l~n(z)1

_{s Kn(lzl)exp(dA;k)exp(l + d-}k_>lzlk₊1).

The()rem 1.

The eigenfunctions ~n of the eigenvalue problem (2.1) are elements of the

a

1 k

space Sat where a

=

k+l snd

a •

_{k+T •}

Proof.

Sin¢e~ is an entire function and since it satisfies (2.6) and (2.7), in

n

view of the criterion of Gelfand and Shilov [2. p. 220J. the result follows.

Theorem 2.

and suppose there is • > 0 such that

k+l

a _n

=

19'(

exp (_'tA;k ) ) •

k/k+l

Then f E Sl/k+l •

Proof.

In (2.6) we can take a > 0 so small that. > a2k+l. Then for some C > 0

7

-and all x E: It

co

k+l

co k+ I 2k I

I

k+ I

S

eLK

exp{-(T - a2 )'A }exp(-a x ).

n=O n n

So If(x)1 S

c'

exp(-alxlk+1) for some C',> O. Further we can take

d > 0 and d < T, so that with the aid of (2.7)

00 If(z)1 S

L

n-O k+1 S exp «(l + d -k)

I

z I k+ 1 )

I

Kn (

I

z

I )

exp ( -(t - d) An2k ) n-O

for some

en

> O. By the criterion of Gelfand and Shilov as used in the

k/k+1

proof of Theorem 1, f E: Sl/k+l •

0

Let ~ be the self-adjoint operator in _L2(1t) defined by

(2.8) _~

==-Then as a corollary of Theorem 2 we have

Corollary 1.

The test function space S L2 (It) ,B

k

k+J

8

-Proof~

The functions ~ are the eigenfunctions of the positive self-adjoint

n - k+1

operator Sk with eigenvalues An2k " Let f €

S,L

2

(lt),B

k

"

Then there

exists h £ L2(lR.) and t > 0 such that

-1'B

f = e k h •

k+]

This provides

(f'~n)

=

exp(-1'A;k )(h,<\In)' So the coefficients (f ,<\In)

k+l '

~ k/k+l

are of the orde1;c exp ( -1' An ) • By Theorem 2 we have f € S l/k+ 1 • 0

We want to prove the converse of Corollary 1:

Theorem 3.

In the proof of this theorem we need some lemmas.

Lemma J.

Let ir,jr be nonnegative integers for r

=

J,2, ••• ,n. Then

where D is the differential operator

~

and where the coefficients Cij(t)

satisfy (j-t)

!

.

,

J.

~

1 ' " i-R. ~ (cij(t) - 0 if R. > min(i,j». etc.

9

-Proof. See Goodman [3,

p.

67].

o

LeIllll.a 2.

Let f be an infinitely differentiable function which satisfies the

f0110-wing inequalities for fixed AtB,C > 0 and

a,e

> 0, a +

e

~

Then for each n € Ii and i, j e: En

oS+l oa oa oS -1

where C_{I -} C, Al • 2 e A, B_{J •} 2 e Band 0 • (a + S) •

~.

n

Let n E :Ii and i,j E ::N • Then by Lemma

it j 1 i j

I'

11

I'

11

I(D x ••• D nx n)f(x) I

:5;

L

Ic •. (1)II(x J- D 1- f(x)l.

1. 1J

With the assumption (2.8) we estimate this series as follows

< C \' I j! rWi-:'!

AU-tIBli-tl

I

'_tla/j-t!ji_nIBli-tl

- L fiT ( '_n) t • _n J k

t k . J x . . 1. x. •

The latter series can be treated as follows

sU;P

10

-We have

With the aid of the inequality n~

And similarly

Combining these results, we derive

cr6+1 cra cra 06 where At • 2 e A, B]

=

2 e B.

o

Lemma 3. k/k+l· For f E St/k+l we have

I

(D2 - x2k)Pf(X)

I

~ K NP 2pk/(k+l) P t p . 012

' t , ...

11

-1 k

Proof. Let a

=

k+T '

B • k+l •

Let f €

SB.

The" there are positive constants A,B,C such that for all

a

X € :R

with t,q - 0,1,2, ••••

Now let p € E. Then

where V (D2,x2k) consists of a sum of (p) combinations of the form

s s

With the aid of Lemma 2 we have

B+] a Q

B

with Al

=

2 e A and B] • 2 e B.

So

,

.

12

-o

Proof of Theorem 3

Because of Corollary 1 we only have to prove the inclusion

Sk/k+1 c: SL (m) 8 •

l/k+l 2 .A , k

k/k+l

So let f Ii: Sl/k+l • Put an ,.. (£,cpn)' n Ii: :N • Then for each P E: :N fixed

With the~aid of Lemma 3 we get positive constants K£ and N

f such that

"(_D2 _{+ 2k)Pf}ll :;; K NP 2pk/(k+l)

x .., ffP •

And

By (2.4) and (2.5)

where ~ only depends on k. Therefore

Finally taking the infinum of the right hand side with respect to p we arrive at

13

-By taking Fourier transforms in Theorem 3 we derive easily.

Theorem 4. Sl/k+l

=

S _ k/k+l L 2(E.) ,Bk d2 k where

1\ .. (- d;i)

+ x • 2. References

I. Bruijn, N .G. De, A theory of generalized functions with

appli-cations to Wigner distributions and Weyl correspondence. Nieuw Archief voor Wiskunde 3, XXI (1973), 205-280.

2. Gelfand, I.M. and G.E. Shilov, Generalized functions, Vol. 2,

Academic Press, New-York (1968).

3. Goo4man~ R.,Analytic and entire vectors for representations of

Lie groups, Trans. Am. Math. Soc. 143, (1969) 55.

4. Graaf, J. ne, A theory of generalized functions based on

holo-morphic semigroups, T.H.-Report 19-WSK-02, Eindhoven University of Technology, 1979.

5. Titcbmarsh, E.C.,Eigenfunction expansions associated with second order