∫ ∫ ∫ A Note on the Convolution in the Mellin Sense with Generalized Functions

MALAYSIAN MATHEMATICAL

SCIENCES SOCIETY

A Note on the Convolution in the Mellin Sense with Generalized Functions

ADEM KILICMAN AND MUHAMMAD REZAL KAMEL ARIFFIN

Jabatan Matematik, Fakulti Sains dan Pengajian Alam Sekitar, Universiti Putra Malaysia, 43400 UPM, Serdang, Selangor Darul Ehsan, Malaysia

Abstract. The classical convolution is given by the following equation

∫

−^∞∞ −

=

∗

= f g x f x t gtdt x

h( ) ( )( ) ( ) ()

and is widely accessible in many literatures including its extension to generalized functions (Gelfand and Shilov [4], A. Zemanian [2]). Another form of convolution is given by the Mellin convolution (i.e. convolution in the Mellin sense) which is given by

( ^∗ ) ⁼

∫

^{∞ ⎜}_⎝^{⎛ ⎟}_⎠^⎞

=

0 ( )

) ( )

( u

u du u g f x x

g f x

h _M .

The theory of the convolution in the Mellin sense for Mellin transformable functions is well known (Butzer and Jansche [3], Srivastava and Buschman [6]). In this work we extend this setting to the generalized functions.

1. Introduction

First of all we review the Mellin transform and the convolution in the Mellin sense for Mellin transformable functions. The following definitions are held in [3].

Definition 1.1. If f : R₊ →C is a function such that f(x)x^s−1 ∈L¹(R₊) for some C

s∈ , where s =c+it and c∈R, t∈R₊ the Mellin transform is defined by the following equation

∫

^{∞ −}

=

= 0

) 1

( )

( )

(f F s f x x dx

M ^s (1)

Definition 1.1 will lead us to the next definition.

Definition 1.2. The space X_c for some c∈R is defined as follows

{

−

( )

+

}

+ → ∈

= f R C f x x L R

X_c : : ; ( ) ^{c 1 1}

and the norm defined on Xc,

( )

∫

^{∞ −}

− =

=

+ 0

1

1 ( )

) (

: f x x ₁ f x x dx

f ^c

R L c X_c

Thus, X_c consists of all f for which the transform M(f) exists for all t∈R₊ and some c∈R, which is operating on the vertical line c×iR, with ( ) .

Xc

f s

F ≤

While the space X_{( )a,b} is given for a,b∈R, a< b by

( )

I

( )

b a c

c b

a X

X

,

, :

∈

=

Thus, the space X_{( )a,b} consists of all f for which the transform M(f) exists for all

∈R+

t and all c∈(a,b), which is operating on an open strip C

iR b a b a

St( , ):= ( , ) × ⊂ .

Example 1.1. Let f(x)= e^−x ∈X_(0,∞). Its Mellin transform is given by the well known the Gamma function,

∫

^{∞ − −}

− = Γ =

0

) 1

( )

(e s e x dx

M ^{x x s} , Re(s) > 0 (2)

However, we can see that e^−x∉X_[0,∞). Thus, convergence on St(a,b) does not necessarily mean convergence on ^St

[ ]

^a,b .

Definition 1.3. The convolution in the Mellin sense f ∗_M gof two given functions C

R g

f, : ₊ → is defined by

(

^∗

)

( ) :⁼

∫

0^{∞ ⎜}_⎝^{⎛ ⎟}_⎠^⎞ ( ) u u du u g f x x

g

f _M (3)

Theorem 1.1. (Butzer and Jansche [3]) If f,g ∈X_c, then the convolution in the Mellin sense f ∗_M g exists (a.e.) on R₊. Also the convolution product

( )

( )

)

(x f g x

h = ∗_M belongs to the space X_c, and one has

c c

c X X

M g X f g

f ∗ ≤ .

Theorem 1.2. (Srivastava, Buschman [6]) (Butzer and Jansche [3]) (Sasiela [5]) If f,g∈Xc, then

(

^{f g}

)

^F(s)G(s)

M ∗_M = (4)

Equation (4) is known as the exchange formula.

Now we let E_p,q be the space of infinitely differentiable functions that satisfies the following conditions.

Let φ be an infinitely differentiable arbitrary testing function in E_p,q. Then there exists any two real numbers p and q such that

( )( ) 0

lim ¹

0

− →

+

→ x^{k p k} x

x

φ

( )( ) 0

lim ⁺¹⁻ →

∞

→ x^{k q k} x

x φ

for p< k+1< q and k = 0,1, 2,_L. Let us define

⎩⎨

⎧

≥

<

= ⁻₋ <

1

;

1 0

) ;

( ₁

1

, x x

x x x

h _q

p q

p

where p<1< q, and

{

^{( )} ( )^{( )}

}

sup )

( _,

0 ,

, h_pq x x^{k k} x

x q

p

k φ φ

γ

>

=

Every γ_k,p,q(φ) is bounded and positively defined. Moreover for particular case

= 0

k , the function γ_0,p,q(φ) is a norm. Next, we prove the following lemma.

Lemma 1.1. Let φ(uw) be an infinitely differentiable function on (0,∞). For q

r

p< +1< and p< k−r+1<q; 0≤ r ≤ k and k, r = 0,1, 2,_L, φ(uw) is a member of E_p,q.

Proof. In order to show that the φ(uw) is a member of E_p,_q we examine its partial derivatives evaluated at the origin and at infinity.

First of all, we fix w and treat φ(uw). Since, there exists two real numbers p and qsuch that p<r +1< q, for each partial derivative with respect to u, we have

( ) ( ) 0

lim ¹

0

∂ →

− ∂

−

→ uw

u u_r

r r p u

φ

( ) ( ) 0

lim ¹ →

∂

− ∂

−

∞

→ uw

u u_r

r r q

u φ

For each partial derivative with respect to ,u that partial derivative possesses partial derivatives with respect to ,w such that with the above mentioned real numbers p and qand p< k−r+1< q, similar to the above equation we have the following relation,

( )

( ) ( ) 0

lim ¹

0

⎟⎟→

⎠

⎞

⎜⎜⎝

⎛

∂

∂

∂

−

−

−

−

→ uw

w u

w _{r k r}

k r

k p

w φ

( )

( ) ( ) 0

lim ¹ ⎟⎟ →

⎠

⎞

⎜⎜⎝

⎛

∂

∂

∂

−

−

−

−

∞

→ uw

w

w u_{r k r}

k r

k q w

φ

For p <1< q, we define

⎩⎨

⎧

≥

≥

<

<

<

= ₋⁻ <

1 , 1

; ) (

1 0

, 1 0

; ) ) (

,

( ₁

1

, uw u w

w u

w uw u

n _p

p q

p

and ^{( ) sup}

{

, ^{( , ) ( )( )}

}

^,

0 , 0 ,

, n_pq u w u^rw^{k r k} uw

w u q

p

k φ φ

λ ⁻

>

>

= where λ_k,p,q(φ) is bounded

and positively defined and similar to the above particular case k = 0, λ_0,p,q(φ) is a norm.

Hence, )φ(uw is truly a member of E_p,q with the above mentioned properties.

We note that the same can be derived if we fix the variable u first and treat φ(uw) as a function of only .w Then we state easily that if φ(uw) is a member of E_p,q for any generalized function f ∈E′_p,q, the Θ(u) = f(w),φ(uw) is also a testing function in

q

Ep_, .

Now we consider the space E′p_,q which is the linear space of continuous linear functionals on

E

_p,q which is zero on the interval(−∞,0). In fact the space E′_p,q is the dual space of

E

_p,q. That is, for every f ∈E′_p,q if and only if the following conditions are satisfied.

For any φ∈E_p,q,

1. f(x),φ(x) is defined.

2. f(x),φ(x) = 0 if φ(x) = 0 for x >0

3. f(x),αφ₁(x)+ βφ₂(x) = f(x),αφ₁(x) + f(x),βφ₂(x) for α,β ∈R 4. f(x),φ_n(x) →0if φ_n(x)→ 0as n→ ∞

If f ∈E′_p,q and φ∈E_p,q f(x),φ(x) is defined by the following integral,

∫

^∞

= 0 ( ) ( )

) ( , )

(x x f x x dx

f φ φ ^≤ 0, ,

∫

0^{∞ ( )}_{( )}

,

)

( _h^{f x}_x dx

q

p pq

φ

γ (4)

where the right hand-side of (4) exists.

Since the theory of generalized functions is a linear theory, we can extend some operations which are valid for ordinary functions to E′_p,q. Such operations are called regular operations such as addition and multiplication by scalar .

For example if f,g ∈E′_p,q and α ∈C, for any φ∈E_p,q then easily one can have, 1. f(x)+g(x),φ(x) = f(x),φ(x) + g(x),φ(x)

2. αf(x),φ(x) =α f(x),φ(x)

2. Convolution of generalized functions in the Mellin sense

Definition 2.1. If f, g∈E′_p,q we define the convolution in the Mellin sense as )

( , ) ( , ) ( ,

, f g gu f w uw

hφ = ∗_M φ = φ (5)

and for φ∈E_p,_q.

Theorem 2.1. If f,g∈E′_p,q, then f ∗_M g∈E_p′_,q for every φ∈E_p,q when q

r

p < +1< where r = 0,1, 2,_L.

Proof: The convolution of generalized functions in E_p′,_q in the Mellin sense is as given by (5). LetΘ(u) = f(w),φ(uw) . For r = 0,1,2,_L we can have the following,

0 )

( )

(

lim 0 ( , )

) ( ,

, 0 )

(

0 ^,

→

≤

Θ

∫

^∞

→ ^r u _{pq n} ^f _u^w_w dw

u ^pq

φ λ

0 )

( )

(

lim 0 ( , )

) ( ,

, 0 )

(

, →

≤

Θ

∫

^∞

∞

→ ^r u _{pq n} ^f _u^w_w dw

u ^pq

φ λ

where λ_0,p,q(φ) and n_p,q(u,w) is as given in Lemma 1.1. Thus, Θ(u) is a testing function in E_p,q. And since, g∈E′_p,q , the right hand-side of equation (5) exists.

Next, let us define the following testing function,

⎪⎩

⎪⎨

⎧

∞

→

∞

→

≤

≤

>

>

=

−

w u

w u

w u

uw uw

s

,

; 0

0 ,

0

; 0

0 ,

0

; ) ( ) (

1

φ (6)

From the equation (6) it follows that φ(uw) is a member of E_p,q for p<r+ s< q and q

s r k

p< − + < with the semi norms imposed on it as given in Lemma 1.1.

Now we give the Mellin transform of the convolution of generalized functions in

q

Ep′_, in the Mellin sense.

Theorem 2.2. Let f(x) and g(x) be Mellin transformable generalized functions in

q

Ep′_, and let,

) ( ] [f F s

M = for p₁ < Re(s) < q₁ and

) ( ] [g G s

M = for p₂ < Re(s) < q₂.

Then f ∗_M g exists, and is also a Mellin transformable generalized function in E_p′_,q and,

[

f g

]

F(s)G(s)

M ∗_M = (7)

for Re(s)∈(p₁,q₁) ∩ (p₂,q₂), where (p₁,q₁) ∩ (p₂,q₂) is not empty and q

s r

p< + < and p< k−r+s < q for 0≤ r ≤ k and k,r = 0,1,2,_L.

Proof. Since p< r+s <q and p< k−r+s <q, then f(x)∗_M g(x),φ(x) exists for any φ∈E_p,_q. The Mellin transform of the generalized function given by

) ( )

(x f g x

h = ∗_M is given by the equation

( )

( ) ( ), ( ) ¹ ( ), ( ),( ) ¹

)

(h = M f ∗_M g = g u ×_M f w uw ^s− = g u f w uw ^s−

M (8)

On using the equation (6) and since p< r+s <q and p <k −r +s< q, the equation (8) exists. Thus it follows that,

(

f g

)

g(u),u ¹ f(w),w ¹ F(s)G(s)

M ∗_M = ^{s− s−} =

for Re(s)∈(p₁,q₁)∩(p₂,q₂), where(p₁,q₁) ∩ (p₂,q₂) is not empty, such that (7) is established.

Corollary 2.1. The convolution of generalized functions in E′_p,q in the Mellin sense is commutative and distributive. That is, for f,g ∈E′_p,q and φ∈E_p,q,

f g g

f ∗_M = ∗_M and h∗_M [f + g]= (h∗_M f) + (h∗_M g).

References

1. A.I. Zayed, Function and Generalized Functions Transformations, CRC Press, 1996.

2. A. Zemanian, Distribution Theory and Transform Analysis, McGraw Hill, New York, 1965.

3. Butzer, Jansche, Mellin transform theory and the role of its differential and integral operators, Proc. Con. On Transform Methods and Special Functions, Varna, 1996.

4. I.M. Gelfand and G.E. Shilov, Generalized Functions Vol. I, Academic Press, 1964.

5. R.J. Sasiela, Electromagnetic Wave Propagation In Turbulence Evaluation and Application of Mellin Transform, Springer-Verlag, 1994.

6. H.M. Srivastava and R.G Buschman, Theory and Applications of Convolution Integral Equations, Kluwer Academic Publishers, 1992.