On a property of the Fourier-cosine transform

On a property of the Fourier-cosine transform

Citation for published version (APA):

van Berkel, C. A. M., & Graaf, de, J. (1988). On a property of the Fourier-cosine transform. (RANA : reports on applied and numerical analysis; Vol. 8806). Technische Universiteit Eindhoven.

Document status and date: Published: 01/01/1988

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Eindhoven University of Technology

Department of Mathematics and Computing Science

RANA88-06 May 1988 ON A PROPERTY

OF

THE FOURIER-COSINE TRANSFORM

by

C.A.M. van Berkel and

1.de Graaf

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

On a property of the Fourier-cosine transform

by

C.A.M. van BeIXel and

J.de Graaf

Eindhoven University of Technology (AMS Classifications 33A65 30015 42A38)

SUMMARY

It is shown that the Fourier-cosine transfonn maps functions of the fonn

t ~ 111(1- 2tanh2_{t)cosh...t ,}

with111an entire analytic function and Rev> 0, bijectively onto the functions

Here 'If is an even and entire analytic function of sub-exponential growth, i.e.

"1_{00 :}sup e-elzl I\jI(z) I<00.

-2-1. Introduction

The Fourier-Jacobitransfonnf H gis defined by

00

g(A)

=

J

f

(tH~a,II)(t)<\"11(t) dt, a >-1 ,~E IR

o with

lll~a,Jl) (t) =zF1 (t(a+~+ l+iA), t(a+~+l-iA);a+ 1;_sinhz t), t>0

and

<\"11

(t)

=(2sinh

t)200+-1 (2cosht)2I*1 .

Ifwe take

f

(t)

=(cosh

t)...-H-i"-Zp~a,fi)(1-2tanhzt), a,~>-1, ~,A,J.l.E IR,

then, following Koomwinder, [K],

g(A)

=

c~" {r(t(~+iJ.l.+1+iA»r(t(~+iJ.l.+l-iA))}.

• WlI(tAZ;t(~+iJ.l.+ 1), t (~-iJ.l.+ 1), t (a+ ~+ 1), t (a- ~+ 1».

Here thep~a,Ii) are Jacobi polynomials and the WII are Wilson polynomials. For the constantc~ see [K]. Ifwe abandon the factor between { } and if we keep the parametersa, ~, ~,J.l. fixed, then it is clear that, via the Fourier-Jacobi transfonn, the space of polynomials is mapped linearly and bijectively on the space of even polynomials.

Let us denote this linear mapping byF~.Now we are in a position to put the following prob-lems.

(i) Extend Fa/l1ij1 bijectively to suitable spaces of analytic functions (entire functions, analytic

functions on [-1, 1], genns of analytic functions at 0, etc.).

(ii) Detennine growth classes of analytic functions (as in [EOl], [EG2]) which are put in bijec-tive correspondence by the extendedF~.

In the present paper we wolk out a pan of this program for the Fourier cosine transfonn,

a

=

~

=

-t.

Even in this very special case the results seemtobe new. The authors expect that the case with general parameters can be dealt with in the same spirit We emphasize that our treat-ment is inspired by Koomwinders fonnula but does not use it.

-3-2. A special infinite upper triangular matrix

In the sequel we take vE C,Rev> 0, fixed. We denoteIN_o=INu {OJ.

Lemma 2.1.

(i) For eachnE IN_othere exist complex numbersCi,1l'O~ j~n,such that

(ii) The numbersCi,1l'O~ j~n <00,satisfy the recurrence relation

(n+t (l +v» (2n+v)Ci,ll+I=Cj-I ...+(2nz

+t

v(l-v»ci,"+ +n (2n-1+2v)Ci,..-I-2n(n-I)ci,"-Z , O~ j~ n+ 1 with boundary conditions

cO,o=l, ci,"=O ifj<Oorj>n.

Proof.

It is obvious that (i) is true forn = 0, then co,o = 1. Now suppose, indutively, that (i) is true for

n

=

0, I, .,. ,N.

Differentiating the expression in (i) twice according toIand evaluating the derivatives using the

induction hypothesis, leads to both assertions (i) and (ii) at once.

0

In the following theorem we gather some properties of the numbersci,'" O~ j~n<00.

Theorem 2.2. (i) (ii) (iii) _ i r(v) cj,j - 2 r(v+ 2j) ' j E INo•

I

ci,"

I~

411 ,

O~ j~

n <00. ci,i • { e(2-logZ)i

}-I

..r;;

2v limc-- = -i"'- J,J / i

r-t

r(v) . Proof.

(i) Taken= j - 1 in the recurrence relation, then (v+2j-l) (v+2j-2)ci,i= 2Ci-IJ-lt hence the

-4-C'

(ii) Putdj ...=...!.:!!..,0:5;j:5;n<00. Then from the recurrence relation, usingRe(v)>0,

CjJ Id· 1

=

I

(2j+1+v)(2j+v) d· + 2n2+t v(1-v) d· + 1+1 ...+1 2(n+1. (I + v» (2n+v) I." (n+1. (I +v» (2n+v) 1+1... 2 2 n(2n-I+2v) . _ 2n(n-l) .

1<

+ (n+1. (I + v» (2n+v) dl₊1...- 1 (n+1. (I +v» (2n+v) d/+1,..2 -2 2 :5; Idj ,.. I+ 2 Idj +1...I+ 2 Idj +1...- 1I+ Idj +1,..-2 I.

Now apply induction.

(iii) Follows from Stirling's formula.

o

We gather the constantsCj...in an upper triangular matrix C

=

[Cj...lj,..=o.The next theorem gives

some results on the inverseC-1ofCwhich is also an upper triangular matrix. The proof does not differ much from the preceding proofs.

Theorem 2.3.

(i) The elementsak,j ,0:5;k:5;j <00ofC-1satisfy

d~

[COSh-tJ

=(cosh-

t).

f

ak,j(2 tanh2t_1)1, tE R.

m

1=0

(ii) The numbersa/,m ,0:5; 1:5;m,satisfy the recurrence relation

ak,j+l

=(k

-t

(I-v» (2k - 2+v)a1-1J+(_2k2

-t

V(I-v» ak,j+

- (k+I)(2k+1+2v)a1+1J+(2k+4) (k+I)ak+2.j,

with boundary conditions

aO,O

=

I, ak,j

=

0 ifk<0ork>j.

(iii) ICjJ. ak,j 1:5;(lIY, 0:5;k:5;j <00,

3. The growth behaviour of the Fourier transform of a class of analytic functions.

Fora

=

P

=

-1. the Fourier-Jacobi transform reducestothe Fourier-cosine transform

-5-g(u)

=~j

f(t) cos(ut) dt.

7t 0

We takef (t) =

~1-

2tanh2t)ecosh-'\'twithell (z) =

i

a"z"an arbitrary entire analytic function.

,,=0

Consider the following formal computation

g(u)

=

~j

eIl(1-2tanh2t) e(cosh-'\'t) cos(ut) dt

=

7t 0

=

~

j

(cosh-'\'t) [

i

a,,(1-2tanh2t)lI] cos(ut) dt

=

7t 0 ,,=0

_12

00 00 (e)

='"

~ ~a"

J

(cosh-'\'t)(l- 2tanh2t)" cos(ut) dt

=

7t ,,=0 0

(0) _

/T

00 , , 0 0 d2; [ ]

=,,-

~ (-1)"a" ~C;,II

J

-zr

cosh-'\'t cos(ut) dt

=

7t ,,=0 ;=0 0 dt

_12

00

[00 ] 00

="

~ ~ ~.(_I)lI+jC;,II all u2;

r

(cosh-'\'t) cos(ut) dt

=

7t)=o ,,=)

b

_IT

2-2 00 [ 00 . ] .

="

~

r( ) r<t

(v+iu»

r(t

(v-iu» ~ ~.(_1)"+)Cj,,, a" u2)

=

7t V )=0 ,,=)

with

'I'(u)

=

i

btl u2ll, btl

=

i

(_I)lI+j Cj,,, a".

,,=0 ,,=;

For the Fourier integralinthis calculation see e.g. [Ol, p. 35.

(0)

Note that, at =, we used the results of lemma 2,1. In order to justify the remaining part of the calculation we proceed as follows.

Introduce the vectors

~

=

column(ao,al ,az, ...)

!!.

=

column(b_{o, b}l ,b2 • •.•) and the infinite diagonal matrix

j

=

diag(1, -I, I, -I, ... , (-1)" , ...).

Now the relation between the, supposed, Taylor coefficients of the functionselland'I'can be writ-ten as

-6-b

=i cia

and

a=i

c-

1

i

b.

-

-

-

-The proof of the following characterization is elementary. Characterization 3.1.

00

(i) Consider the Taylor series ~(z)

=

L

a,.z". ~ is an entire analytic function iff

,.=0

00

(ii) Consider the Taylor series'If(z)

=

L

b,.z2A.'Ifiseven entire and sub-exponential, Le.

,.=0

V'e.>O : SUP I'If(z)Ie""i!lal <00 ,

ael:

iffV't>o :(n2Aelll

b,.)':=oe 12•

Inthe next theorem we derive some funcamental estimates for the matrices C and C-1 •

Theorem 3.2.

For eacht>0 there exists't>0 such that the infinite matrices

8(t,'t):=diag(n2A

e"')i

C

i

diag(e-"~)

E(t, 't):=diag(e"')i C-1_{j diag(e-"~n-}2A) are bounded as 12-operators.

Proof. We have

Ie·_J."(t_,'t)I=j.2j eft I

c·

_J,J

.1.1

Cj.,.

le-"~

c· ._J,J

I - ( ) I kt I I 1 .-2j -j~

':'t,j t,'t

=

e Cj.jat,j •

-1--1

_{c· .} j e .

J,J

Ifwe take't>t+3, the wanted result follows with the aid of theorems 2.2, 2.3 and the

esti-mate

00

IIKII~

L

sty> I KjIIl

k_"-J'=k

for the 12-operator norm UK II of aninfinitematrixK.

Finally, our main result.

7

-Theorem 3.3.

The mapping F_!. ._!.,6... which maps the space of polynomials bijectively on the space of

•

•

even polynomials can be extended to a bijective continuous linear mapping between the space of entire functions and the space of even entire functions of subexponential growth.

Proof.

Lett > O. Consider

diag(n2Aelit)b

_-

=diag

(n2Aelit)i Cia

_-

=

For't>t+3 the operator between { } is bounded in1₂(theorem 3.2),funherdiag(e"~)f!E 1₂

for all't > 0 (characterization 3.1 (i». So diag(n2Aellt

_)!!.

E 1_{2 ,}

Therefore'I'is entire and of subexponential growth (characterization 3.1 (ii».

The inverse

F:t

._p...'which correspondstothe equalityf!

=

i

C-1j

!!.'

can be dealt with in a similar way.

Thus all formal calculations at the beginning of this section become justified. I]

Corollary 3.4.

The Fourier transform of ell (1-2tanh2t)cosh-t, ell entire, has the form

r

(t

(v+iz»

r

(t

(v - iz» 'I'(z),'I'entire, even,'I'of sub-exponential growth and vice versa. Corollary 3.5.

Comparison with the general formula in section 1 shows

N' (v) N [ N ]

= .

2N

1: 1:

(_I)/I+jCj./Ia,. x2j

(-4f j=O /I=j

with

(V)2N

=

(v+2N -I)(v+2N -2) ... (v)

=

..;;.,r.:..;..(~...;..;7~=)

-'-) anda,. such that

N p~t,&)(z)

=

1:

a1 zl.

-8-References

[EG1] Eijndhoven, SJ.L. van and J. de Graaf, Trajectory spaces, generalized functions and unbounded operators.LectureNotes in Math. 1162, Springer-Verlag, Berlin etc., 1985. [EG2] Eijndhoven, S.J.L. van and J. de Graaf, Domains of exponentiated fractional Jacobi

operators: Characterizations, Oassifications, Expansion results. To appear in Construc-tive Approximation.

[K] Koomwinder, T.H., Special orthogonal polynomial systems mapped onto each other by the Fourier-Jacobi transform. C. Brezinski, e.a. (editors) Polyn6mes orthogonaux et applications. Lecture Notes in Math. 1171, Springer-Verlag, Berlin etc., 1985, pp. 174-183.

[0] Oberhettinger, Fritz, Tabellen zur Fourier transformation, Springer-Verlag, Berlin, Gottingen, Heidelberg, 1957.

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