Gelfand-Shilov spaces of infinitely differentiable functions onthe positive real lineCitation for published version (APA):van Berkel, C. A. M., & Eijndhoven, van, S. J. L. (1992). Gelfand-Shilov spaces of infinitely differentiablefunctions on the positive real line. (RANA : reports on applied and numerical analysis; Vol. 9207). EindhovenUniversity of Technology.

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EINDHOVEN UNIVERSITY OF TECHNOLOGYDepartment of Mathematics and Computing Science

RANA 92-07April 1992

GELFAND-SHILOV SPACES OFINFINITELY DIFFERENTIABLEFUNCTIONS ON THE POSITIVE

REAL LINEby

C.A.M. van BerkelS.J.L. van Eijndhoven

Reports on Applied and Numerical AnalysisDepartment of Mathematics and Computing ScienceEindhoven University of TechnologyP.O. Box 5135600 MB EindhovenThe NetherlandsISSN: 0926-4507

Gelfand-Shilov spaces of infinitelydifferentiable functions on the

positive real line

by

C.A.M. VAN BERKEL AND S.J.L. VAN EIJNDHOVEN

Abstract

In this paper a number of characterizations of the spaces Sa ,even , S~ven and S~,evenconsisting of all even functions in the Gelfand-Shilov spaces Sa, SfJ and S~, respectively,is presented. Introducing Goo GfJ and G~ as the spaces of all Coo_ functions f on 1R+with the property that x 1-+ f(x 2 ) belongs to Sa ,even , S~ven and S~,even, each of thementioned characterizations leads to a description of the functions f in Gal GfJ and G~

in terms of growth conditions on f and its derivatives f(l) 1 1 E IN. As such the paperyields refinements and extensions of results in a recent paper by Duran [Dul].

AMS Classifications 46F12, 46FIO, 26F05

Introduction

The Gelfand-Shilov spaces have been subject of investigation by a lot of mathemati­cians in quite different fields. We mention Gong-Zhing Zhang [Zh] who showed theconnection between the spaces S:, Q 2:: ~, and Hermite pansions, Kashpirovskii [Ka],who proved the intersection result S~ = Sa n SI3, De Bruijn [Br], who employed thespace S;f: as a test space for a theory of generalised functions and Goodman [Go],

who showed that the spaces Sille and si/le are analyticity domains of unitary represen­tations on L 2 (JR) of certain nilpotent Lie groups. More recently, Van Eijndhoven [EI]characterized the Gelfand-Shilov spaces as intersections of analyticity domains of theoperators IQl1

/a and JPI1113 where Q denotes the self-adjoint operator of multiplication

by x in L2(JR) and P the self-adjoint differentiation operator id/dx in L2(JR), thusextending Goodman's results.Furthermore, Ter Elst and Van Eijndhoven [EE] proved that the spaces S~'1::Z' are theGevrey domains of certain symmetric differential operators in L 2(JR).

For f E S~ its Fourier transform :Ff belongs to S$. So the spaces S: are Fourierinvariant and therefore suitable as test function spaces in distribution theories involv­ing the Fourier transformation. The spaces Sa,even, S~ven and S~.even playa similar rolewhen dealing with the Hankel transformations. In [EG] it was proved that S:,even for~ ~ Q < I remains invariant under some modified form of the usual Hankel transfor­mations as defined in [Mos] , p. 397. An extended version of this result can be found in[Du2] and in [E2].In [EB] a characterization of the spaces Sa,even, S~ven and S~.even in terms of the growthbehaviour of a COO-function on JR+ and of its Hankel transforms was proved. It turnedout that the Hankel transformations are bijections from Sa,even onto S':ven and fromS~,even onto SN,even' A characterization of these spaces by means of the operators x and~ tz in L 2 (JR+) was proved by [Dul].Duran's paper is partly based on results in [EB]. However these results do not seemfully exploited. Moreover Duran's characterizing conditions can be somewhat weak­ened and put in a context closer to the results in [EB] by employing properties of theHankel transformations as stated therein.

The paper is divided into three sections. In Section I we introduce some notation andpresent some auxiliary results. In Section 2 characterizations of the Schwartz space S+of COO-functions on JR+ and the space Seven of all even functions in the Schwartz spaceS of Coo functions on JR. In Section 3 we consider the spaces Sa,even, S~ven and S~.even

and we present a number of characterizations of these spaces. Besides we introduceand describe the spaces Ga , GI3 and G~ (notation in accordance with Duran's).

I

1. Notation and auxiliary results

The test space in the theory of tempered distributions is the space S of all COO_ functionson JR of rapid decrease. So an infinitely differentiable function 1 on JR belongs to S iffor all k,l E INo,

In order to get a feel for the strength of the theory of tempered distributions it ishelpful to have several characterizations of S. The characterizations (1.1), (1.2) and(1.3) we present here, are known, ef. [Go], [Jo], lSi], [El], or can be derived easily fromthe mentioned literature.

(1.1) Let 1 be an infinitely differentiable function on JR.

- f E S iff xkf E L 2 (lR) and f(l) E L2 (lR) for all k, 1E INo;

- f E S iff xkf E L2(IR) and f(l) E Loo(lR) for all k,l E INo.

In the above characterizations, growth conditions and differentiability conditions occurseparately.

Let :F denote the Fourier transformation on L2(JR),

1 /00 .(:Ff) (x) = y"'j;i f(y) e-'ZJ/ dy .-00

Then:F is a unitary operator on L 2 (lR) and:F diagonalizes the differentiation operator,

:F1' = ix:Ff .

With the Fourier transformation, S admits the following characterization.

(1.2) Let f be a square integrable function on lR.

- f E S iff xkf E L 2 (lR) and xl:Ff E L 2(lR) for all k,l E INo;

- 1 E S iff x"1 E Loo(lR) and x l :F1 E Loo(lR) for all k,l E INo.

Let Q denote the self-adjoint operator of multiplication by the identity function inL 2(lR) with maximal domain D(Q) and let P = :FQ:F* denote the differentiationoperator in L2(JR) with D(P) = :F(D(Q)). Then characterization (1.2) says

2

where

00 00

Doo(Q) = n D(Qn) and Doo(P) = n D(pn).n=1 n=1

In the third characterization of S the Hermite basis in L 2(IR) is used. For n E INo theHermite functions 'l/Jn are defined by

They form an orthonormal basis in L 2(lR). The following characterization of S wasproved:

(1.3) Let j be a square integrable function on lR.

- j E S iff sup n 1c l(f, 'l/Jn)L3(n) I< 00 for all k E INO'nERo

The Hermite functions 'l/Jn satisfy the differential equation ( - ::3 + x2) 'l/Jn = (2n +1) 'l/Jn, n E INo, so that characterization (1.3) can be reformulated as

As we see from characterization (1.2), S is Fourier invariant. So its dual space S', i.e.the space of tempered distributions, is Fourier invariant, too.

From characterization (1.1) it follows that each tempered distribution can be writ­ten as the sum of a regular and an irregular distribution: For each c) E S' there existbounded functions 91,92 on IR and k, I E IN such that

00 00

c)(f) = f x1c j(x) 91(X) dx + f j(l)(x) 92(X) dx, j E S .-00 -00

From characterization (1.3) it follows that each tempered distribution can be repre­sented by a Hermite pansion: For each c) E S',

00

c) = L <p('l/Jn) 'l/Jnn=O

to be understood as

3

00

~(f) = I: (f, tPn)L2(B) ~(tPn), f E S .n=O

In the paper [EG:I.] the space Seven consisting of all even functions f in S, was studied,and the following description of Seven was derived.

(1.4) The even extension of an infinitely differentiable function f on IR+ belongs to Siff for all k,l E JNo,

Another characterization was obtained via the Laguerre basis (lL~)neNo which is or­thonormal in the Hilbert space XII = L2(IR+, X211+l dx) where II ~ -~ arbitrarily.Therefore, define

( )

1/2ILII(x)= 2r(n+1)

n r(n+II+1)

with L~ the n-th Laguerre polynomial of order II,

For II = -1/2 we have

1

lL~"2 = (-1t V2 tP2n, n E JNo .

_1So replacing tPn by lLn 2 in (1.3) we obtain a characterization of Seven in terms of

1

expansions with respect to the basis (lL~"2). Since for 0'., (3 ~ -i,

L~ = t (0'. - ~)m L~_m ,m.m=O

as a quite natural consequence we have, cf. [E2]:

(1.5) Let II ~ -~. The even extension of f E XII belongs to Seven iff for all k E !No,

sup n" I(f, lL~)xvI < 00 •nENo

With Jv , the II-th order Bessel function, introducing the Hankel transformation on XIIby

4

00

(lH"f) (X) = / (xy)-" J,,(xy) f(y)y2"+1dy, f E X"'o

we have

lH"JL~ = (-1t JL~ ,

cf. [MOS], p. 244. So lH" is a unitary operator on X" and, by (1.5). Seven re­mains invariant under lH" for each II ~ - ~. (Observe that lH_! is the Fourier-cosine­transformation.) A characterization of Seven in terms of Hankel transforms was provedin [EB]:

(1.6) Let II ~ -~. The even extension of a measurable function f on IR+ belongsto S iff for all k, 1E INo,

For II = -~ this characterization is a reformulation of (1.3). For future use we shallreprove (1.6) with Loo(JR+) replaced by X". First we present some auxiliaries.

(1.7) LEMMA

Let II ~ - ~ and let f E X" such that also X,,+3/2 f E X". Then f E L1(JR+ , x2"+l dx)and lH"f is continuous.

ProofUsing Cauchy's inequality,

00

/ If(y)l y2"+1dyo

00 00

~ (/ 1{1 +y)1 y2"+1 f(y)1 2dy)1/2 (/ (1: y)2 dy)1/2o 0

~ Ilfllx" +2I1x"+3/2fllx" ,

so that f E L1(JR+ , x2"+l dx).

Since zi J-i(z) = /! cos z and since for II ~ -~,

-" 1 () 2 1 (1)" /1 ( 2),,_1 ()dz "Z = 1r1/ 2 r(1I + 1/2) 2 1 - t 2 cos zt t,o

cf. [MOS, p. 79], we have sup Iz-" J,,(z)1 < 00, and so the integralz~O

00

/ (xy)-" J,,(xy) f(y) y2"+l dyo

5

converges absolutely and uniformly on 1R+.

From [EB, Proposition (1.3)] we take the following result:

o

(1.8) PROPOSITION

Let -! :::; J.t < v and let f E L2(IR+, (1+t2 )a t 211+1 dt) with Q > v+1. Then Illllf E X,..and for almost all x > 0,

o

We come to the following adaption of Lemma (1.5) in [EB].

(1.9) LEMMA

Let -l :::; J.t < v and let W(x) be a nonnegative function on IR+ such that (1 +x )-(11-,..+1) W( x) is nondecreasing on [b, (0) for some b> O. Let f E XII such that

for all k, IE INo and suppose in addition that W(x) Illllf E XII' Then

ProofLet J.t E 1R with -~ :::; J.t < v. Since x'Illllf E XII for all I E INo, it follows from Lemma(1.7), that f = IllII(IllIIJ) is continuous on IR+. Continuity of f together with thefact that x 1c f E XII for all k E INo, yields x 1c f E X,.. for all k E INo. Thus we obtaincontinuity of Ill,..f on 1R+ by Lemma (1.7).We conclude that

Furthermore, since x'JHIIf E XII for all lEINo,

and so replacing f by JHIIf in Proposition (1.8),

2,..-11+1 /00(Ill f) ( ) (t2 - X2 )1I-,..-1 (/Hllf) (t) t dt .

,.. X = f( v - J.t) ." ." ." ."z

6

The proof is completed by the following estimations:

00 00

=11(1 +X)-(II-I-'+l) W(X) 1(e - X2t-I-'-1 (lHllf) (eH del2X21-'+1 dxb z

00 00

$ 1II (1 + e)-(II-",+l) W(e) (e - x2t-",-1 (lHllf) (e)e del2x21-'+l dxb z

00 00

$ 1(I 1(1 +e)-(II-I-'+l) W(O (lH1If) (0\2 (e - x2t-I-'-1 ell+3 de)·b z

00·(1 (e - x2t-I-'-1 e-211- 1 dO x21-'+l dxz

00 00

= l B(v - JL, JL +1) 1x-1(/1(1 +et(II-I-'+l) W(O (lHlI!) (012.b z

00

= l B(v - JL, JL + 1) 11(1 + et(II-I-'+l) W(O (lHlI!) (01 2 ell+3.b

e·(1 (e - x 2t-",-1 X-l dx) deb

o

As a consequence of Lemma (1.9) there is the following extension of characterization(1.6).

(1.10) THEOREM

Let v ~ -~ and let f E X;.to Then the even extension of f belongs to S iff for allk,l E INo,

7

ProofLet the even extension of I belong to S. Then by (1.6),

for all k, 1E lNo, from which the necessity of the condition follows. For sufficiency let forall k,l E lNo, x 1cI E Xv and x'lHvI E Xv' If v = -~, characterization (1.2) yields the

wanted result (Recall that for an even function I E L 2(lR), (Fn!R+ = IH-!uIR+))·If v > -l note first that from the fact that x'lHvI E Xv, 1 E lNo, it follows that I iscontinuous and therefore that x 1cI E X_! for all k E lNo, from the fact that x 1c I E Xv,

2

k E lNo. Applying Lemma (1.9) with W(x) = x P , p> v +3/2 and J.t = -l yields

So with p = 1+ v +3/2, 1E lNo,

o

For J.t, v ~ -~ the linear mappings IH~IHv are bijective on Seven. By Proposition (1.8)for I E Seven and -l :s J.t < v

We see that for v ~ -~ and k E lNo,

The operator IHv+aIHv, v ~ -l, a ~ -v -l interprets the fractional differentiation(- ~ ~r·

8

2. The spaces S+ and Seven

(2.1) DEFINITION

The space S+ consists of all infinitely differentiable functions on JR+ with the propertythat for all k, I E JNo,

sup Ix 1e f(l)(x) I< 00 .eeB+

There exist relations between the space S+ and the spaces S and Seven as indicated inthe next theorem.

(2.2) THEOREM

Let f be an infinitely differentiable function on JR+.

(i) f E S+ iff there exists 9 E S such that I = g\.R+ '

(ii) I E S+ iff the function h : x ~ f(x 2 ) belongs to Seven.

Proof

(i) Let f E S+. From Borel's theorem, see [Zu, Chapter 1, Exercise 1], the existenceof a function 'P on JR with compact support can be derived such that

Now define 9 on JR by

{

I(x) ,g(x) =

'P(x) ,

x> 0 ,

x::;O.

Sufficiency of the condition is trivial.

(ii) Let h on JR be defined by hex) = f(x 2 ), X E JR. Then for alII E JNo,

and so the statement ~ollows from characterization (1.4) of Seven. o

The connections between S+, S and Seven, as just mentioned and the characterizationsof S and Seven yield characterizations of S+ accordingly. For this, introduce the Hilbertspace YII = L2(JR+,XII dx), the orthonormal basis (L:~)nE.J\To in 1";"

9

and the Hankel-Clifford transformation 'H.., on x"

00

('H..,1) (x) = ~ ! (xytt.., J..,((xy?/2) f(y) y'" dy .a

Then for fEY.." h defined by h(x) = f(x 2), X > 0, belongs to X.., with

('H..,1) (x2) = (IH..,h) (x) and v'2 IIiIlY" = IIhllx" ,

so that

'H..,£~ = (-1)" £~ .

(2.3) THEOREM

I Let f be infinitely differentiable on 1R+.

- f E S+ iff x1e f E Loo (IR+) and f(l) E Loo (IR+) for all k,l E INa.

II Let n 2:: -~ and let fEY..,.

- f E S+ iff x1e fEy;, and x''H..,f E y;, for all k,l E INa;

- f E S+ iff x1e f E Loo(IR+) and x''H..,f E Loo (IR+) for all k,l E INa.

III Let v 2:: -~ and let fEY..,.

- f E S+ iff for all k E INo,

sup n1e I(I, £~)y"I < 00 .

nERo

Proof

I From the definition of S+ necessity of the condition follows. For its sufficiency letf satisfy x 1e f E Loo(IR+) and f{l) E Loo (IR+) for all k,l E INa. Then there is afunction 'P on 1R with compact support such that

10

Define the function 9 on JR by

{

f(x) ,g(x) =

ep(x) ,

x> 0,

x ~ o.

Since ep has compact support, x"g E Loo(JR) and g<'} E Loo(JR). So by characteri­

zation (1.1), 9 E Sand f = glll+ E S+.

II and III. These assertions are consequences of the relation between Seven and S+ asstated in Theorem (2.2) (ii) and of the characterizations of Seven given in (1.5),(1.6) and in Theorem (1.10). 0

Remarks

- For f E YII the function h : x 1-+ f(x 2 ) belongs to XII and (lHllh) (x) = ('HII!) (x2).

So by Proposition (1.8), for f E S+ and 11, it ~ -~ with 11 > fl,

Therefore, for each 11 ~ -~ and k E !No,

f E S+ .

- The space S+ remains invariant under the Hankel-Clifford transformations 'HII , 11 ~1

-2'

- The space S+ is interesting also because of its behaviour with respect to the Laplacetransformation. It maps S+ onto the space of all analytic functions on the open righthalf plane which are infinitely differentiable on the imaginary axis. See [Du2,3].

In the next characterization we connect the space S+ with the operators x" and

X'/

2 (ctz)'·(2.4) THEOREM

Let 11 ~ -!. For an infinitely differentiable function 9 on 1R+ the following conditionsare equivalent

(i) 9 E S+ ;

11

(iii) X(k+I)/2g<l) E YII fOl all k,l E INo ;

ProofThe implications (i) =? (iii) =? (iv) and (i) =? (ii) =? (v) need no explanation. Weprove that (iv) =? (i) and (v) =? (iv). Assume 9 satisfies condition (iv).Define 1 on (-1,00) by I(x) = g(x + 1). Then xle1 E YO, k E INo, since

00

::; J I(x + l)le-!lIg(x + 1)12 (x +It dxo00::; JIx le"g(x)1 2

XII dx ::; IIxle"gIlY" ,1

with kll = k for v ;:::: 0 and kll = k + 1 for -~ < v < O. Further l(l) E YO for 1 E IN,smce

00::; J(x +1)l+lIlg(l)(x + 1)1 2 dxo

::; IIx'/2g(l)IIY" .

Since 1 is infinitely differentiable on (-1,00) there is an infinitely differentiable function'" on JR with compact support such that ",(l)(0) = 1(l)(0). Define h on JR by

Then

{

I(x) ,h(x) =

",(x) ,

x;::::o,

X < O.

00

J Ix le h(x)1 2 dx < 00 and-00

00J IM')(x)12 dx < 00 .-00

So by (1.2), h E S and hence 1 E S+.

For 1E INo we have by Cauchy-Schwartz's inequality,

12

So, using 1 E S+, we obtain g(l) E L1(m+ , yl+v dy). It follows that 'H'+vg(') is continu­ous with

00

(*) ('H'+vg('») (x) = ~ f (xy)-!(l+v) l'+v((xy)1/2)g(l)(y)yl+vdy .o

In the next part of the proof we show that we can apply integration by parts inthe above integral expression. First observe that (x + 1)(l+V)/21(l-1) E Loo(JR+) andtherefore x(l+V)/2g(I-1) E L oo ((l, 00)), 1 E IN. Further, for 0 < x < 1 and 1 E IN wehave

1

+2 f ly!¥g(l-l)(y)lll:v y!¥-l g(l-l)(y)+y!¥g(I)(y)ldyIII

1

:::; IgC'-1)(1)12 + (I + v) f yl-1IgC'-1)(y)1 2yVdy+III

1

+2 f ly(I-1)/2 gCl-1)(y)ll y'/2 gC')(y) IyV dyIII

Hence for each 1E IN,

Since e-p. lp.(e) is bounded on m+ for all 11 ~ -1/2, it follows that for each x E m+,

tends to 0 as y 10 or y i 00.With the recurrence relation

13

we obtain through integration by parts

Repeating this procedure yields

Since 'Hv+l is unitary on Yv+1we have proved that for all 1E IN,

00 00! x' I('Hvg) (x)1 2XV dx = 221 ! x' I('Hv+lg(l») (x)1 2

XV dxo 0

00

=221 ! xl lg(l)(x)1 2XV dx < 00 •

o

We see that for all k,l E INo,

whence 9 E S(1R+) by Theorem 2.3.II and the proof that (iv) implies (i) is complete.

Let 9 satisfy condition (v). Then clearly f on IR+ defined by f(x) = g(x+1), x E IR+,belongs to S+. Hence for all k, 1E lNo,

00! Ix le f(l)(x)1 2dx < 00.

o

It follows that for all k,l E INo,

00! Ix(Ie+I)/2g(I)(x)12dx ~ sup IXI/ 2g(l)(x)1 2+o 0:$:1'9

00

+! (x + 1)1e+1 If(I)(x)12dx < 00 ,

o

whence 9 satisfies condition (iv) with 11 = O. o

RemarkCharacterizations of S+ as in the preceding theorem are not in Duran's paper [Du1].In fact Duran states that such a characterization cannot be valid. But his examplef(t) = vii e-t does not work since it does not satisfy condition (ii) in Theorem (2.4):For this consider its second derivative,

14

Clearly t f(2) is not bounded on 1R+.

The final characterization of S+ presented here, is a consequence of Theorem (2.4)and the following result.

(2.5) LEMMA

Let v ~ -~ and let / be an infinitely differentiable function on 1R+ with X1c/ 2

/ E 1';,and f(l) E Yv for all k,l E lNo. Then for all k,l E lNo,

ProofThe proof is by induction with respect to I. For a > v + 1,

z

xa If(x)1 2 = 2 Re f [aea

-1 J(e) +ea 1'(e)] J(e) de

o

::; 2a lIe(a-v-l)/2 fll~v + 2 Ilea - v fllyv 1I1'IIyv

and so for all f3 > 0,

x/3 f( x) -+ 0 as x -+ 00 .

Moreover, for x > 1,

z

11'(x) - 1'(1)1 = If f"(t) dtl ::; 11/"lIyv qv(X)1

where

if v ~ 0 ,

if - ~ ::; v < 0 .

Now let k E IN. Then

00

f x1c+v 11'(x)12 dx = x1c+v f(x) f'(X)I~1

15

oc-! {(k +v) X1e+"'-1 P(x) + x1e+'" 1"(x)} l(x) dx1

:::; 11(1)111'(1)1 + (k + v) Ilx"-1111 111'IIY" + IIx"ll1y" 1I1"lly" ,

and

1 1! X1e+'" 1l'(x )12 dx :::;! x'" If'(x )12 dx :::; II l'II}" ,o 0

so that

oc! X1e+'" 11'(x)12 dx < 00 •

o

We find that X k / 2l' E Y", for all k E INo.So with 11 = If we have for all k, 1E INo

which yield, repeating the previous step, x"/21" = x"/21{ E Y"" and so on. 0

(2.6) THEOREM

Let v ~ - ~ and let 1 be an infinitely differentiable function on IR+. Then 1 E S+ ifffor all k, 1E INo,

x"/21 E YII and JCZ) E Y", .

ProofThe necessity of the condition is evident from Theorem (2.4).So let 1 satisfy the stated condition for some v ~ -~. Then Lemma (2.5) says thatx"/21(Z) E YII for all k,l E .iNo. It follows that x k / 21 E YII and xZ/21(Z) E Y", for allk,l E INo. So 1 E S+ by Theorem (2.4). 0

The above characterizations of S+ and the relation between S+ and Seven yield neces­sary and sufficient conditions on an infinitely differentiable function 1 on IR+ to havean even extension to S. They are additional to the ones given before.

(2.7) THEOREM

Let v ~ -! and let 1 be an infinitely differentiable function on IR+. Then the followingstatements are equivalent.

16

(i) The even extension of f belongs to Sj

(ii) xlr+l (~ tz)' f E Lco(]R+) for all k,l E INo ;

(iii) x lc f E Lco(]R+) and x' (~ tz)' f E Lco(]R+) for all k,l E INo ;

(iv) xlr+l (~ tz)' f E X" for all k,l E INo ;

(v) x lc l E X" and x' (~ tz)' I E X" for all k,l E INo ;

(vi) x lc (~ tz)' I E X" for all k,l E INo ;

(vii) x lc f E X" and (~tz)' I E X" for all k, I E INo .

Finally we present two very elegant relations which put corresponding results in [Du1]in a wider context.

(2.8) LEMMA

(i) Let I E S+. Then for all k,l E INo,

IIx(lc+I)/2 (d:)' Illy" = 2lc-1IIx(lc+I)/2 (tz)lc 'H"fIlY" .

(ii) Let I E Seven' Then for all k, lEINo,

II x(lr+l) (~ tz)' Illx" = 2lc- 1 Ilx lc+1 (~ tzr lH"/llx" .

ProofSince for J.L 2: -l, 'HI-' is unitary on YI-" we have for a > -(v + 1/2),

Thus we derive

The proof of (ii) runs the same.

17

o

3. Gelfand-Shilav spaces Se even and G~I

The spaces Sa, Sf3 and S~ were introduced by Gelfand and Shilov as subspaces of S.Let us mention their definition and some of their properties. For a ~ 0 the space Saconsists of all j E S for which there are constants A > 0 and B, > 0 depending on 1and j, only, such that for all k,l E lNo,

sup Ixkj(l)(x)1 ~ BIAk(k!t .illER

Sa is nontrivial for all values of a; So consists of all infinitely differentiable functionson JR with compact support.For 13 ~ 0 the space Sf3 consists of all j E S for which there are constants B > 0 andAle > 0 depending on k and j, only, such that for all k,l E lNo,

sup Ix le j(I)(x)1 ~ AleB'(l!)f3 .illER

For all values of 13, 5 f3 is nontrivial; for 13 < 1, 5 f3 consists of entire analytic functions;51 consists of functions which are analytic on a strip around the real axis where thewidth of the strip depends on the particular function.

For a ~ 0 and 13 ~ 0 the space S~ consists of all j E S for which there are con­stants A, B, C > 0 depending on j, only, such that for all k,l E lNo,

sup Ixlej(I)(x)1 ~ CAleB' (k!t(l!)f3.illER

The space 5~ is nontrivial if a +13 ~ 1 with a > 0 and a = 0 and 13 > 1. Kashpirovskiiproved the intersection result

Refinement of this result can be found in [El]. We observe that for a + 13 ~ 1, a > 0and 0 ~ 13 < 1 the space 5~ consists of entire functions j for which there are constantsJ{, a, b > 0 such that

Ij(x + iy)1 ~ J{ exp(_alxI 1!a + b!yI1!1-(3) •

(3.1) THEOREM

Let a > 0 and 13 > 0, and let j be a square integrable function on JR.The statements in each of the following triples are equivalent

I (i) j E Sa ,

18

III (i)

(ii)

(ii) there is t > 0 such that exp(tlxl l/

Q) f E Loo(IR) and

for alII E INo, x':Ff E Loo(IR) ,

(iii) there is A > 0 such that for all k E INo

sup Ixkf(x)l::; Ak+l(k!)QillER

and for alII E INo, sUPIllER Ix'(:FJ) (x)1 < 00.

II (i) f E Sf3 ,

(ii) for all k E INo, xkf E Loo(IR) and there is t > 0 such that exp(txl/ f3 ):Ff E

Loo(IR) ,

(iii) for all k E INo, sUPIllER Ixkf(x)\ < 00 and there is B > 0 such that for alllEINo,

sup Ix'(:FJ) (x)1 ::; B'+l(l!)f3 .illER

f E S~ ,

there is t > 0 such that

exp(tlxl l/

Q

) f E Loo(IR) and exp(tlxl l/ f3 ):Ff E Loo(IR) ,

(iii) there is A'B> 0 such that for all k, lEINo,

sup Ixkf(x)l::; Ak+I(k!tillER

and

sup Ix':Ff(x) I ::; B'+I(l!)f3 .illER

ProofIn all three cases I, II and III equivalence of (ii) and (iii) follows from simple estima­tions based on properties of the function x k exp(-tx l / lI

) for 1I > 0, X > 0 and t E 1R+.The remaining part of the proof can be found in [E1], p. 140. 0

(3.2) THEOREM

In Theorem (3.1) in all statements Loo(IR) can be replaced by L2 (IR) and therefore,the Loo-norm by the L 2-norm.

ProofThis is a consequence of the Sobolev inequality

19

and other straightforward estimations. o

III (i)

(ii)

In [EB], we studied the spaces Sa,even, S~ven and S~.even' The role played by the Fouriertransformation in the characterization of the spaces Sa, Sf3 and S~ is taken by theHankel transformations JHII , II ~ -~.

(3.3) THEOREM

Let II ~ - ~ and let a > 0, (3 > o. For f E XII the statements in each of the followingtriples are equivalent.

I (i) f extends to an even function in Sa j

(ii) there exists t > 0 such that exp(tx1/a) f E Loo(JR+) and for all i E INo,

x' JHIIf E Loo(JR+) ;

(iii) there exists A > 0 such that for all k E INo,

sup Ix 1ef(x)l:S; Ale+l(k!)az>o

and for alIi E INo, sUPz>o Ixl(JHIIJ) (x)1 < 00.

II (i) f extends to an even function in Sf3 j

(ii) for all k E INo, x 1ef E Loo(JR+) and there exists t > 0 such that

exp(tx1/ f3 ) !Hllf E Loo(JR+) j

(iii) for all k E INo, supz>o Ix le f(x)1 < 00 and there exists B > 0 such that foralIi E INo,

sup Ix'(lHlIJ) (x)1 :s; B'+l(l!)f3 .z>o

f extends to an even function in S~ j

there exists t > 0 such that

exp(tx1/a) f E Loo(JR+) and exp(tx1

/ f3 ) lHIIf E Loo(JR+) ;

(iii) there exists A, B > 0 such that

sup Ix1ef(x)1 :s; A1e+l(k!t2>0

and

sup Ix'(JHIIJ) (x)1 :s; B'+l(l!)f3 .z>o

Next we show the same characterization of Sa ,even , S~ven and S~.even with Loo(JR+)replaced by XII and with the Loo-norm replaced by the XII-norm. Via these new char­acterizations the properties of the Hankel transformations JHII as stated in Lemma 2.8

20

III (i)

(ii)

can be applied fully.

(3.4) THEOREM

Let a > 0, f3 > 0 and let v ;:::: -~. For I E XII the following conditions are equivalent.

I (i) I extends to an even function in Sa ,

(ii) there exists t > 0 such that exp(tx1/ a ) I E XII and for alII E INo, x'7-llli E

XII'(iii) there exists A > 0 such that for all k E INo,

IIxleIlIx" ~ AIe+l(k!)a

and for alII E INo, II x' IIIIII IIx" < 00.

II (i) I extends to an even function in S13 ,

(ii) for all k E INo, x le I E XII' and there exists t > 0 such that

exp(tx1/ 13 ) III III E XII ,

(iii) for all k E INo, IIx le Illx" < 00 and there exists B > 0 such that for alllEINo,

Ilxl III III II x" ~ B'+l(l!)13 .

I extends to an even function in Se ,there exists t > 0 such that

exp(tx1/a

) f E XII and exp(tx1/ 13 ) IllIII E XII ,

(iii) there exist A > 0 and B > 0 such that for all k,l E INo,

and

ProofIn the triples I, II and III, the equivalence of (ii) and (iii) follows from the inequality

and a straightforward estimc..tion of Ilxnh111:1:" in

21

I Let I extend to an even function in Sa. Then by Theorem (3.3) there exists T > 0such that exp(TXl/Ol ) IE Loo(m+) and so for 0 < t < T, c = T - t,

II exp(txl/

Ol) Illx" ~ II exp( -exl/Ol)llx" sup exp(Txl/Ol ) II(x)1 .

:.:>0

Further, the even extension of I belongs to Seven so that x1c lEIvI E Xv, k E INa,by Theorem (1.10). We conclude that (i) implies (ii).For the converse, let I satisfy condition (ii) for some t > O. Then as a consequenceof Theorem (1.10) the even extension of I belongs to Seven' Hence for alll E INa,xl IH lIE X_! and so I is continuous and bounded on (0,00). In its turn this

-2 2

implies exp(tx l / Ol) IE X_i. We conclude that the even extension Ie of I satisfies

2

and for all lEINo,

Hence Ie E SA by Theorem (3.2).

II Let I extend to an even function in Sf3. Then from Theorem 3.3 we see that(IHvf)e E Sf3 (Observe I = IHv(IHvf).) So from 1. (i) ¢} (ii), (IHvf)e E Sf3 if andonly if there exists t > 0 such that

and for all k E INa,

III The even extension Ie of I belongs to S~ if and only if Ie E SA and Ie E sf3. Sothe equivalence (i) ¢} (ii) is a straightforward consequence of I and II. 0

We arrive at the point to present characterizations of SOl,even, S~ven and S~,even whichare refinements and extensions of similar results derived by Duran [Dul].

(3.5) THEOREMLet a > 0, {3 > 0 and let 1I ;::: -~' For an infinitely differentiable function on m+ thestatements in each of the following triples are equivalent.

22

I (i) The even extension of I belongs to Sa ,

(ii) there exists A > 0 such that for all k E INo,

IIxleIlIx" ~ AIe+l(k!t

and for all lEINo,

IIxz(~ tzY Illx" < 00 ,

(iii) there are A > 0, Bz > 0 depending only on I and I such that for allk, lEINo,

II x lc+Z (~ tz)' IlIx" ~ BIAIe(k!)a .

II (i) The even extension of I belongs to S{3 ,

(ii) for all k E INo, IIxle llx" < 00 and there exists B > 0 such that for alII E INo,

IIxl (~ tz)lllx" ~ BI+1(1!){3 ,

(iii) there exist B > 0, Ale > 0 depending only on k and I such that for allk,1 E INo,

II xlc+l (~ tz)' IlIx" ~ Ale B' (l!){3 .

III Here, in addition, we assume Q' +(3 ~ 1.

(i) The even extension of I belongs to S~ ,

(ii) there exist A > 0, B > 0 such that for all k,1 E INo,

lIxle/llx" ~ AIe+l(k!)a and Ilxl (~ tz)' Illx" ~ B'+l(I!){3 ,

(iii) there exist A > 0, B > 0, C > 0 such that for all k,1 E INo,

Ilxlc+l (~ te)' Illx" ~ CAkB'(k!t (1!){3 .

ProofIn Lemma (2.8), taking k = 0 we proved the following relation: For all v ~ -~ and forall I E Seven,

So in each of the cases I, II and III equivalence of (i) and (ii) is a consequence ofTheorem (3.4). Also in all three cases (iii) implies (ii). So there remains to show that(ii) implies (iii).Let us first make some general observations. Let I E Seven and let k ~ 1. Then anapplication of Leibniz's differentiation rule yields

23

= (X 2(le+l) (! 1_)' f, (1 d)' f)_ ...., ;; d:r: Xv

<~ (1) 2; f(k +1+ II +1) I( 2(le+l-;) (1 J!.)21-; f f) I- L-' f(k +1+ +1 _ ') x z d:r: ,xv

;=0 J II J

For k > I we apply Lemma (2.8),

and then, replacing f by IH,J and interchanging roles of k and 1 above, we derive

From the considerations above we obtain the following estimation:Let f E Seven and let k, IE lNo. Put r = min{k, I}, then

I For f satisfying I(ii) we get by (*)

Ilxle+l (~ tz)' flit ;:;

24

II For I satisfying II(ii) we get similarly from (*)

~ (21+.8(1 + B))21+l(l!?.8 o~~<1 22k+

vj! Ilx2le

-j IlIx"

_3_

III Finally for I satisfying III(ii) the estimation (*) yields

( .,)1-0-13 (2k) -0 (21)-13max J. . .O~j~r J J

o

As we have shown, the space S(JR+) consists of all COO-functions I on JR+ with theproperty that the function x 1-+ l(x2 ), X E IR, belongs to Seven- Accordingly we intro­duce the spaces G20tl G2.8 and G~~ Here we follow Duran's notation in [Dul].

(3.6) DEFINITION

Let I be an infinitely differentiable function on IR+_

III f E G~~ :¢:> x 1-+ l(x2) E S~,even _

25

By definition G2a , G2/3 and G~~ are subspaces of S+. The spaces G2a and G2/3 arenontrivial for all values of Q, f3 ~ 0, G~~ is nontrivial if Q + f3 ~ 1, Q > 0 and f3 > 1,Q= O.Theorems (3.3), (3.4) and (3.5) lead directly to characterizations of G2a , G2/3 and G~~

be replacing x 2 by X, ~ ~ by ~, IHv by ?-lvand Xv by Yv'

(3.7) THEOREM

Let Q > 0, f3 > 0 and let 11 ~ -l. Let I be a function on JR+, IE Yv'

I Each of the following conditions is necessary and sufficient for I to belong toG2a ·

(i) There exists t > 0 such that exp(tx1/ 2a ) I E Loo(JR+) and for alII E INa,X

'/

2?-lvl E Loo(JR+),

(ii) there exists A > 0 such that for all k E INa,

sup Ix le /2 l(x)1 :::; AIe+t(k!)a:1:>0

and for alII E INa, sUP:l:>o IX' /2(?-lvf) (x)1 < 00,

(iii) there exists t > 0 such that exp(tx1/ 2a ) I E Yv and for alII E INo, x'/2 ?-lvl E

Yv ,

(iv) there exists A> 0 such that for all k E INo,

IIxle/

2Illy" :::; AIe+t(k!)a

and for alII E INa, Ilx'/2 ?-lvl II y" < 00 ,

(v) there exists A > 0 such that for all k E INo,

Ilxle/

2Illy" :::; AIe+t(k!t

and I is infinitely differentiable on JR+ with for alII E INo,

Ilxl/

2 (~)' Illy" < 00 ,

(vi) I is infinitely differentiable on JR+ and there exist A, B, > 0 depending onlyon I and I such that for all k, I E INa,

Ilx(k+l)/2 (~)' Illy" :::; AleB,(k!)a .

II Each of the following conditions is necessary and sufficient for I to belong toG2/3.

(i) For all k E INo, x le/2I E Loo(JR+) and there exists t > 0 such thatexp(tx1

/ 2/3) ?-lvl E Loo(IR+),

(ii) for all k E INo, sUP:l:>o Ix le /21(x)1 < 00 and there exists B > 0 such that foralII E INo,

26

sup IX' /2 ('Ha..!) (X) I ::; B'+

1 (l!)/3 ,lIl>O

(iii) for all k E INo, xle/2IE YI/ and there exists t > 0 such that exp(tx1/213 ) 'HI/I EYI/,

(iv) for all k E INo, Ilxle/ 2Illy" < 00 and there exists t > 0 such that for all1E INo,

Ilx' /2 'HI/Illy" ::; B'+l (1!)13 ,

(v) for all k E INo, IIxle/ 2IllY" < 00 and I is infinitely differentiable on lR+ withfor all 1E INo,

IIx'/2 (~)' Illy" ::; B'+l(l!)13 ,

(vi) I is infinitely differentiable on lR+ and there exist Ale, B > 0 dependingonly on k and I such that for all k,l E INo,

II x (Ie+I)/2 (~)' Illy" ::; AIe B' (l!)13 •

III In addition we assume that a + j3 ~ 1. Then each of the following conditions isnecessary and sufficient for I to belong to G~~.

(i) There exists t > 0 such that

exp(tx1/

2Q ) I E Loo(JR+) and exp(tx1/ 213 ) 'HI/I E Loo(IR+) ,

(ii) there exists A > 0 such that for all k,l E INo,

sup Ixle/2I(x)1 $ AIe+l(k!)Q and sup IX'/2('HI/f) (x)1 ::; A'+

1 l!13 ,~>o ~>o

(iii) there exists t > 0 such that

exp(tx1/

2Q )IE YI/ and exp(tx1/ 213 )'HI/I E Yv ,

(iv) there exists A > 0 such that for all k,l E INo,

Ilxle/2IllY" ~ AIc+l(k!r~ and Ilx'/

2'HI/Illy" ::; A'+1 (1!)13 ,

(v) there exists A > 0 such that for all k E INo,

IIxle/2Illy" $ AIe+l(k!t ,

I is infinitely differentiable on lR+ and there exists B > 0 such that for all1E INo,

Ilx' /2 (~)' Illy" ::; B'+l(l!)13 ,

(vi) I is infinitely differentiable on lR+ and there exist A, B, C > 0 such thatfor all k,l E INo,

IIx(le+l)/2 (~)' Illy" ::; CAIeB'(k!)Q (1!)13 .

27

Finally we shall show that in Theorem (3.7) condition (vi) can be replaced by thecorresponding sup-norm condition.For this, let f E S+. A straightforward estimation shows

Further,

co

= ~(_~)' X'/2 f (xy)-1/2 J,(y'XY) (11.0 J) (y) yl dy .

o

Since, d. [MOS], p. 78,

sup IJ,(OI ~ 1 ,eER

we derive henceforth, applying Cauchy-Schwartz's inequality

co

(**) IX' /2f(l)(x)1 ~ (~)' f y'/21(11.0 J) (y)1 dy

o

~ (~)'-l (lI y'/211.ofllyo + Ily(IH)/211.ofIIYo)

= G)'-l (1I yl/2 f(l) II Yo + lIy(IH)/2 f(1+2)llyo) .

Further, for k E IN, 1E INo,

z

(* * *) Ix(1c+l)/2 f(l)(x)1 2 = 2 Re f {e(1c+1)/2 f(l)(O}· {(k +1)/2 e(1c+l)/2-1 f(l)(Oo

:::: 2i (C ~ 1) 1((HI-1)!'f(l) (0I' + I('t' f(I)(() 11(('+ll!'fll+l)(0 I) d(o

Starting from the inequalities (*), (**) and (* * *) and applying some straightforwardmanipulations the following results turn out valid.

(3.8) THEOREM

Let a > 0, (3 > 0 and let f be an infinitely differentiable function on IR+.

28

I f E G2a iff there exist A, B, > 0 depending only on f and 1 such that for allk,l E INa,

sup IX(1e+1)/2 j(I)(x)1 ::; AleB,(k!t .z>o

II j E G213 iff there exist Ale, B > 0 depending only on j and k such that for allk,l E INa,

sup Ix(le+I)/2 j(l) (x) I ::; AleB'(1!)13 .

z>o

III j E G~~ iff there exist A, B, C > 0 such that for all k,1 E INa,

sup IX(1e+I)/2j(l)(X) I ::; CAleB' (k!t(I!)I3.z>o

Here in addition we assume that Q' + f3 2: 1.

29

o

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[Du3] A.J. DURAN, The Stieltjes moment problem for rapidly decreasing functions.Proc. Am. Math. Soc. 107, 731-741 (1989).

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31