Hiroshima Math. J. volume 34:1

34 (2004), 117–126

On products of distributions involving delta function and its partial derivatives

Sandra Monica Molina

(Received June 6, 2003) (Revised September 30, 2003)

Abstract. Li Banghe and Li Yaqing defined in [1] the product ST as a hyper- distribution (we call hyperdistributions to the complex linear functional of DðRⁿÞ in

rC⁰) forSandT inD⁰ðRⁿÞand they calculateddd. In this paper, we shall obtain expressions for the productsd_qx^q

idand ^q

qxid_qx^q

idfori¼1;. . .;n, and for evenn, these

products have the Hadamard finite part nonzero.

1. Introduction

In this section we shall define the product ‘‘’’. This definition involves a harmonic representation of distributions and Non-Standard Analysis. Let C and R be nonstandard models for the complex field and the real field respec- tively, and let r be a positive infinitesimal. Let Y denote the set of all infin- itesimal in C. Let

rC ¼ fxAC: for some finite integer n;jxj<rⁿg; ^rC⁰¼^rC=Y:

We call hyperdistributions the complex linear functional of DðRⁿÞ in ^rC⁰. Definition 1.1. Let TAD⁰ðRⁿÞ and uðx;yÞ be a harmonic function in R_þ^nþ1¼ fðx;yÞ:xARⁿ;y>0g so that limy!0þ uðx;yÞ ¼T in the sense of D⁰ðRⁿÞ. Then uðx;yÞ is called a harmonic function associated with T or har- monic representation of T.

Lemma 1.2. Let S;T AD⁰ðRⁿÞ and SS;^ TT be harmonic representations of S^ and T respectively, and SS;^ TT denote the nonstandard extensions of^ SS;^ TT respec-^ tively. For any fADðRⁿÞ,

hSSðx;^ rÞTT^ðx;rÞ;fðxÞi A^rC:

Lemma 1.3. Let S;T AD⁰ðRⁿÞ. If SS^1;SS^2 and TT^1;TT^2 are two harmonic representations of S and T respectively, then for any fADðRⁿÞ, we have

2000 Mathematics Subject Classification. 46F05, 46F10, 03H05, 46F20.

Keywords and phrases. nonstandard analysis, distribution, products.

hSS^2ðx;rÞTT^2ðx;rÞ;fðxÞihSS^1ðx;rÞTT^1ðx;rÞ;fðxÞiA0 Let F:^rC! ^rC⁰ be the homomorphism modulo Y. We define

Definition 1.4. Let S;TAD⁰ðRⁿÞ and SS;^ TT be harmonic representations^ of S and T respectively. We call the complex linear functional of DðRⁿÞ !^rC⁰

hST;fi¼FðhSSðx;^ rÞTT^ðx;rÞ;fðxÞiÞ the product of S and T.

The product ST is a generalization of the usual product for continuous func- tions onRⁿ. In the following, we enunciate the fundamental properties of the product:

I. Commutativity: ST ¼TS, with S;TAD⁰ðRⁿÞ.

II. Bilinearity:

ða1S1þa2S2Þ ðb₁T1þb₂T2Þ ¼ X²

k;m¼1

akb_mðSkTmÞ with a_k;b_mAC and S_k;T_mAD⁰ðRⁿÞ for k;m¼1;2.

III. Localizability: Let U be an open subset of Rⁿ and let Sk;TmA D⁰ðRⁿÞ for k;m¼1;2 so that S1n_U ¼S2n_U and T1n_U ¼T2n_U. Then ðS1T₁Þn_U ¼ ðS2T₂Þn_U.

IV. Let f AC^y, TAD⁰ðRⁿÞ and f T be the usual multiplication.

Then f T ¼ f T.

V. Let f and g be continuous functions on Rⁿ. Then f g¼ fg.

VI. The Leibnitz formula: Let S;TAD⁰ðRⁿÞ. Then q

qxi

ðSTÞ ¼ qS qxi

TþSqT qxi

:

2. The product d_qx^qd

i

Li Banghe and Li Yaqing obtained in [1], the following formula for dd with dAD⁰ðRⁿÞ:

dd¼X^½n=2

j¼0

X

siANtfog Tsi¼j

r^2jnCs1;...;sn

q^2jd qx^2s₁¹. . .qxn^2sn

; ð1Þ

where

Cs1;...;sn¼ 2

Gⁿ₂þjYⁿ

i¼1

Gsiþ¹₂ ð2siÞ!

ð^y

0

t^2jþn1 c_n²ð1þt²Þ^nþ1 dt and c_n¼^pðnþ1Þ=2

Gð Þ^nþ1₂ , r a positive infinitesimal.

We can simplify the expression for the coe‰cients Cs1;...;sn by taking into account the following formula for the Beta function:

bðp;qÞ ¼ ð1

0

x^p1ð1xÞ^q1dx¼ ð^y

0

y^p1 ð1þyÞ^pþq dy

¼2 ð^y

0

t^2p1

ð1þt²Þ^pþq dt ð2Þ

where p;qAR^þ. Next, we obtain by taking into account commutativity, the Leibnitz formula and (2), the following expression for d_qx^qd_i:

d qd qx_i¼1

2 q

qx_ifddg

¼X^½n=2

j¼0

X

siANtfog Tsi¼j

r^2jnDs1;...;sn

q^2jþ1d

qx₁^2s1. . .qx^2s_i ^iþ1. . .qxn^2sn

; ð3Þ

where

Ds1;...;sn¼Gⁿ₂ jþ1 Qn

i¼1Gsiþ¹₂ 2Qn

i¼1ð2siÞ!n!c_n² : ð4Þ We observe that the Hadamard finite part of d_qx^qd

i is nonzero for even n, because it is obtained for j¼ⁿ₂ in the second term of the formula (3), so then

Hpf d qd qx_i

¼ X

siANtfog Tsi¼n=2

Ds1;...;sn

q^nþ1d

qx₁^2s1. . .qx_i^2siþ1. . .qx^2snⁿ

; ð5Þ

where

D_s1;...;sn¼ Qn

i¼1Gs_iþ¹₂ 2Qn

i¼1ð2siÞ!n!c_n²: ð6Þ

3. The product _qx^qd

i_qx^qd_i

An expression for the product _qx^qd

i_qx^qd_i can be obtained by evaluating the following integral:

ð

Rⁿ

cqd qx_i qd

qx_iðx;rÞ qdc qx_i qd

qx_iðx;rÞ fðxÞdx; ð7Þ here, we consider

c qd qxi

qd qxi

¼ rðnþ1Þxi

cnðjxj²þr²Þ^ðnþ3Þ=2: ð8Þ

a harmonic representation for _qx^qd

i

AD⁰ðRⁿÞ, fADðRⁿÞ andr the positive infin- itesimal considered in section 1. We know that

fðxÞ ¼X^nþ2

j¼0

X

ðk1;...;knÞAN₀ⁿ Tki¼j

x^k₁¹. . .x_n^kn k₁! . . .k_n!

q^jf

qx₁^k1. . .qx_n^knð0Þ þ jxj^nþ3cðxÞ ð9Þ

is the Taylor expansion of ftonþ2 th order, where cis continuous forx00 and bounded near x¼0. Then, we obtain in (7), by taking into account (8), (9) and suppfHfx:jxj<ag that

ð

Rⁿ

qdc qxi

qd qxi

ðx;rÞ cqd qxi

qd qxi

ðx;rÞ fðxÞdx

¼ ð

jxj<a

r²ðnþ1Þ²x_i² c²_nðjxj²þr²Þ^nþ3

X^nþ2

j¼0

X

ðk1;...;knÞAN₀ⁿ Tki¼j

x₁^k1. . .x_n^kn k1! . . .kn!

q^jf qx₁^k1. . .qxn^kn

ð0Þ 8>

>>

<

>>

>:

9>

>>

=

>>

>; dx

þ ð

jxj<a

r²ðnþ1Þ²x²_i

c_n²ðjxj²þr²Þ^nþ3jxj^nþ3cðxÞdx: ð10Þ The last term of the second member of the equality (10) is an infinitesimal number and we can omit it, in fact:

Let M¼supfcðxÞ:jxj<ag and we consider the polar coordinate in Rⁿ: x₁ ¼ rcosy₁

x₂ ¼ rsiny₁ cosy₂ ...

...

...

xn1 ¼ rsiny₁ siny₂. . .sinyn2 cosyn1

x_n ¼ rsiny₁ siny₂. . .sinyn2 sinyn1

where

0ay₁;y₂;. . .;yn2ap; 0ayn1a2p: ð11Þ

We denote dWthe volume element of the unitary sphere Sⁿ¹. Then we have dx¼rⁿ¹drdW with

dW¼sinⁿ²y1sinⁿ³y2. . .sinyn2dy1. . .dyn1: Let Bn ¼^2pn=2

Gð Þⁿ₂ be the ðn1Þ-dimensional measure of the ðn1Þ-sphere unitary Sⁿ¹. We obtain

ð

jxj<a

r²ðnþ1Þ²x_i²

c_n²ðjxj²þr²Þ^nþ3jxj^nþ3cðxÞdx

ar²ðnþ1Þ²

c²_n M ð

jxj<a

jxj^nþ5 ðjxj²þr²Þ^nþ3 dx

¼r²ðnþ1Þ² c_n² M

ð

Sⁿ¹

dW ða

0

r^nþ5

ðr²þr²Þ^nþ3rⁿ¹dr

r¼rt¼ rM⁰

ða=r 0

t^2nþ4

ð1þt²Þ^nþ3dt; ð12Þ

where M⁰¼^ðnþ1Þ2MBn

c_n² , M⁰AR.

Now, we have ða=r

0

t^2nþ4

ð1þt²Þ^nþ3 dt¼ ð1

0

t^2nþ4

ð1þt²Þ^nþ3 dtþ ða=r

1

t^2nþ4

ð1þt²Þ^nþ3 dt: ð13Þ Moreover

ða=r 1

t^2nþ4 ð1þt²Þ^nþ3 dt

<

ða=r 1

1

t² dt<y: ð14Þ Next, from (12) we obtain:

rM⁰ ða=r

0

t^2nþ4

ð1þt²Þ^nþ3 dtA0

since rA0. Then, we consider in (10) the terms in the Taylor expansion of f until nþ2 th order

ð

jxj<a

r²ðnþ1Þ²x_i² c_n²ðjxj²þr²Þ^nþ3

X^nþ2

j¼0

X

ðk1;...;knÞAN₀ⁿ Tki¼j

x₁^k1. . .x_n^kn k₁! . . .k_n!

q^jf qx₁^k1. . .qxn^kn

ð0Þ 8>

>>

<

>>

>:

9>

>>

=

>>

>; dx

¼X^nþ2

j¼0

X

ðk1;...;knÞAN₀ⁿ Tki¼j

r²ðnþ1Þ² c_n²k1! . . .kn!

ð

jxj<a

x₁^k1. . .x_i^kiþ2. . .x_n^kn ðjxj²þr²Þ^nþ3 dx

q^jf qx₁^k1. . .qxn^kn

ð0Þ: ð15Þ

We consider for each janþ2, ðk1;. . .;knÞAN₀ⁿ with k1þ þkn¼ j, the following terms:

r²ðnþ1Þ² c_n²k1! . . .kn!

ð

jxj<a

x₁^k1. . .x_i^kiþ2. . .x_n^kn

ðjxj²þr²Þ^nþ3 dx: ð16Þ First, we consider i¼n. For that, we calculate the following terms:

r²ðnþ1Þ² c_n²k1! . . .kn!

ð

jxj<a

x₁^k1. . .x_i^ki. . .x_n^knþ2

ðjxj²þr²Þ^nþ3 dx₁. . .dx_n: ð17Þ

Applying polar coordinate in Rⁿ and by taking into account that Pn

i¼1k_i¼ j, we obtain in (17) the following formula:

r²ðnþ1Þ² c_n²k1! . . .kn!

ð

jxj<a

x₁^k1. . .x_i^ki. . .x_n^knþ2

ðjxj²þr²Þ^nþ3 dx1. . .dxn

¼ r²ðnþ1Þ² c_n²k1! . . .kn!

ða 0

r^k1þþknþ2

ðr²þr²Þ^nþ3rⁿ¹dr ðp

0

cos^k1y₁sin^jk1þ2þn2 y₁dy₁

ðp

0

cos^k2y₂sin^{jðk1þk2Þþ2þn3}y₂dy₂

ðp 0

cos^kn2yn2sin^{jðk1þk2þþkn2}Þþ2þnðn2þ1Þ

yn2dyn2

ð2p

0

cos^kn1 y_n1sin^{jðk1þk2þþkn1}Þþ2þnðn1þ1Þy_n1dy_n1: ð18Þ The integrals of the type Ðp

0 cos^kysin^rydy are zero for k odd, then we consider even k₁;. . .;kn2. Moreover, the integrals of typeÐ2p

0 cos^k ysin^rydy are nonzero only for even k and r. Then we consider even ki for i¼ 1;. . .;n. Let ki¼2si, then Pn

i¼1ki¼Pn

i¼12si¼ j, we obtain in (18) r²ðnþ1Þ²

c²_nð2s1Þ! . . .ð2snÞ!

ð

jxj<a

x^2s₁¹. . .x_i^2si. . .x_n^2snþ2

ðjxj²þr²Þ^nþ3 dx1. . .dxn

¼ r²ðnþ1Þ² c_n²k1! . . .kn!

ða 0

r^jþnþ1 ðr²þr²Þ^nþ3 dr

G^2s1₂^þ1

G^{j2s1þ2þn2þ1}₂ G^jþ2þn2₂ þ1

G^2s2₂^þ1

G^jð2s1þ2s2₂^Þþ2þn3þ1 G^j2s1þ2þn3₂ þ1

G^2sn2₂^þ1

G ^{jð2s1þ2s2þþ2sn2}Þþ2þnðn2þ1Þþ1 2

G ^{jð2s1þ2s2þþ2sn3}Þþ2þnðn2þ1Þ

2 þ1

2

G^2sn1₂^þ1

G ^{jð2s1þ2s2þþ2sn1}Þþ2þnðn1þ1Þþ1 2

G ^{jð2s1þ2s2þþ2sn2}Þþ2þnðn1þ1Þ

2 þ1

¼ r²ðnþ1Þ² c_n²k1! . . .kn!

ða 0

r^jþnþ1 ðr²þr²Þ^nþ3 dr

2

Qn1

i¼1 Gsiþ¹₂ G ^j

Pⁱ

r¼12srþnðiþ1Þþ3 2

G ^jþ2þn2

2 þ1

Qn2

i¼1 G ^j

Pⁱ

r¼12srþnðiþ1þ1Þþ2

2 þ1

: ð19Þ

Now,

ða 0

r^jþnþ1

ðr²þr²Þ^nþ3 dr^r¼rt¼ r^jn4 ða=r

0

t^jþnþ1

ð1þt²Þ^nþ3 dt ð20Þ and,

Qn1

i¼1 Gsiþ¹₂ G ^j

Pⁱ

r¼12srþnðiþ1Þþ3 2

G^jþ2þn2₂ þ1 Qn2

i¼1 G ^j

Pⁱ

r¼12srþnðiþ1þ1Þþ2

2 þ1

¼snþ¹₂ Qn

i¼1Gsiþ¹₂

G^jþn₂ þ1 : ð21Þ

Next, we obtain from (17) and the above calculations:

r²ðnþ1Þ² c_n²ð2s1Þ! . . .ð2snÞ!

ð

jxj<a

x₁^2s1. . .x^2s_i ⁱ. . .x_n^2snþ2

ðjxj²þr²Þ^nþ3 dx₁. . .dx_n

¼ r²ðnþ1Þ²

c²_nð2s1Þ! . . .ð2snÞ!r^jn4 ða=r

0

t^jþnþ1

ð1þt²Þ^nþ3 dt2snþ¹₂ Qn

i¼1Gsiþ¹₂

G^jþn₂ þ1 : ð22Þ We observe that

ða=r 0

t^jþnþ1 ð1þt²Þ^nþ3dt¼

ðy 0

t^jþnþ1 ð1þt²Þ^nþ3 dt

ðy a=r

t^jþnþ1 ð1þt²Þ^nþ3 dt;

and,

r^jn4 ð^y

a=r

t^jþnþ1 ð1þt²Þ^nþ3 dt

ar^jn4 ð^y

a=r

t^jn5dt¼ a^jn4 njþ4:

Therefore, this last term is a finite number. Next, we obtain from (22), r²ðnþ1Þ²

c_n²ð2s1Þ! . . .ð2snÞ!

ð

jxj<a

x₁^2s1. . .x^2s_i ⁱ. . .x_n^2snþ2

ðjxj²þr²Þ^nþ3 dx₁. . .dx_n

¼ ðnþ1Þ²

c²_nð2s1Þ! . . .ð2snÞ! r²r^jn4 ð^y

0

t^jþnþ1

ð1þt²Þ^nþ3 dtr²r^jn4 ð^y

a=r

t^jþnþ1 ð1þt²Þ^nþ3 dt

" #

2s_nþ¹₂ Qn

i¼1Gs_iþ¹₂

G^jþn₂ þ1 ð23Þ

and

r²r^jn4 ðy

a=r

t^jþnþ1

ð1þt²Þ^nþ3 dtA0:

Next,

r²ðnþ1Þ² c_n²ð2s1Þ! . . .ð2snÞ!

ð

jxj<a

x₁^2s1. . .x^2s_i ⁱ. . .x_n^2snþ2

ðjxj²þr²Þ^nþ3 dx₁. . .dx_n

¼ ðnþ1Þ²

c²_nð2s1Þ! . . .ð2snÞ!r^jn2 ð^y

0

t^jþnþ1

ð1þt²Þ^nþ3 dt2snþ¹₂ Qn

i¼1Gsiþ¹₂

G^jþn₂ þ1 : ð24Þ By taking into account the formula (2), we obtain:

ð^y

0

t^jþnþ1

ð1þt²Þ^nþ3 dt¼1 2

G^jþnþ2₂

Gⁿ₂^jþ4

Gðnþ3Þ : ð25Þ

We obtain from (24) and (25) that:

r²ðnþ1Þ² c_n²ð2s1Þ! . . .ð2snÞ!

ð

jxj<a

x₁^2s1. . .x_i^2si. . .x_n^2snþ2

ðjxj²þr²Þ^nþ3 dx1. . .dxn

¼

ðnþ1Þ²r^jn2snþ¹₂

G^njþ4₂ Qn

i¼1Gsiþ¹₂

c²_nð2s1Þ! . . .ð2snÞ!ðnþ2Þ! ð26Þ Finally, we obtain from formulas (10) and (26) for the case i¼n that:

qd qxn

qd qxn

¼^½ðnþ2Þ=2X

j¼0

r^2jn2 X

siANUf0g Tsi¼j

C_s1;...;sn q^2jd

qx₁^2s1. . .qxn^2sn

; ð27Þ

where

Cs1;...;sn¼ðnþ1Þ²G^n2jþ4₂ s_nþ¹₂ Qn

r¼1Gs_rþ¹₂ Qn

r¼1ð2srÞ!c_n²ðnþ2Þ! : ð28Þ

In the same way, the following formula is obtained for i¼1;. . .;n1:

qd qxi

qd qxi

¼^½ðnþ2Þ=2X

j¼0

r^2jn2 X

srANUf0g Tsr¼j

C_s1;...;sn q^2jd

qx₁^2s1. . .qxn^2sn

; ð29Þ

where

Cs1;...;sn¼ðnþ1Þ²G^n2jþ4₂

s_iþ¹₂ Qn

r¼1Gðsrþ¹₂ Qn

r¼1ð2srÞ!c_n²ðnþ2Þ! : ð30Þ

The Hadamard finite part is nonzero for evenn. In fact, we obtain in formula (29) that:

Hpf qd qxi

qd qxi

¼ X

siANUf0g Tsi¼ðnþ2Þ=2

Cs1;...;sn

q^2jd qx₁^2s1. . .qxn^2sn

; ð31Þ

where

Cs1;...;sn ¼ðnþ1Þ²siþ¹₂ Qn

r¼1Gsrþ¹₂ Qn

r¼1ð2srÞ!c_n²ðnþ2Þ! : ð32Þ

Acknowledgement

The author would like to express her thanks to Professor Li Bang-He for their helpful remarks and suggestions.

References

[ 1 ] Li Banghe and Li Yaqing, Nonstandard Analysis and Multiplication of Distributions in any dimension, Scientia Sinica (Series A) Vol. XXVIII No. 7, (1985), 716–726.

[ 2 ] Li Banghe and Li Yaqing, On the Harmonic and Analytic Representations of Distributions, Scientia Sinica (Series A). Vol. XXVIII No. 9, (1985), 923–937.

[ 3 ] Liu Shangping, Distributions in D⁰ðRⁿÞ as boundary values of harmonic functions, Scientia Sinica (Series A), Vol. XXVII No. 9, (1982).

[ 4 ] Li Banghe, Nonstandard Analysis and Multiplication of Distributions, Scientia Sinica, Vol.

XXI No. 5, (1978), 561–585.

[ 5 ] Li Ya-Qing, The harmonic product of dðx1;. . .;x_nÞ and dðx1Þ and two combinatorial identities, Hiroshima Math. J.30 (2000), 371–376.

[ 6 ] A. Robinson, Nonstandard Analysis, North-Holland Publ., Amsterdam, 1966.

[ 7 ] W. A. J. Luxemburg, Non-Standard Analysis, Mathematics Department, California In- stitute of Technology, Pasadena, California, 1973.

[ 8 ] S. Albeverio, R. Hoegh-Krohn, Jens Erik Fenstad, Tom Lindstrom, Nonstandard Methods in Stochastic Analysis and Mathematical Physics, Academic Press, 1986.

[ 9 ] Laurent Schwartz, The´orie des Distributions, Ed. Hermann, Paris, 1966.

Departmento de Mathematica Facultad de Ciencias Exactas y Naturales

Universidad Nacional de Mar del Plata Funes 3350, CP. 7600

Mar. del Plata, Argentina e-mail: smolina@mdp.edu.ar