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

(1)

A characterization of the spaces $S^{k/k+1}_{1/k+1}$ by

means of holomorphic semigroups

Citation for published version (APA):

Eijndhoven, van, S. J. L., Graaf, de, J., & Pathak, R. S. (1983). A characterization of the spaces

$S^{k/k+1}_{1/k+1}$ by means of holomorphic semigroups. SIAM Journal on Mathematical Analysis, 14(6), 1180-1186. https://doi.org/10.1137/0514092

DOI:

10.1137/0514092

Document status and date: Published: 01/01/1983

Document Version:

Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)

Please check the document version of this publication:

• A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website.

• The final author version and the galley proof are versions of the publication after peer review.

• The final published version features the final layout of the paper including the volume, issue and page numbers.

Link to publication

General rights

Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain

• You may freely distribute the URL identifying the publication in the public portal.

If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement:

www.tue.nl/taverne Take down policy

If you believe that this document breaches copyright please contact us at: openaccess@tue.nl

providing details and we will investigate your claim.

(2)

SlAMJ.MATH. ANAL.

Vol. 14,No.6,November1983

(C)₁₉₈₃_{Society for}_Industrial_{and Applied}_Mathematics

0036-140/83/1406-0010$01.25/0

A CHARACTERIZATION

OF

THE SPACES

BY MEANS OF HOLOMORPHIC

SEMIGROUPS*

S.J. L. VANEIJNDHOVEN, J. DEGRAAF

"

ANDR. S. PATHAK

Abstract. The Gel’fand-Shilovspaces$ff,a _1/(k+_1),

_t=

k/(k

+

1),are special cases ofageneral type oftestfunctionspacesintroducedby de Graaf.Wegive a self-adjointoperatorsothat thetestfunctions in those$ffspacescanbeexpandedin terms ofthe eigenfunctions of that self-adjoint operator.

AMS-MOSsubject classification(1980).Primary 46F05, 35K15

1. Introduction.

De

Bruijn’s theory of generalized functions based on a specific

one-parameter semigroup ofsmoothing operators

[1]

was generalized considerably by deGraaf

[4]. In

briefthis extended theorycan be described asfollows:

In

a Hilbert

space

%

considertheevolutionequation

where 9 is a positive, self-adjoint operator, which is unbounded in order that the

semigroup

(e-tt)t>_

0 is smoothing.

A

solution u of

(1.1)

is called a trajectory if u

satisfies

(1.2i)

(1.2ii)

Vt>0V’r>O"

_{e-au(t)-u(t+r),}

Vt>0"

_u(t)%.

Thelimit

_limt,oU(t

)

doesnotnecessarilyexistin

!

6r _The _elements _of

Thecomplex vectorspaceof all trajectoriesis denotedby

,t--,a

arecalledgeneralizedfunctions.

The test function space

,a

is the dense linear subspace of consisting of

smooth elements of the form

e-tXh,

where

h%

and t>0; we have

,:

LJt>oe-tt(6).

Thedenselydefined inverseofe-tgxisdenotedbye

tgx. For

each

,

there exists z>0 such that

e*a

makes sense. The pairing between

_$,a

and

,a

is

definedby

(1.3)

Here (.,.)

denotes the inner product in

%.

Definition

(1.3)

makes sense for >0 sufficientlysmall, and due to the trajectory property (1.2i) it doesnot depend on the specificchoiceofr.

For

further results concerning thistheorywereferto

[4].

Theaimofthe present

paper

is toshow that for certain Gel’fand-Shilovspaces

$

[2]

there exists an operator 9 such that

-c,a.

This leads to the result that the elements of the dual of

ff

canbe interpreted as trajectories.Furthermore, wefindthat

afunctioninthe studied

$ff-spaces

canbedevelopedin aseriesofcertainorthonormal functions.

*Receivedby theeditorsFebruary 19,1982.

DepartmentofMathematics,TechnologicalUniversity, Eindhoven, the Netherlands. One of the authors

(SJLVE)wassupportedbyagrant from the NetherlandsOrganizationforthe Advancement ofPureResearch

(Z.W.O.).

(3)

CHARACTERIZATION OFSPACES BYHOLOMORPHICSEMIGROUPS 1181

2. Eigenfunctionexpansions of testfunctions in

.

Letusconsider thefollowing

eigenvalueproblemin

dE

(2.1)

_dxEY+()

xEk)y

O,

where is a real number and k a positive integer.

It

is well-known that the operator

-d2/dx2+x

2k _has _a _{point spectrum and the} _set _of _eigenvalues

₍₎

_is _{real, positive}

andunbounded.

In

thesequelweshallregardit asordered with

_hn+l

kn,

n--0,

1,,--Thecorrespondingnormalizedeigenfunctions

_{(Pn }}

form acompleteorthonormalbasis in

_E2(R).

So bythe Riesz-Fischer theoremevery

_f

22()

canberepresentedby

(2.2)

f=

n--0 where

a,

(

f,

q,

)

isan

e2-sequence.

First of all we gather some of the estimates for the eigenvahes

X,

and the

eigenfunctions

q,

oftheproblem

(2.1),

andthen characterize

_{

p,

_}

aselementsofcertain

ff-spaces.

We

take

,(x)>0

for large positive values of x, cf. Titchmarsh [5,Chap.

VIlil.

From

Titchmarsh

[5,

p.

144]

wehave

(2.3)

,,,--O(n2/(+’)),

n-)m.

Accordingto Titchmarsh wehave thefollowingestimatesfor the normalized

eigenfunc-tions

(2.4)

_{I (x)l 3}

2

hln+V4k

forallxR, nN

[5,p.

168],

(2.5)

[q,.(x)l<-q,,,(xo)exp{-f(u’--X,,)!/du}

for/-->/0-->hk/2

[5,p.

165].

Wetake

_XO--(kn)

l/2k. From

astraightforwardcalculation itfollows that

2

{

[x[k+l

}

[q’(X)

--<

-5

hi"+

3/4kexp

4 k+l

for

Ixl_>2X./2k.

For

anynumber a, 0<a<

1/4(k+ 1),

wehave

(2.6)

[,l,,,(x)l

<-g,,exp(-alxl"+),

x,

where

2

)l.+

(2k+l

(k+l)/2k

K,,

_-5

3/4kexp

a).,,

).

The eigenfunction

k,(x)

can be extended to an entire function

,(z).

We

want to

estimate

pn(z)

in thecomplex plane. First weproduce an estimate for

Ik,(0)l.

Let

_>0

denote a point at which

p,2

reaches its absolute maximum.

We

have

_0ln/2k.

Integrate

the equality

d

)2

d

(4)

1189, S.J.L.VANEIJNDHOVENJ.DE GRAAF AND R. S. PATHAK

from 0to

.

Acrude estimateyields

2

1 +

2k

_kln

+3/2+3/4k

I;(o)1-<

₃

Next,following thetechniqueofTitchmarsh

[5,

p.

172]

itcanbeshownthat

k,(z)=Y()(z) +

_E

_{(Y(m)(z)--Y(m-I)(z)},}

ZC. m--I

Here

y(O)(z)

k,(O)

+

z

k(O)

and

y(m)(z

),

m

-->

1,canbe obtained from

With

y(m)(z)---y(O)(z)-’l-

(s2k--.n)y(m-l)(s)(W--s)ds.

lY(">(z)-Y<m-’>(z)l<ly(>(z)l{Izl:++x"}

_(2m)t

weget theestimate

Here

I+(

z

)1

K.(lz

I)exp

(Iz[++

-I

_x’/lz

I).

4. ₊_3/4k _1/2

K.(lzl)-

_-h.

(1

+(1

+2k)x’./ll)_>ly<(z)l

Nowletd>0. Then

exp(

_X’f+

_Iz

)

_<

exp(

_d-klz[

k

+’)

whenever

Izl_>d./++

and

exp(

x’+/=Iz

I)_<

exp(

d,(

+l)/2k

₎

whenever

_Izl

d +/z _Thus_we_have

(2.7)

I+(

z

)1- K.(lz [)

exp(

_dX(

+,)/2k

₎

_exp(1

₊

_d-k

_)lzl

++l

THEOREM 1. The eigenfunctions

_of

the eigenvalueproblem

(2.1)

areelements

_of

the space

,

wherea

_1/(k

+

1)

and

_fl

k/(

k

+

1).

Proof.

Since

_k,

isan entirefunctionand since itsatisfies

(2.6)

and

(2.7),

in view of thecriterion ofGel’fandandShilov

[2,

p.220],the result follows. []

THEOREM2.Let

_{f 2(),}

and supposethere is >0 such that

a,,--O(exp(--’rh(n+"/Zk)).

Then

f$

k/k+l l/k+l"

(5)

CHARACTERIZATION OF SPACES BYHOLOMORPHICSEMIGROUPS 118

Proof.

In

(2.6)

we can take a>0 sosmall that ’>a2

k+.

Then forsomeC>0and all x

[f(x)l

X

tan[lqn(x)l

n---0

<-C

X

Knexp{--(z--a2’+l)(n+’)/2k}exp(--alx[’+l)

n--O

So If(x)lC’exp(-alxl

k+l)

for some C’>0. Further we can take d>0 and d<-, so

thatwiththeaidof

(2.7)

If(z)l

<

₂

lanllq(z)l

n--0

_<exp((1

+d-)lzl

+l)

_X

K,,(lzl)exp(-(z-d)2t(k,

+’)/2’)

n-O -<C"

exp((1

+d-’)lzl

+’)

for some C">0.

By

the criterion of Gel’fand and Shilov as used in the proof of

Theorem

f$k/k+

_l/k+l" [’-1

Let9_kbethe self-adjoint operatorin

2()

definedby

d2

(2.8)

9_k

₊

x2k"

dx2

Thenas acorollaryofTheorem 2wehave

COROLLARY 1. The test

_function

space

_e2(a),,

is included in

/+/+

.

Here

k----(k)(k+l)/2k.

Proof.

The functions

q,

are theeigenfunctionsofthe positiveself-adjointoperator

3

with eigenvalues

(n

k+l)/2k. Let

_fge),,.

Then there exists

_h2(

)

and z>O

such that

f

e-,3,h.

This provides

(f,p,,)-exp(--A(+/2’)(h,p,).

So the coefficients

(f,,)

are of the order

exp(-,h(

+

_l)/2g).

_By

_{Theorem 2}_we

havef

g/’+

l/k+

"

[-’] Wewant toprovetheconverseofCorollary1"

THEOREM3.

__k/k+l "2( ), 1/k+

Intheproofof thistheoremweneed somelemmas.

LEMMA 1.Let

Mr,Jr

benonnegative integers

_for

r-1,2,. .,n. Then

Oi’xJ’Di2...Di"xJ"-

_{cij(l)xlJ-tlDli-tl,}

1[

whereDis the

_differential

operator

d/dx

andwhere the

_coefficients

_{ci( l)}

satisfy

J!

I%(1)1-<

_(j-l)!

(6)

1184 S.J.L. VANEIJNDHOVEN,J.DEGRAAF ANDR. S. PATHAK

Weusemulti-indices, and

_{Ii1-i+ i2}

+

+in,

_i!"-il!i2!

in!,

etc.

Proof.

See

Goodman[3, p.

67].

LIMMA

2. Let

_f

be an infinitely

_{differentiable function}

which

_satisfies

the following inequalities

_for

_fixed

A, B,

C >0 anda,

_fl

> O,a

+

_fl

>__1:

(2.9)

[(xDtf)(x)[<_CABtkl/t,

k,l-O, 1,2,

.

Then

for

each n Nand ,j f

(Di’x

j’

Di"xJ"f

)(

x)lx

C

alnl(llJlUllil

all

whereC C,

A

=2a+leA,

B

=2elB

and

e-(a+B)

-.

Proof.

Letn M andi,j

n.

Thenby

Lemma

Withtheassumption

(2.9)

weestimatethisseriesasfollows:

I<--min(i,j)

_<CE

1

J!

1!

(j--l)!

li--11!

<_CAUlBII,

_{1--( (j-l)’}

li-ll’

Ij-ll

i-Thelatterseriescanbetreatedasfollows

l<--min(i,j)

I--T.

(j-

1)!

li-

ll!

[j-

1[

u-zlli-

_II

_<sup sup

J

i,i<_I,I

[i-ll

t(IZl

!)

I-1

Ij-ll!(IZlt)

-1

We

have

(

)-’

,<_j

I11

With the aid oftheinequalityn

<

n

<

n

!e

n"

(1i1!)

a

_lal’-’l

<:

(li-II

!)ot

li--l

<--2lilellil([i[

!)[

(li-l[)(-)[li-tl<--2lile[lilli[/lil

andsimilarly

l-ll

(7)

CHARACTERIZATION OF SPACES BY HOLOMORPHIC SEMIGROUPS 1185

Combiningtheseresults,we derive

(

D

xj’

_D

i’,_xJ,

f

)

(

x

)

<

ca

_l

_nlil

_lj

’t all 2a+

eaA

B

2"e#B.

E] whereA

LEMMA

3 For

_f

q /k+ /k+ wehave

f(x)i<-KN p

p-O,

l,2,

whereKandNare

_fixed

positive constantsdependingon

_f

Proof.

Let

a

_1/(k

+

1),

_fl-

k/(k

+

1). Let

_f

_ff.

Then there are positive con-stants

A, B,

Csuch that for all x R

withl,q=0, 1,2,....

Now

let pq_N.Then

1(

X Dqf

)(

X

)1--<

calnqlatq

oq,

p

X

s--0

where

Vs(D2,

x

2k)

consists ofasum of

(P)

combinations oftheform

(D

2

_{)il(x2k )Ji}

_(D

2

_)in(x2k

_)Jn

where

_i

+

_i--s

andj

+

+j=p-s.Withtheaidof

Lemma

2wehave

with

_A

20+

e"

and

_B

2"eB.

So

](DX--x2)Pf(xl[

C

p

C s=O

=C(A

Substituting the values of and

_B

itfollows that

[(D

2 _x2

)P

f(

x

)1

C(

A2

+

B2

)p

p2pk/(

k+1) where

_A

=((2k)"A)

and

_B

2B.

Proof of

Theorem 3.

Because

of

_Corolla

weonlyhavetoprovethe inclusion

So

_letf

/+

/+. Put

a

(f,),

nN.Then foreachpN fixed

(8)

1186 S.J.L. VANEIJNDHOVEN,J.DE GRAAF AND R. S. PATHAK And

By (2.4)

and

(2.5)

n--0,1,2,-. 8

tln

+5/4k

_Ckkln

+3/4k <--

+

-3 wherec_konly dependsonk.Therefore

la.l--<c

kln+

5/4k

_t-p

_gf

_N[p2pk

/(k+1)

Finally taking theinfinumofthe right-handside withrespecttopwearrive at

lanl<_c,Kf X,+

s/4kex

p

(2fie-tNf-1/2/3)

_kln/’2O

with/3--k/(k

+

1).

Fromthistheassertionfollows.

By

takingFouriertransformsinTheorem 3wederiveeasily

THeOreM4.

$1/+

_--$e

k/k+l (n),

where

k-((-d2/dx2)

k

₊

_x2)

+

_)/.

REFERENCES

[1] N. G. DE BRUIJN, A theory _ofgeneralized_functionswith applications to Wigner distributions and Weyl correspondence,Nieuw Archief voorWiskunde,3,XXI (1973),pp. 205-280.

[2] I. M. GEL’FANDANDG.E. SHILOV,Generalized Functions, Vol.2,AcademicPress, NewYork,1968.

[3] R. GOODMAN,Analytic andentire vectors_forrepresentationsofLiegroups,Trans. Amer.Math. Soc.,143

(1969),pp. 55-76.

[4] J.DEGRAAF, A theoryofgeneralized_functionsbasedonholomorphicsemigroups,T.H.-Report79-WSK-02, EindhovenUniversity ofTechnology,Eindhoven, theNetherlands,1979.

[5] E. C.TITCHMARSH,Eigenfunction ExpansionsAssociated withSecondOrder_DifferentialEquations, PartI,