The Hankel transformation and spaces of type W

The Hankel transformation and spaces of type W

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Eijndhoven, van, S. J. L., & Kerkhof, M. J. (1988). The Hankel transformation and spaces of type W. (RANA : reports on applied and numerical analysis; Vol. 8810). Technische Universiteit Eindhoven.

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RANA 88-10

June 1988

THE HANKEL TRANSFORMATION AND SPACES OF TYPE W

by

SJ.L. van Eijndhoven and

MJ. Kerkhof

Reports on Applied and Numerical Analysis

Department of Mathematics and Computing Science Eindhoven University of Technology

P.O. Box 513 5600 MB Eindhoven The Netherlands

THE HANKEL TRANSFORMATION AND SPACES OF TYPE

W

by

S.J.L. van Eijndhoven and M.J. Kerkhof

In their celebrated monographs on generalized functions, [GS 2-3J, Gelfand

and Shilovhave introduced spaces of type

S

and of type

W.

Through functional

analytic methods i t has been proved in the papers [EG 1-2J that the spaces a

S

_a give rise to Hankel invariant test function spaces. In [PaJ, Pathak

sug-gested a considerable extension of certain results in [EG 1-2J. Unfortunate-ly, the proofs of some results in [PaJ are incorrect, cf. Appendix. In this paper, we present correct proofs and add some new results.

1. The Hankel transformation

Let J denote the first type Bessel function of order ~. The corresponding

~

Hankel transformation of a function ~ on (0,00) is defined by

00

'"

( 1. 1) (TIl ~) (x)

~

°

In this paper we consider a modified version of this transformation, viz.

the transformation TIl defined by

~.

(1. 2) (TIl ~) (x) ~

00

°

Observe that TIl

~

-(~+~) ~+~

x TIl x Abusing terminology we call also TIl a

~ ~

Hankel transformation.

Here we study the case ~ ~ -~.

2. The Young inequality

We consider the following collection of functions on [0,(0):

(2.1) K {M E C2([0,00» I M(O) = M' (0) = 0, M' (00) = 00 and M"(x) > 0, x >

oL

The fUnctions which belong to

K

are subadditive,

x

For each M E

K

we define its Young dual M as follows: Let m M' and let

+

m denote reciprocal of m. We put

x

(2 .3) M (x) x ~

a .

a

x xx

Then clearly M E

K

and M M The proof of the following result is left

to the reader.

(2.4) Lemma.

Let M E

K.

Then for all x,y ~

_a

x

xy ~ M(x) + M (y)

with equality if and only if y

So we have

x

inf [-xy + M (y)J

y~a

m(x)

-M(x) .

M' (x) •

We mention the following classical example. Let M (x)

a Then we have x M (y) a (1 _ a) y l / 1-a

and Lemma 2.4 yields the Holder inequality

l/a (1 ) l/1-a

xy ~ ax + - a y .

3. Spaces of type

W

l/a . h a l

ax w~t < a < •

Fix M,Q E

K.

In [GS 3J the following function spaces are defined.

(3.1) Definition.

Let a >

a.

The space W consists of all COO-functions tp on

:m.

which satisfy:

M,a

Va , V 3 a V Tn:

<a <a £EJN C_n ,>

XE..L<'-)CIa

1<+>(£) (x) I :0; C

n ,exp[-M(a' Ixl)J

)CIa

By

We

we denote the space of all even functions on

W

- 3

-(3.2) Definition.

Let b > O. The space

W~,b

consists of all entire functions

~

which satisfy Vb' >b '1kElN 3C >0 'I ZEtr

k,b'

"-W ~,b .

By e we denote all even funct~ons in W~,b .

(3.3) Definition.

O

W

~,b f . f

Let a > , b > O. The space consists of all entire unctl0ns ~ or M,a

which 'I 'I 3 ' I :

O<a'<a b'>b C, b'>O ZE~

a ,

::; C , b' exp{ -M (a' IRe Z I) + ~ (b'

11m

Z I )} . a ,

By

We~,b

we denote the space of all even functions in

W~,b.

M,a M,a

(3.4) Definition.

The spaces

W ,

W~

and

W~

are defined by

M M u a>O u b>O ~ ~

and correspondingly the spacesWe

M,

We

and WeM.

4. Spaces of type S

u

a>O, b>O

Let S denote the space of rapidly decreasing functions, i.e.

~

E S:- '1

k ElN 'In ... ElN sup

Ixk~(£')

(x) I < co • xEIR

By Se we denote the space of all even functions in S. In [GS

2J

the following subspaces of S are introduced.

(4.1) Definition.

Let a > 0, A > O. The space S A consists of all functions ~ E S with the

a,

property that '10 ~ 'In 3 0 'I :

<A <A ... E IN C~ > XE IR

The space Se consists of all even functions in S .

a,A a,A

(4.2) Lemma.

1/

a

For 0 < a < 1, put M : x ~ ax , x > O. Then we have

a with a S a,A W a n d Se M ,a a,A a -1 (A exp(a)) . Proof. Cf. [GsJ, p. 172. (4.3) Definition.

We

M ,A a S

S

6,B

S

Let > 0, B > O. The space consists of all ~ E with the property

that V ~ V 3 V

O<B<B kErn C~ >0 xElR B,k

The space Ses,B consists of all even functions in SS,B.

(4.4) Lemma. For 0 < S < 1, put Q S: y + (1 - S)yl/1-S, Y > O. Then SeS,B where b B exp(S). Proof. Cf. [GS 2J, p. 210. (4.5) Definition.

Let a,S> 0 and A,B > O. The space S6,B consists of all

~

E S with the pro-a,A

perty that VO<A<A VO<B<B 3

_c

o;:- ~>O VkErn V£'Ern VXElR l-l.,B

The space SeS,B consists of all even functions in SS,B.

a,A a,A

o

5

-(4.6) Lemma.

Let 0 < a < 1 , 0 < S < 1 and A,B > O. Then

SS,B WltS ,b SeS,B WeltS,b

a,A M ,a a,A M ,a

a a

where a (A exp (a) ) -1 and b B exp (S) .

Proof. Cf. [GS 2J, p. 212.

o

(4.7) Definition.

Let a > 0, S > O. The spaces

S ,

SS and SS are defined by

a a

S u S

,

SS u SS ,B and SS u SB,B

a

A>O a,A B>O a A>O, a,A

B>O and correspondingly Se

,

Se B and Se S

a a

5. Some properties of Bessel functions

In this section we recall some relevant properties of the Bessel functions

J and the Hankel functions H(l) and H(2) . As a general reference we use the

v v v

book of Magnus et al. [MOSJ, p. 60 ff.

(5.1) (Recurrence) rela-tions

Let C denote one of the functions J , H(l)and H(2).

v v v v (S.1.a) (S.1.b) ( - --d 1 d )

m -v

[z

c

(z)J z z V Further we have zv-m

_c

(z) v-m m -\i-m (-1) z C (z) v+m m 0,1,2, ... (S.1.c) H ( 1) (-x) v - ( v+ 1 ) 1T i ( 2) ( )

=

e H x x > 0, Re v > -~ , (S.1.d) J v v

(5.2) Integral representations -v+1 -l:! 1 (5.2.a) z

-v

J (z) v 2 1T f (v + l:!)

I

2 v-~ (1 - t ) cos zt dt, Re v > -l:!, z e ~ .

o

1T (5.2.b) J (z) n ,-n 1.

1T

J

exp[iz cos tJcos nt dt, n

=

0,1,2, ... , z e ~ ,

o

(5.2.c) _z -v _{J v+n (z) = 2} -v;; _{1T ( -1.)} , n

_r

_{(v + l:!) f (n + 2} f(2v)f(n + 1) _v) 1T

J

exp[iz cos tJc (v) (cos t)sin 2v (t)dt

n

o

where v

1

0, v > -l:!, n polynomial.

0,1,2, ... and where c(v) denotes the n-th Gegenbauer

n (5.2.d) H (1) (z) v -l:! (~1TZ)

r

(v +~) exp[i(z - ~V1T - ~1T)J iC ooe

I

o

where v > -l:!,

10

1

< l:!1T and -l:!71 +

0

< arg(z) < %71+

o.

On the basis of these i'ntegral representations the following estimations can be derived in a straightforward manner.

(5.3) Growth order estimations

(5.3.a) J (x)

=

O(xv)

,

x 4- 0

,

J (x)

=

O(x -l:!)

_,

x -+ 00

.

v v

(5.3.b) V

v>-~ V nelNu{O} 3 B >0 V ze(:' Iz-vJ v+n (z) 1 :-:; B v,n expl Im v,n (5.3.c) Let v ~ ~. Then 3 V I 1 .

IZ~H(l)

(z) 1 :-:; A exp[-Im zJ . A>O zecr,z>1· v v v v ( 1 ) , 1 I

The function z ~ z H (z) 1.S bounded on z :-:; 1. v (S.3.d) Let -l:! < v < ~. Then 3 V A >0 zeq: v I z l:!H (1) (z) I :-:; A exp[ -1m zJ . v v zl

.

7

-6. Auxiliary results

Throughout this section we take v > -~.

(6.1) Lemma.

Let ~ EO Se be analytic wi thin a strip T

=

{z EO <I:

I

11m z

I

< c}, 0 < c ::; 00

c

(T <1:). Let there be a positive bounded continuous function g on [O,c)

00

such that

Then for all

n,

0 <

n

< c, and all x > 1

(H ~) (x) v

Observe that for c

Proof.

00

-00

00 all

n

> 0 may be taken.

Let 0 < n < c. Consider the contour C

n,

R, € III -R+in~---~~--~~~---~ IV v ... I .... -R -€ R

Let x > 1. Then the function G defined by x,v

G (w ) = (xw) - v H (1) (xw) ~ (w) w 2 v+ 1

x,v

v

R+in

II

is analytic within the contour C . So

f

G (w)dw = O.

n,R,€ C x,v

n,R,E:

Because of the growth conditions imposed on ~, the estimations (5.3 c-d) and

( 1 )

the regularity properties of H at z = 0, the contributions of II, IV and

v

_00

(6.2) Lemma.

I

-v (1) . . . 2v+1

(x(t; + in)) Hv (x(t; + IT)))<.O(t; + In) (t; + In) d';

lim lim E-I-O Rtoo R

I

(x';) - v (H

~

1) (x';) + H

~

2) (x';) ) <.0

(E~)

.; 2 v+ 1 d'; 2

I

_{(x.;)-VJv (x.;)<.O(.;)t;2V+1 d .;}

o

o

Let <.0 E Se satisfy the conditions stated in Lemma 6.1. Then for each v > -~,

ill <.0 is bounded on IR. In addition, there exists D > 0 such that for all

v v x E IR with

I

x

I

> 1 I (ill <.0) (x) \ $ v D v inf g(n)exp[-\x\n] . O<n<c Proof.

In [EG 1] we have proved that ill <.0 E Se. Therefore ill <.0 is bounded on IR and

v v

we need only investigate its growth behaviour at 00. So let x > 1 and let

o

< n < c.

First we consider the case -~ < v < ~. By (S.3.d) there exists A > 0 such that

v

\ (x (.; + in))

~H

( 1) (x (t; + in)) \ $ A exp [ -x n] .

v v

Further, there exists C > 0 such that

v, <.0 Thus we obtain (*) \~ _ 00 I -v-~ $ ~A C x g(n)exp[-xn] v v,<.O

f

1 2 d'; . 1 + t;

9

-Next, we consider the case v ~ ~.

We set I

=

{~

(

IR

x,n Ix(~ + in) 1 ~ 1}. For ~ ( I x,n we have

1 (x

(~

+ in))

~H

( 1) (x

(~

+ in)) 1 :<::: v

( 1 )

A exp[-xn]

v

On Izl < 1 the function zvH (1) (z) is bounded. So for some constant A(2) > 0

v v

indepent of x and n, we have

(2)

:<::: A exp[-xn],

v

Further, there exist c(l) ,c(2) > 0 such that v,\O (j)

I(~ + in)(j)(~ + in) 1

Thus we obtain ( **)

I

(. .. ) d~ 1 :<::: I lR \1 x,n x,n (1) (1) -v-~

f

:<::: ~A C x g(n)exp[-xnJ v v,(j) 1

---=-

d ~ + 1 +

~2

f

IR \1 I x,n 1 ---:-2 d~ . 1 + ~ x,n ~

i

I x,n

From (*) and (**) and Lemma (6.1) we derive that for all v > ~, there exists D > 0 such that V 1 V

v,(j) x> n,O<n<c

1 (IH (j)) (x) 1 :<:::

D

g(n)exp[-xnJ

v v,(j)

Thus the resul t follows, because IH (j) is even.

(6.3) Lemma.

Let ~ £ Se be analytic within the strip T , 0 < c ~ 00 Suppose there exists

c

a function g on [O,c) which is positive bounded and continuous such that

:3 _M£ K 3 _a>

a

_{Vk £lli U} {O} 3 _{C >}

a v

_1',;=c,+lnET L ' :

k c

Then JH ~ extends to an entire function. Moreover, there exists D > 0

v v,~

such that for all x,y E IR

and, in particular, for x £ IR with Ixl > 1

inf g(n)exp[-lxlnJ . O<n<c

Proof.

The growth conditions imposed on ~ ensure that lHv~ can be extended to an

~ (1)

even entire function. There exists a constant C >

a

such that

v,~

So with (S.3.b) we obtain a constant c(l) > 0 such that

v,~ 00

I

(lH ~) (2)

I

v

o

00

o

(1 ) [ x (

I

lm 2

I )

J

~ D exp M v,~ a where 2 £ q: and 00

o

- 11

-The function w ~ w-vH (w) is analytic on the complex w-plane cut along the

v

negative imaginary axis. So for all ~ E IR and all n, 0 < n < 00 the function

is analytic on the region {z E ~ I Re z > 0, 1m z >

a}.

Now by standard arguments we obtain (cf. Lemma (6.1»

(IH _v 4) (z) _.

-00

00

for all Z E ~ with Re z > 1 and 1m z >

o.

First we consider the case -~ < v < ~. Then we have the estimate

00 _00 00 _00 (2 ) ₊ MX (y) ] ::; D g(n)exp[-xn v,4) a

where Z x + iy, x > 1 and y >

o

and where

00

D (2) c(2)

f

(1 + ~2)-ld~

.

v,4) v,4)

_00

Next, we let v ~ ~. We fix Z E ~ with Re z > 1 and lm z > O. Also, let

o

~ n < c. We set

I

=

{~ E IR

I

Iz(~ +

in)1

~ 1}.

z,n

Then following (5.3.c) there exists a constant A > 0 such that

v

Iz(~

+

in)~H(l) (z(~

+ in» I ::; A exp[-(xn +

y~)J

,

v v

and

Iz(~

+ in)vH(1)

(z(~

+ in»

I ::;

A exp[-(xn +

y~)J

,

v

v

So we get the following estimation

~ E I

z,n

~

i

I

f

I z,n :'S: A C(3)

Izl-V-~g(n)exp[-xnJ

v V,t!) and, similarly, (****)

f

IR \1 z,n 00 - 0 0 00 _00

We observe that the constants 0(1) and 0(2) do not depend

v ,

to

v , (D

By (**)-(****) we derive that for all v > -~ there exists

on z and n.

(2)

a constant D

v,tP

such that for all n, 0 < n < c and all z E ~ with 1m z > 0 and Re z > 1

inf g(n)exp[-n Re zJ . O<n<c

By (*) and (~) the wanted result follows.

7. The Hankel transformation and spaces of type

W

Let IF denotes the Fourier transformation

00 (IFlO) (x) 1

&

f

-iyx tP(y)e dx _00

In [GS 3J, the following results have been proved

IF (W ) M,a x 1 M W 'a

w

x l ' Q

'b

x 1 M ,

-W

a x 1 Q -'b > 0

- 13

-Since IH 1 equals the Fourier cosine transformation with the aid of the

-~

above relations we get

(7. 1) Corollary. IH I.-,We ) - . M,a x 1 M

'a

We x 1 t-1 -'a We x 1

st

'i;

o

In this section we prove similar results for the transformations IH , v > -~.

v

First, we present the following auxiliary result

(7.2) Lemma.

Let M €

K,

let a > 0 and let ~ € Se. Then we have

~ E We ~ V V 3 V :

M,a O<a'<a 9odNu{O} C_n ,>0 xElR

"",a

I

_«x -1 _D) £ _(?)(x)

I

_{~Cn ,exp[-M(a' x)]}

I I

"",a

where we set D d dx Proof.

The proof is a consequence of the following relations

n-1 k \' (-1) (n - 1 + k) ! -1 n+k n-k L. k ( 1 _ k).' ( x ) D k=l 2 k! n

-[~n]

( - l ) k l_n! n-2k -1 n-k

I

x (x D) • k =0 k! (n - 2k)!

Our results are contained in the following lemmas.

(7.3) Lemma.

Let v > ~. Let M E

K

and let a > O. Then we have x -1

IH ( We ) c We M ,a

v M,a

Proof.

Take a fixed ~ E We Following the preceding lemma, for each a' with

M,a

o

< a' < a and each £ E IN u

{a}

there exists C_n , > 0 such that

N,a

1«x-1D)£~)(x)1

~

C_n ,exp[-M(a'lxl)]. N,a

_1 £

So the function IH «x 'D) ~) extends to an entire even function for all v

Now let 0 < a' < a" < a, let k E IN and zEd:. Then we have

k Z (TIl ~) (z) v 00

o

00

r

-v -1 k 2v+k+1 J (z~) J v+k (z~) «~ D~) tp) (~) ~ d~

o

where we inserted the recurrence relations (S.2.a). With (S.3.b) we get the estimate 00

1/(TIlv~)

(z) I

~

f

Iz~l-vIJv+k(zO

II

«~-lD~)k~) (~)

1~2v+k+1d~

o

00

~

C B

f

exp[~IIm

zl -

M(a,,~)]~2V+k+1d~

. k,a" v,k

o

Now the inequality -M(a"~) ~ -M(a'~) - M«a" - a')O together with Young's inequality yield where I zk (TIl tp) (z) I

~

v x Im z Dk ,exp[M (-,,-)] ,a a 00 Dk ,a ' Ck ,a "B v, k

f

exp[-M«a" -

a')~)]t;2v+k+ldt;.

o

x -1 . k ' k WeM,a

.0

- 15

-(7.4) Lemma.

Let v > ~. Let Q €

K

and let b > O. Then we have

We

Proof.

x -1 Q ,b

L t_{e ~} In _€

W

_e Q,b _{. T en} h _{~ lS} ' _{an entlre} ' f_{unctlon Wl} ' 'th _t h _{e property t at} h

v

V 3 V

b'>b k€lNu{O} Ck,b'>O Z€(J:

Let b' > band £ € IN U {a}. The recurrence relations (S.1.a) induce the equality

-1 £ £

(x D) lli lO

=

(-1) lli oW

x v v+~

Now we can apply Lemma (6.2) where we take c

=

00 and g(n)

=

exp[Q(b'n) J with

n > O. It follows that there exists a constant D£,b' > 0 such that for all x > 1

I (

(x-1D) £TU _=VIn) (x) _~

I

_ < D _{£,b' exp}[()_" (b' ) _{n - Xn} ] _.

Since for all x > 1

inf [Q(b'n) - xnJ T»O

i t follows that

I

((x -1 D) £ lli <p) (x)

I

v

Hence IHv<P E

We

x 1 because of Lemma (7.2).

Q 't)

(7.5) Lemma.

Let M,Q € K and let a,b > O. Then we have

x 1

M ,

-e

We

a

x 1

Proof.

Let

~

E Wen,b and let 0 < a' < a and b' > b. Then for each k E IN there M,a

exists C

k > 0 such that for all ~,n E IR

So we can Lemma 6.3 with g(n) = exp[n(b' n)], n > O. It follows that lli LO ex-v tends to an entire function and there exists a constant C , > 0 such that

v,a I

I (ill ~) (x + iy)

I

~ C

v v,a',b'

for all x,y E IR and

( lit )

_I

(ill ~) (x + iy)

I

~ C

v v,a',b' eXP[MX(Jx+)] inf exp[-a Ixln +

~

(b'

n

)]

n>O

C

v,a',b' for all x,y E IR with Ixl > 1.

Now (*) and (lit) yield: 3

c

v,a' ,b'

v

V

XEIR YEIR I (lli

~

)

(x + iy) I

~

C ' b' expC-nx (lbx!) + MX(Jx+)] . v v,a , a Summarizing we have (7.6) Theorem.

Let v ~ -~. Let M,n E K and let a,b > O.

lli (We ) v M,a

Proof.

We observe that lli 2 v We We We 1. x M x ~ x M x n 1 'a lli (We ) v M 1 lli v (We~)

'b

1

'a

_{lli v}

_(We~)

1

,

'b

x WeM We x n x WeM _x ~

o

o

- 17

-8. The Hankel transformation and spaces of type S

As a consequence of Theorem (7.6) and Lemmas (4.2), (4.4) and (4.6) we have

(8. 1 ) Theorem.

Let v ::=! -l:!. Let A,B >

°

and let

_°

< a, S < 1

lli

(Se

)

v a,A

=

Sea,A , _{lli (Se )}

v a Sea

lli (Se S,B)

v Se S,B lli v (Se S) = Ses

lli (SeS,B) Sea,A _{lli (Se S) Se}a

S

v a,A S,B v a

In this final section we prove some results for the limiting case a S

=

0,1. We have the following Paley-Wiener type of result.

(8.2) Theorem.

Let v > -l:! and let A > 1. We have

lli (Se O )

=

SeO,A v ,A Proof.

°

Se . 0,1 or 00

Remark: SeO consists of all even C -functions with support contained in

,A

[-A,A]. SeO,A consists of all even entire functions with the property that for each k E IN there exists C >

°

such that

k,A

l(x+iY)\P(x+iy)1 $ Ck,Aexp[Alyl], x + iy E ll:.

c: Let lD E Seo . Then lli extends to an even entire function; for all

,A v

k E 1N ~ {a} and all Z E ~

A

k

z (lli lP) (z)

v

J

(zE;) -vJv+_k

(z~) «~-1D~)klP) (~) ~2V+k+ld~

°

A straightforward estimation yields

A

f

I

«~-lD~)klP) (~) 1~2V+k+ld~

°

S O,A

whence 1H tp E e .

\i

S O,A

~: Let tp E e . Then tp satisfies the conditions of Lemma (6.1) with c = 00

and g: n -+

all Ix I > 1

exp[AnJ, n > O. So there exists a constant D

\i

I

(lH tp) (x)

I

\i S D \i inf (exp[An - IxlnJ)

n>O

for Ixl :5 A for Ixl > A

> 0 such that for

So the support of the COO-function IH tp is contained in [-A,AJ where

\i

~

A

=

max { 1 , A } .

(8.3) Theorem.

Let \i > -~ and let 0 < a :5 1. Then we have

Proof.

1

Remark: Se , 0 < a :5 1, consists of all functions ~ E Se which are analytic

a

on a strip T , 0 < c < 00 and for which there is a > 0 such that

c sup X+iYET

c

Itp(x + iy) lexp[-aalxI1/aJ < 00 •

Here a and c depend on the choice of tp. a

Se

1, 0 < a < 1, consists of all even entire functions with the property

that 3 3 3 V

a>O b>O C >0 X,YEIR a,b

Itp(x + iy) I s C exp[-alxl + b(l - a) lyll/1-aJ

1

In [EG 2J i t has been proved that IH \i (Se 1) Se1

1. So we only have to

consi-o

der 0 < a < 1.

1

c: Let (..0 E Se . Then tp is analytic on a strip T , c > O. On this strip (J)

sa-a c

tisfies: 3 V 3 V

a>O kEINU{O} C >0 X+iYET

k,a c

- 19

-So we can apply Lemma (6.3) where we take g _ 1 and M(x) Then we get for IRe

zl

> 1

l/a ax x > O. inf exp[-nIRe

zlJ

O<n<c so that

x

11m

zl

I I I (TIl tp) (z) I $ D exp[M ( ) - c Re z ] • v a,c a a

Also, following Lemma (6.3) there exists D > 0 such that for all z E ~ a,c

I

(TIl tp) (z)

I

$

D

v a,c x 11m

zl

exp[M ( ) ] a a x 1/1-a a

Since M (y)

=

(1 - a)y we obtain TIl to E Se

l •

V'

~. Let tp E Se~. Then there exists a,b > 0 such that Vk IN {OJ 3 _{E U C >} 0 V _~=s+1nc-' _EC :

k

I I I 11/1-a

Itp([, + in) I $ C

k exp[-a [, + b(l - a) n ]

Due to the growth and regularity properties of J i t follows that TIl tp can

v v

be extended to an analytic function wi thin the strip T = {z E <r I 11m z I < a}. a

In T , the function TIl tp satisfies

a v

I

ern

tp) (z) 1$ D (a - 11m z l ) - l

v v

for some D > O. Now let z E T with Re z > 1. Then for all n > 0 we have

v a

00

(ill tp) (z)

v

f

(z([, + irll)

-VH~l)

(z ([, + in) )<.0([, + in) (t,; + in) 2V+ld[,

- 0 0

With the same techniques as used in the proof of Lemma (6.3) we find a con-stant C

b

,v

> 0 (independent of z and n) such that

1/1-a I (TIl tp) (z) I $ C

b exp[-n Re z + b(l - a)n ]

v

,v

Taking the infimum over n > 0 we get

I (TIl t,O) (z) I $ C

b exp[- +-(Re z) l/a

J

v ,v b -a

1 Since ill tp is even, (*) and (**) yield TIl tp E Se .

Remark. Theorems (8.2) and (8.3) are not stated as such in [PaJ.

Appendix.

In this appendix we list some errata in Pathak's paper [PaJ. First, we note

h · d f W W n,b W n,b

t at lnstea 0 the spaces e , e and e Pathak uses the spaces

M,a M,a and U \.I,M,a n,b u \.I Un,b \.I,M,a

=

{x -+

x]..1+~tO(x)

Correspondingly, instead of ill the transformation ill is used, cf. (1.1).

]..1 ]..1

The major errata are the following:

2

n

-1

- On p. 92 of [PaJ, it is stated that for M(x) = n(x) = ~x the space U ,a

]..1,M,a

equals the space T(X,A ) introduced in [EG 1J. This observation is

incor-]..1

rect, because T(X,A ) consists of all functions to with the property that

-(\.l+~) ]..1

to(x) = x W(x) where W extends to an entire function satisfying:

VO<a<~

3

_c

>0 a

Iw(x + iy) I ~ C exp -ax + [ 2 _1 y2

_{J .}

a a

- In Theorem (5.2) of [PaJ, p. 94, it is stated that

ill [un,b] c U

\.l Il x 1

\.l,Q

'b

which corresponds to the statement in Theorem (7.2) of the present paper.

The proof of this statement in CPa] is incorrect. Indeed from the inequa-lity

r-l

~

A

I

(r+l)C exp[Q((b + p) Iyl) - ulyl]

q n=O n np

where y E IR is arbitrary, it cannot be concluded that

x 1

C

q6 exp[-n ((b - 6)u)J

-1 1

where (b + p) =

b -

6, on the basis of Young's inequality. To this end,

- 21

-ulyl ~ )l x (u/(b + p)) + )l ((b + p) I y I )

and hence

-ulyl + )l ( (b + p) Iy I) ~ -)l x (u/(b + p))

.

(See CPa], p. 95. )

In CPa], p. 96, the following formula is given

1jJ(u + it)

o

where 00

o

Formula (*) is false, 00

I ~(x

+ iy) ((x + iy) (y +

it))~Jp((X

+ iy) (u + it))dx

2

-~x

in general. E.g. take p

=

-~ and ~(x)

=

e (To

this end observe that

It

J (t) =

I~

cos t). Hence, the proof of Theorem

-~ 'IT

(5.9) in CPa] is incorrect.

Finally, with respect to p. 97 in CPa] we observe that using Young's

ine-quality as indicated yields 1

Is-P-~1jJ(s)1 ~

cP exp{)l([b + p]y) + M((u + l)/p) +

8p

Il

I I \' 2(n-r) -1 1 I

+ )l(p Y )}

L

x exp{-M((a - 8)x) + M(yx) + )l(y t ) }

r=O

[EG 1]

[EG 2J

[GS

2J

[GS 3J

[MOSJ

Eijndhoven, S.J.L. van, and J. de Graaf, Some results on Hankel invariant distribution spaces, Proc. Koninklijke Nederlandse Aka-demie van Wetenschappen, Series A, 86(1), pp. 77-87.

---, Analyticity spaces of self-adjoint operators subjected to perturbations with applications to Hankel invariant distribu-tion spaces, SIAM J. Math. An., 17(2), 1986.

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

of fundamental and generalized functions, Ac. Press, New York, 1968.

Genera,lized functions 3: Theory of differential equa-tions, Ac. Press, New York, 1968.

Magnus, W., F. Oberhettinger and R.P. Soni , Formulas and theorems for mathematical physics, Springer, Berlin etc., 1966.

[PaJ Pathak, R.S., On Hankel transformable spaces and a Cauchy problem,