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The expansion and Fourier's transform ofManuel Aguirre Téllez aa Howard University , Piso, Tandil, Argentina , 7000, Pinto 399, 3er.Published online: 03 Apr 2007.

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Integral Transforms and Special Functions, @ 1995 OPA (Overseas Publishers Association) 1995, Vol. 3, No. 2, pp.113-134 Amsterdam B.V. Reprints available directly from the publisher Published under license by Photocopying permitted by license only Gordon & Breach Science Publishers SA

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THE EXPANSION AND FOURIER'S TRANSFORM OF 6('-')(m2 + P)

Manuel Aguirre TELLEZ

Facultad de Ciencias Exactas Universidad Nacional del Centro de la Provincia de Buenos Aires, Pinto 399, 3er. Piso, (7000) Tandil, Argentina

(Received October 14, 1994)

In this paper we show two results. At the first section we obtain an expansion in series (of ~ y l o r ' s type) of the distributions dk-')(m2 + P ) where m is a positive real number and P is defined in (I), and using that expansion, at the paragraph 4.1 we get the Fourier's transform of 6(k-') (m2 + P).

Applying properties that relate residue and the Fourier's transform of the families of distri- butional functions ( P f i ~ ) ~ and P i for X = - q - k, k = 0,1,2 ..., we prove that the formula of Fourier's transform of dk-l)(m2 + P) obtained at 4.2 is equivalent to the formula 6 of the Gelfand and Shilov page 294.

KEY WORDS: operations with distributions, Fourier transforms in distribution spaces, trans- forms of special functions

MSC (1991): 46F10,46F12, 44A20

1. INTRODUCTION

In [4], page 294, formula (6) appears the Fourier's transform of 6 ( k - 1 ) ( m 2 + P ) in terms of the functions of Bessel.

In this paper we obtain the same result in a very different way: using the expan- sion of the type of Taylor's series of 6(k-1)(rn2 + P).

Also, our method permits to apply it in order to obtain multiplicative products and products of convolution where the ~ ( ~ - ' ) ( n i ' + P) is involved.

Next, we will give the list of definitions and formulae used to obtain those results. Let P be a quadratic form in n variables defined by

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where n = p + q ; the distributions (m2 + P zt i0)' are defined by

E > 0 , m a positive real, X is a complex number and 1xI2 = XI + . . . + a:. It is useful to state an equivalent definition of the distribution (m2 + P & i0)' .

In this definition appears the distributions

and

From [3] , page 566 and from [ I ] , page 58 we have respectively

and

The distributions ( P f i0)' are defined by

a n d h a v e p o l e s a t p o i n t X = - 5 - k , k = 0 , 1 , 2 , . . . . From [4] , page 279, formula ( 2 ) and ( 2 ' ) we have,

where

and

We note that the following results are valid (c.f. [4], page 256 and 278) ,

(-1)k-I Res P: = - ~ $ ~ - l ) ( p ) , a = - k

k = 1 , 2 , ... ( k - I ) !

d k ) ( p + ) = ( - l ) k k ! Res P: A=-k-1 ( 1 2 )

and for odd n , as well as for even n and k < 5 - 1 ,

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THE EXPANSION AND FOURIER'S TRANSFORM OF dk - ' ) ( rn2 + P) 115

([41, page 2501,

g ( r , s) = J p i n ( p ) d ~ ( q ) (15)

([4], page 249), dQ(p) and d ~ ( q ) are elements of surface on the unit sphere in Rp and Rq respectively.

Therefore, from ( l l ) , (12) and (13) and considering

(-l)k-' Res r ( X + l ) = - A=-k

k = l , Z , . . , (k - 1) '

we have,

dk- ' ) (P+) = lim Pi for n odd A + - k r(X + 1)

and P: n dk-')(P+) = lim for even n if k < -.

A+-k r ( ~ + 1) 2

Classically, the Fourier's transform is denoted by A and defined by

where < x , y >= xlyl + . . . + xnyn Now, from [4], page 284 we have,

and

where

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p + q = n (dimension of the space) and (Q f i ~ ) ~ is defined by (7). On the other hand, from [4], page 257, 259 and 260 we have,

Res P: = 0 if p is even and q odd a = - q - k (27) k = 0 , 1 , 2 , . ..

and ,A - ( -1)q/2~f Lk 6 if p is odd and q even. - 22k k! q5 + C) (28)

k = 0 , 1 , 2 , ... Where L~ is the k-th iteration of the differential operator of the form

such operator is often called ultrahyperbolic. From [5], page 192 and [4], page 277 we have: If n is odd.

If n is even:

a) p and q both are even

e ~ y T 4 Res ( ~ f i 0 ) ' =

A=-* - , , 22k k! r(; + k) Lk6.

k = o , l , a , ...

b) p and q both are odd Res ( P k i 0 ) ' = 0 .

a=- q - k k = 0 , 1 , 2 , ...

Also we may note from [4] page 278 that if n is odd as well as if n is even and k < $ we have,

Res ( P f i0) '=0. A = - k

k = 1 , 2 , ... (33)

Finally considering properties of Fourier transform (see [4] , page 190) and from [6] , page 40, the following formula is valid

k = 0 , 1 , 2 , . . . . Where Q = Q(y) is defined by (26) and L is the operator ultrahyperbolic defined by (29).

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THE EXPANSION AND FOURIER'S TRANSFORM OF d k - l ) ( m 2 + P) 117

Along the following paragraph we will need the result announced at Lemma 1 and Lemma 2.

Lemma 1 . T h e following proper t ies are valid:

Res ( P f i ~ ) ' } " = Res { ( P f i ~ ) ' ) ~ , {),=-+-k A=- 4-k

for n odd.

Proof.

a) From (28) and considering (34) we have,

T + e ~ y Res ( ~ f i 0 ) ' ) " =

{ k - + - k 22k k! r($ + k) ( ~ ~ 6 ) "

if the dimension n is odd, as well as if n is even and p and q are even.

From (20), (21), (24) and taking into account the properties,

Res r ( t ) = - r = - k ,

k = 0 , 1 , 2 , ... k!

and

(see formula (8))

we have,

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R e {(P * iO)*IA = lim {(A + + k)r(A + !)2"t2* A=- 4-k A+-4-k 2

From (39) and (42) we obtain a) b) From (42) we have,

and

From (43) and (44) we obtain b). c) It is a direct consequence of b). d) From (22), (25) and considering (40) and (41) we have,

Res {pi}" = 2-2k~ j7" -1 n (- l lk A = - a - k

2 I'(1 - - - k)-(2i)-'(-1)~ 2 k!

x [ef (Q - io)* - e - F ( Q + io)k]

Res {PaA = O i fp i seven a n d g o d d A=-&-k (46)

and considering the formula

7F I'(z)I'(l - z ) = -

s i n ( z ~ ) (PI, page 344) (47)

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THE EXPANSION AND FOURIER'S TRANSFORM OF 6(k-1)(m2 + P) 119

if p is odd and q is even.

From (27), (28) and considering (34) we have,

Res {P: )" = 0 if p is even and q odd A = - = - k

and

(-1)476 Res {P:}" =

X = - " - k ( - l ) k ~ k i f p i s o d d a n d q i s e v e n . (50)

22k k! r(; + k)

Lemma 2. Let Bx(Q f i0,q) the distribution defined by

then the following properties, are valid:

a) B-q- r (Q&iO,q)=O i f n i s e v e n

ii) B- -,. (Q f i O , Q) = 0 i f p is odd and q even and

i i i )B- %-,(Q f iO, q) = 0 if p is even and q odd.

Proof. From (20), (21) and (24) we have,

where

R {(P - i0)*}" - Res {(P + i0)*IA X = - V - r A=-"-,

2-2rr5 - n - Res r ( A + - ) B - 2 - r ( Q f i 0 , q ) .

r ( % + r) A=-2-r 2

Considering the Lemma 1 part a) and (40) we have,

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Res {(P - i ~ ) ' } A - Res { ( P + i ~ ) ' } A A=- q - , A=-3-r

From (55) and considering (32) we obtain that

B-q- r (Q&iO,q ) = 0 if q and p are odd.

On the other hand, considering (41) we have,

From (57) we obtain

B- 4 - r ( Q f iO, q) = 0 when q is even independently of p. (58)

From (22) , (25) and (40) we have,

Res { P : ' ~ ) " A=- 3-7, v=o,1>2, ...

- - 2 n + ~ ( - 3 - ~ - ~ ) ( - - 1 n-a Res I '(A+p+l) n 2 A=-4-r-1

Res P:~' = Res P"~ = 0 if p is even and q odd, (60) A = - + - r - l x=-q-,-l + +

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THE EXPANSION AND FOURIER'S TRANSFORM O F dk- ' ) (m2 + P ) 121

From (59) and (60) and considering the Lemma 1, part d we obtain

- r ! - - { Res P$"}̂ - ( + I ) ~ = - f - r - l

= 0 if p is even and q odd.

From (56), (58) and (61) we conclude that

i) B- P-r(Q f iO, q) = 0 if q and p are odd (n even) (62)

ii) B-+-,.(Q f iO, q) = 0 if q and p are even (n even) (63)

iii) B- 2-r(Q k iO, q) = 0 if q is even and p is odd (n odd) (64)

and

iv) B-r-l(Q f iO, q) = 0 if q is odd and p is even (n odd) (65)

In this formula p denotes the number of positive terms in the canonical form of P a n d p + q = n

We will prove the following lemma.

Lemma 3. Let (m2 + Pk i0))" be the distribution defined by (2) then the following formula is valid:

if P 2 m2 and a # u - - k, k = 0 ,1 ,2 ...; where a is a complex number.

Proof. Let be ~ , ( m ~ + P k i&(x12) = (m2 + Pf i ~ ( x ( ~ ) ~

and G,(m2 + P f iO) = (m2 + P f iO),

then from (2) we have,

lim ~ , ( m ~ + P k i&1xI2) = Ga(m2 + P f iO) E-0

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122 M.A. TELLEZ

From [2], Vol.1, page 101, formulae 1, 2 and 4, we have,

From (67) and considering (70) and for P 2 m2, we have,

Ga(m2 + P f ia1xI2) = (m2 + P f i ~ ( x ( ~ ) ~

(71) Let G i be the sequence defined by

Therefore, for Real (a - v) > 0, we have,

and by analytical continuation the formula (73) is valid for every cu (complex num- ber) so that cu # v - - k, k = 0,1,2, ... .

On the other hand,

lim ~ i ( m ~ + P f i&lx12) = Ga(m2 + P A i&lx12) j-cc (74)

because the terms of the sequence are locally integrable in every compact KCRn f o r c u f v - t - k , k = 0 , 1 , 2 ,....

Therefore,

w

lim (lim ~ $ ( m ~ + P i i ~ I x 1 ~ ) ) = j -cc E-0

v = O r ( a + 1 - v) v!

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THE EXPANSION AND FOURIER'S TRANSFORM OF 6(k -1 ) (m2 + P) 123

From (75)) (67) and (68) we conclude:

' (m2 + P f i ~ ) , = G,(m2 + P & iO)

lim (lim GL(m2 f ie1xI2)) j-m E-+O

Theorem . Let k be a positive integer and let n be the dimension of the space, then the following formulae are valid:

03

(m2)v 6(k+v-I) 6("l)(m2 + P ) = - (P+) if n is odd v ! (77)

v=o

and a - k - 1

(m2)Y 6(k tY-1) (~+) i fn is even and v < 5 - k . (78) 6(k-')(rn2 + P) = - v !

u=O

Proof. From (5), we have,

Otherwise, the following formula is valid:

n r ( z ) r ( - t ) = -- cosec z.lr (c.f. [ 2 ] , page 3).

t (80)

We obtain, from (79) and (80),

e ( " - l ) ~ i ( m ~ + p - ~ o ) Q - 1 - e-(a- l )~a ( m2 +- P + iO),-'

From the above formula and considering the formula (76), we obtain,

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124

On the other hand.

and

with v = 0,1 ,2 , ... . Therefore, from ( 1 8 ) and considering ( 8 0 ) , ( 8 3 ) and ( 8 4 ) we have,

Replacing the formula ( 8 5 ) into ( 8 2 ) we obtain,

From ( 8 6 ) and considering the formula

k = 1 , 2 , ... we have,

From ( 8 8 ) and considering ( 6 ) we obtain,

6("l)(rn2 + P ) = lim (m2 + P):

A--k r(x + 1 )

From ( 8 9 ) and considering ( 1 7 ) we arrive at the following formula,

(rn2)" 6(k-11(m2 + p ) = -6(kt"-1) ( P + ) for n odd and k = 1 , 2 , ... . (90) v!

v=O

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THE EXPANSION AND FOURIER'S TRANSFORM OF 6(k -1 ) (m2 + P) 125

For the case n even, from (18) we have,

where k = 1,2 , ... and v = 0,1,2, ... . Considering the following conditions, k + v < a , n even, n > 4 and k = 1,2 , . . .

we have, n n n - k , - - k + 1 , - - k + 2 ,.... v ' z 2 2 (92)

Therefore, from (89) and taking into account (92) we arrive at the following formula,

i f k + v < ;, n e v e n , n > 4 a n d k = 1 , 2 ,.... The formulae (90) and (93) coincides with (77) and (78 ) respectively.

4. THE FOURIER TRANSFORM OF s(~-')(P*) AND 6(k-1)(m2 + P )

4.1.

From (22) and considering (25) we have,

where 2 2 2 Q = Q(y) = y,2 + . . . + Yp - YPtl - . " - Yp+pl (95)

(Q f i ~ ) ~ is defined by (7) and

,(A, n) = 2 2 A t n a 9 1 ' ( ~ + 34)(2i)-' 2 (96)

Now, taking into account (8) and (80) we have,

= (2ri)-' [e-"'(p + i ~ ) " ee r i (p - io)*]. r(A + 1)

(97)

From (94), (96) and (97) and using (20), (21) and (24) we obtain the property,

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From (94) and (98) and considering (17) and (18), for odd n as well as for even n and k - 1 < 5 - 1, we have the following formula,

where

n-2k n-2 c (-k,n) = 2 i~ 2 F (- - k)(2i)-'

2 (100)

and k = 1 ,2 , ... . Similarly considering the formulae

([4], page 279) where 6ik)(p) is defined by (14) and k = 0,1 ,2 , ... we have,

{6(" ' ) (~-)) A = (-l)k-l { 6 ( " - ' ) ( ~ + ) } ~ (103)

for odd n , as well as for even n and k - 1 < 5 - 1. From (77), (78) and (99) we have the following results

if n is odd, and

(105) for even n and f? < 5 - k.

Remining that the symbol A indicates the Fourier transform and

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THE EXPANSION AND FOURIER'S TRANSFORM OF d k - l ) ( m 2 + P) 127

The Fourier transform of 6(k-1)(m2 + P ) appears in [4], page 294, formula (6) in terms of Bessel functions.

In this paragraph we will show the equality of the Fourier transform obtained a t Section 4.1 and the Gelfand's result published in [4], page 294, formula (6),

(107)

x [e- f (Q - i ~ ) - " ~ & ( c d m ) - e y (Q + i ~ ) - ' l ~ I ~ ~ (cd - ) ]

where s = 5 - t , n dimension of the space,

for A # integer,

for l = 1,2, ...,

(Q 4~ i ~ ) ' is defined by (7) and

2 2 2 c 2 + P = c2 + P ( x ) = c2 + x : + . . .+ x p - X p + 1 - . . . - X p t g

In (109) $ ( x ) is defined as $ (x ) = 8 and for integer values $ ( x ) is given by

$(k) = -7 + 1 + + . . . + &, k = 2,3, ...; where y is Euler's constant. Now we obtain the same result by Gelfand and Shilov by a direct proof, essentially

using the expansion of dk)(rn2 + P ) . In order to do it , we will consider two cases: 1) n is odd. 2) n is even.

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128

Case n odd

M.A. TELLEZ

From (112) and (110) and considering the property

([6], page 23), formula (66) for A, p and X + p # -; - r, r = 0 ,1 ,2 , ... we have,

where

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THE EXPANSION AND FOURIER'S TRANSFORM OF 6("-')(m2 + P) 129

and

From (116) and considering the Lemma 2, formula (64) we have,

if q is even and p is odd.

On the other side considering the formula (41) we have,

Now we rewrite (116) using (117) and applying the formula (65) of Lemma 2 we

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130

obtain

= 0 if q is odd and p is even.

From (117) and (119) we have that

X ( Q T iO, n ) = 0 if n is odd. (120)

Therefore from (120) we obtain

A

(6'-'(e2 + P ) } = Y ( Q r iO, n ) if n is odd. (121)

Where Y ( Q iO, n ) is defined by the equation (115).

On the othe side considering the formulae (47) and (87) we have,

T 1 r(? - t ) r ( l - (: - t ) ) n - - = ( - l ) ' r ( - - t - r ) . sin(; - t ) ~ I'(r - (5 - t ) + 1 ) I'(T - ( 5 - t ) + 1 ) 2

(122) Putting c2 = m 2 , t = k, k = 1,2, ... in (121) and (115) and considering (122) we

obtain

x [ e - ~ ( Q - io)'+*-f - e + Y (Q + i0)'+*-4 ] for n odd.

(123) The formula (123) proves our assertion for the case n odd.

Case n even.

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THE EXPANSION AND FOURIER'S TRANSFORM OF 6 ( k - 1 ) ( m 2 + P) 131

From (107) and (109) we have,

where s = 2 - t ,

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132 M.A. TELLEZ

(127) Now, we rewrite (125), (127) using (41), (113) and applying the formulae (62)

and (63) of Lemma 2 we obtain,

x [e- " (Q - iO)' - e (Q + i ~ ) ~ ] ) 00

(128) C ~ t 2 r

= ( - - ~ ) ~ + l C log -Q$ B- $-,(Q i iO, q)

r=O 25+2rr!r(s + r) 2

= 0 for n even and

= 0 for n even.

(129)

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THE EXPANSION AND FOURIER'S TRANSFORM OF 6( k - l ) (m2 + P ) 133

From (124) and considering (128) and (129) we have,

A {6(t-1)(c2 + p)} = ( - l ) t + 1 i 2 ' n ~ c ~ ~ ~ iO, q) if n is even. (130)

Where 12(Q iO, q) is defined by the equation (126).

From (130) and (126) and using (113) we get,

x [ e - t ( ~ - i O ) ~ - ' - e f ( Q + ~ O ) ~ - ' ] forneven.

Putting c2 = m2, t = k, k = 1,2, ..., and s = - t in (131) we obtain

for n even.

Where c(-k - r, n) is defined by the equation (106).

The formula (132) proves our assertion for the case n even.

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134 M.A. TELLEZ

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Tnstituto Argentino de Mated t ica , CONICET, Buenos Aires, Argentica, (1988).

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