# On Modified Mellin Transform of Generalized Functions

Authors: S. K. Q. Al-Omari; Adem Kılıçman
Journal: Abstract and Applied Analysis
 (2013)

---

## Abstract

We investigate the modified Mellin transform on certain function space of generalized functions. We first obtain the convolution theorem for the classical and distributional modified Mellin transform. Then we describe the domain and range spaces where the extended modified transform is well defined. Consistency, convolution, analyticity, continuity, and sufficient theorems for the proposed transform have been established. An inversion formula is also obtained and many properties are given.


---

## Body

## 1. Introduction

The Mellin transformμ of a suitably restricted function over ℝ+((0,∞)) was defined on some strip in the complex plane [1], where many of the transform properties are obtained by applying change of variables to various properties of the Laplace transformation. The Mellin transform is extended to distributions in [1] and to Boehmians in [2].By combining Fourier and Mellin transforms, the obtained transform is called Fourier-Mellin transform which has many applications in digital signals, image processing, and ship target recognition by sonar system and radar signals as well.The modified Mellin transform of a suitably restricted functionf over ℝ+ was introduced by [3]
(1)μfm(y)=:μmf(y)=:∫ℝ+f(x)yxy-1dx‍
as a scale-invariant transform. Then, as earlier, combining the modified Mellin transform with the Fourier transform gives the Fourier-modified Mellin transform  [3, Equation 16].The Mellin-type convolution product of two functionsf and g is given by
(2)(f⋎g)(x)=∫ℝ+‍f(τ)g(xτ)τ-1dτ.
From [1] it has been noted that
(3)μ(f⋎g)(y)=μf(y)μg(y).
Utilizing the Mellin-type convolution product the following theorem is essential for our next investigations.Theorem 1 (convolution theorem).
Letℒ1 be the Lebesgue space of integrable functions and f,g∈ℒ1; then
(4)μm(f⋎g)(y)=μfm(y)μg(y),
where μfm and μg are the Mellin-type and Mellin transforms of f and g, respectively.Proof.
By the definition of the Mellin and modified Mellin transforms we have(5)μm(f⋎g)(y)=∫ℝ+‍(f⋎g)(x)yxy-1dx=∫ℝ+‍(∫ℝ+‍f(τ)g(xτ)τ-1dτ)yxy-1dx=∫ℝ+‍yf(τ)τ-1(∫ℝ+‍g(xτ)xy-1dx)dτ.
Employing Fubinis theorem, then the substitution z=x/τ together with simple computations establishes that
(6)μm(f⋎g)(y)=μfm(y)μg(y).
Hence the theorem is proved.


## 2. Modified Mellin Transform of Distribution

Letμa,b be the space of smooth functions φ over ℝ+ such that [1]
(7)γk(φ)=supx∈ℝ+|ξa,b(x)xk-1dkdxkφ(x)|
is finite, k=0,1,2,…, where
(8)ξa,b(x)≜{x-a,for0<x≤1,x-b,for1<x<∞,
and a,b∈ℝ+.Thenμa,b is linear space under addition and multiplication by complex numbers. μa,b can also be generated by the multinorms (γk)0∞ which turns to be a countably multinormed space.Denote byμ´a,b the complete strong dual space of μa,b; then it is assigned the weak topology. Let ξ(ℝ+) be the space of smooth functions over ℝ+; then for any a,b∈ℝ, μa,b is dense in ξ(ℝ+) and the topology assigned to μa,b is stronger than that induced on μa,b by ξ(ℝ+) and is identified with a subspace of  μ´a,b.The straightforward conclusion is that the kernel function(yxy-1) of μfm is a member of μa,b for a≤Rey≤b.This usually leads to the following definition: letf∈μ´a,b; then the distributional transform μfm^ of f is defined as(9)μfm^(y)≜〈f(x),yxy-1〉,y∈Ωf,
where f∈μ´a,b and Ωf={y∈ℂ:a<Rey<b}.Theorem 2 (analyticity theorem).
Letf∈μ´a,b; then μfm^ is analytic and
(10)dkdykμfm^(y)=〈f(x),yxy-1lnkx+kxy-1lnxk-1〉,
where k is nonnegative integer and y∈Ωf.It is easy for reader to see thatμfm^ is injective and linear from μ´a,b into μ´a,b.The Mellin-type convolution product off,g∈μ´a,b is given as
(11)〈f⋎g,φ〉≜〈f(x),〈g(t),φ(tx)〉〉,
where φ∈μa,b.From (11) it is clear that f⋎g is a member of μ´a,b, for a≤b.Therefore, denote byϑ(ℝ+) the space of test functions of bounded support over ℝ+; then the convolution product of f∈μ´a,b and g∈ϑ(ℝ+) can be given as
(12)(f⋎g)(x)=〈f(t),1tg(xt)〉,
where x∈ℝ+.


## 3. Boehmians

Let𝒢 a group and 𝒮 a subgroup of 𝒢. We assume that to each pair of elements f∈𝒢 and ω∈𝒮 is assigned the product  f*g such that(1)
ifω,ψ∈𝒮,  then ω*ψ∈𝒮 and ω*ψ=ψ*ω;(2)
iff∈𝒢 and ω,ψ∈𝒮, then (f*ω)*ψ=f*(ω⋆ψ);(3)
iff,g∈𝒢,ω∈𝒮, and λ∈ℝ, then
(13)(f+g)*ω=f*ω+g*ω,λ(f*ω)=(λf)*ω.Let Δ be a family of sequences from 𝒮 such that(1)
iff,g∈𝒢,(δn)∈Δ,  and f*δn=g*δn(n=1,2,…), then f=g, for all n;(2)
if(ωn),(δn)∈Δ, then (ωn*ψn)∈Δ. Elements of Δ will be called delta sequences.Consider the class𝒜 of pair of sequences defined by
(14)𝒜={((fn),(ωn)):(fn)⊆𝒢ℕ,(ωn)∈Δ},
for each n∈ℕ.An element((fn),(ωn))∈𝒜 is called a quotient of sequences, denoted by [fυ/ωn], if fn*ωm=fm*ωn, for all n,m∈ℕ.Two quotients of sequencesfn/ωn and gn/ψn are said to be equivalent, fn/ωn~gn/ψn, if fn*ψm=gm*ωn, for all n,m∈ℕ.The relation~ is an equivalent relation on 𝒜. The equivalence class containing fn/ωn is denoted by [fn/ωn]. These equivalence classes are called Boehmians. The space of all Boehmians is denoted by β1.The sum of two Boehmians and multiplication by a scalar can be defined in a natural way(15)[fnωn]+[gnψn]=[fn*ψn+gn*ωnωn*ψn],α[fnωn]=[αfnωn]=[αfnωn],α∈ℂ, space of complex numbers.The operation* and the differentiation are defined by [fn/ωn]*[gn/ψn]=[(fn*gn)/(ωn*ψn)] and 𝒟α[fn/ωn]=[𝒟αfn/ωn].The operation* can be extended to β×𝒮 as follows. If [fn/ωn]∈β1 and ω∈𝒮, then
(16)[fnωn]*ω=[fn*ωωn].δ-Convergence. A sequence of Boehmians (βn) in β1 is said to be δ convergent to a Boehmian β in β1, denoted by βn→δβ, if there exists a delta sequence (ωn) such that
(17)(βn*ωn),(β*ωn)∈𝒢,∀k,n∈ℕ,(βn*ωk)⟶(β*ωk)asn⟶∞,in𝒢,for  everyk∈ℕ.The following is equivalent for the statement ofδ-convergence:  βn→δβ(n→∞) in β1 if and only if there is fn,k,fk∈𝒢 and (ωk)∈Δ  such that βn=[fn,k/ωk], β=[fk/ωk] and for each k∈ℕ,fn,k→fk as  n→∞ in 𝒢.A sequence of Boehmians(βn) in β1 is said to be Δ convergent to a Boehmian β in β1, denoted by βn→Δβ, if there exists a (ωn)∈Δ such that (βn-β)*ωn∈β1, for all n∈ℕ,  and (βn-β)*ωn→0 as n→∞ in  β1. See [2, 5–15].


## 4. Modified Mellin Transform of Boehmian

In this section we discuss the modified Mellin transform on spaces of Boehmians. Consider the groupμ´a,b(ℝ+) and ϑ(ℝ+) as a subgroup of μ´a,b(ℝ+). Let ⋎ be the operation between μ´a,b(ℝ+) and ϑ(ℝ+) and Δ the set of delta sequences given by [2](1)
∫ℝ+‍φn(t)dt=1, for all n∈ℕ;(2)
∫ℝ+‍|φn(t)|dt≤𝔪, for all n∈ℕ, for some 𝔪>0;(3)
suppφn⊂(an,bn), for all n∈ℕ for some 0<an<bn<∞ with an→1, bn→1 as n→∞. Let β1 be the Boehmian space obtained from  μ´a,b,ϑ(ℝ+) and Δ; then β1 will serve as the domain space of  μm↷.Our next objective is to construct a range space, sayβ2, for μm.↷Let(18)Δℝ+e={μφn:(φn)∈Δ},ϑe={μφ:φ∈ϑ},ℓm={μfm^:f∈μ´a,b}.
For g∈ℓm,ψ∈ϑe, define
(19)(g⋏ψ)(y)=g(y)ψ(y).
We have the following theorem.Theorem 3.
Letg∈ℓm and ψ∈ϑe, then g⋏ψ∈ℓm.Proof.
Letg and ψ belong to ℓm and ϑe; respectively. Then there are f∈μ´a,b and φ∈ϑ such that g=μfm and ψ=μφ, respectively. Therefore, by the convolution theorem and (19) we get that
(20)(g⋏ψ)(y)=g(y)ψ(y)=(μfm^μφ)(y)=μm(f⋎φ)(y).
Since f⋎φ∈μ´a,b, it follows g⋏ψ∈ℓm.
Hence we have proved the theorem.Theorem 4.
Letψ1,ψ2∈ϑe; then ψ1⋏ψ2∈ϑe.Proof.
By definition ofϑe we can find φ1,φ2∈ϑ such that ψ1=μφ1 and ψ2=μφ2.
Therefore, by [1],
(21)ψ1⋏ψ2=ψ1(y)ψ2(y)=μφ1(y)μφ2(y)=μ(φ1⋎φ2)(y).
But since φ1⋎φ2∈ϑ, we get ψ1⋏ψ2∈ϑe. Thus we have the theorem.Theorem 5.
Letg1,g2∈ℓm and ψ∈ϑe; then (g1+g2)⋏ψ=g1⋏ψ+g2⋏ψ and (αg)⋏ψ=g⋏(αψ)=α(g1⋏ψ).Proof.
Is straightforward.Theorem 6.
Letgn→g in ℓm,ψ∈ϑe; then gn⋏ψ→g⋏ψ as n→∞ in ℓm.Proof.
Can easily be checked.Theorem 7.
Letg∈ℓm and (ψn)∈Δℝ+e; then g⋏ψn→g as n→∞.Proof.
By (19) we have
(22)(g⋏ψn)(y)=(μfm^μφn)(y),
where f∈μ´a,b and (φn)∈Δ are such that μfm^=g and μφn=ψn, for all n∈ℕ.
Sinceμφn(y)→1 as n→∞ on compact subsets of ℝ+, (22) implies that (g⋏ψn)(y)→μfm^(y)=g(y), for all y, as n→∞. Hence we obtain the theorem.Theorem 8.
Let(ψn),(θn)∈Δℝ+e; then ψn⋏θn∈Δℝ+e.Letψn=μαn,θn=μσn; then taking into account the fact that (ψn⋏θn)(y)=(μαn⋏μσn)(y)=μ(αn⋎σn)(y), since αn⋎σn∈Δ, this theorem follows.The Boehmian spaceβ2 is therefore constructed.In addition, scalar multiplication, differentiation, and the operation⋏ in β2 are defined similar to that of usual Boehmian spaces.Eachg∈ℓm can be identified by a member of β2 given as
(23)g⟶[g⋏ψnψn]asn⟶∞,
where ψn∈Δℝ+e.Definition 9.
The extended modified Mellin transformμm↷:β1→β2 is defined by
(24)μm↷(β)=[μfnm^μφn],∀β=[fnφn]∈β1.Theorem 10.
The extended modified Mellin transform is well defined.Proof.
The proof of this theorem is straightforward. See [11–13].Theorem 11 (consistency theorem).
The extended modified Mellin transformμm↷ is consistent with the distributional μfm^(μfm^:μ´a,b→μ´a,b).Proof.
For everyf∈μ´a,b, let β be its representative in β1; then β=[(f⋎φn)/φn], where φn∈Δ, for all n. Then it is clear that φn is independent of the representative, for all n.
Therefore(25)μm↷(β)=μm↷([f⋎φnφn])i.e.=[μfm^μφnμφn]i.e.=[(μfm^)μφnμφn]
which is the representative of μfm^ in μ´a,b.
Hence we have the proof.Theorem 12 (necessity theorem).
Let[gn/ψn]∈β2; then the necessary and sufficient condition that [gn/ψn]is to be in the range of  μm↷ is that gn belongs to range of μfm^ for every n∈ℕ.Proof.
Let[gn/ψn] be in the range of  μm;↷ then of course gn belongs to the range of (μfm^), for all n∈ℕ.
To establish the converse, letgn be in the range of μfm^, for all n∈ℕ. Then there is fn∈μ´a,b such that μfnm^=gn,n∈ℕ.
Since[gn/ψn]∈β2,
(26)gn⋎ψm=gm⋎ψn,
for all m,n∈ℕ. Therefore,
(27)μm(fn⋏φn)=μm(fm⋏φn),∀m,n∈ℕ,
where fn∈μ´a,b and φn∈Δ.
The fact thatμfm^ is injective, μfm^:μ´a,b→μ´a,b, implies that fn⋏φm=fm⋏φn,m,n∈ℕ.
Thusfn/φn is quotient of sequences in β1. Hence, [fn/φn]∈β1 and
(28)μm↷([fnφn])=[gnψn].
Hence the theorem is proved.Theorem 13 (generalized convolution theorem).
Letβ=[fn/φn]∈β1 and γ=[κn/ϕn]∈β1; then
(29)μm↷(β⋎γ)=μm↷([fnφn])⋏μm↷([κnϕn]).Proof.
Assume that the requirements of the theorem satisfy for someβ and γ∈β1; then using Definition 9 and the operation ⋏ yields
(30)μm↷(β⋎γ)=μm↷([fn⋎κnφn⋎ϕn])=[μm(fn⋎κn)μ(φn⋎ϕn)]=[μfnm^⋏μκnmμφn⋏μϕn]=[μfnm^μφn]⋏[μκnmμϕn].
Therefore
(31)μm↷(β⋎γ)=μm↷([fnφn])⋏μm↷([κnϕn]).
This completes the proof.Theorem 14.
The extended modified Mellin transformμm↷:β1→β2 is bijective.Proof.
Assumeμm↷[fn/φn]=μm↷[κn/ϕn]; then it follows from the concept of quotients of sequences that μfnm^⋏μϕm=μκmm^⋏μφn. Therefore, μm(fn⋎ϕm)=μm(κm⋎φn). The property that μm is one to one implies fn⋎ϕm=κm⋎φn. Therefore,
(32)[fnφn]=[κnϕn].
Next to establish that μm↷ is onto, let [μfnm^/μφn](∈β2) be arbitrary; then μfnm^⋏μφm=μfmm^⋏μφn for every m,n∈ℕ. Hence fn,fm∈μ´a,b are such that μm(fn⋎φm)=μm(fm⋎φn), for all m,n∈ℕ.
Hence, the Boehmian[fn/φn] belongs to β1 and satisfies
(33)μm↷[fnφn]=[μfnm^μφn].
This completes the proof of the theorem.Now we introduce(μm↷)-1 as the inverse transform of μm↷, where
(34)(μm↷)-1([μfnm^μφn])=[(μfnm^)-1(μfnm^)(μφn)-1(μφn)],
for every [fn/φn]∈β1.Theorem 15.
Let[μfnm^/μφn]∈β2 and ϕ∈ϑe, then
(35)(μm↷)-1([μfnm^μφn]⋏ϕ)=[fnφn]⋎ϕ,(36)μm↷([fnφn]⋎ϕ)=[μfnm^μφn]⋏ϕ.Proof.
We prove (35) and omit the proof of (36) due to its similarity. Given [μfnm^/μφn]∈β2 and ψ∈ϑ such that ϕ=μψm then employing (34) yields
(37)(μm↷)-1([μfnm^μφn]⋏ϕ)=(μm↷)-1([μfnm^⋏ϕμφn])=[(μfnm)-1(μfnm^⋏μψm)(μφn)-1(μφn)].
Using (19) gives
(38)(μm↷)-1([μfnm^μφn]⋏ϕ)=[(μfnm)-1(μfnm^μψm)(μφn)-1(μφn)].
Hence the convolution theorem gives
(39)(μm↷)-1([μfnm^μφn]⋏ϕ)=[(μfnm)-1(μfnm(fn⋎ϕ))(μφn)-1(μφn)].
Thus
(40)(μm↷)-1([μfnm^μφn]⋏ϕ)=[fnφn]⋎ϕ.
Proof of the second part is similar.
This completes the proof of the theorem.Theorem 16.
μ
m
↷
:
β
1
→
β
2 and (μm↷)-1:β2→β1 are continuous with respect to δ-convergence.Proof.
Letβn→δβ in β1 as n→∞; then we establish that μm↷βn→μm↷β as n→∞. Let fn,k and fk be in μ´a,b such that
(41)βn=[fn,kφk],β=[fkφk]
and fn,k→fk as n→∞ for every k∈ℕ.
The continuity ofμfn,km^ implies μfn,km^→μfkm^ as n→∞. Thus,
(42)[μfn,km^μφk]⟶[μfkm^μφk]
as n→∞ in β2. This proves continuity of μm↷.
Next, letgn→δg∈β2 as n→∞; then we have gn=[μfn,km^/μφk] and g=[μfkm^/μφk] for some μfn,km^,μfkm^∈ℓm, where μfn,km^→μfkm^ as n→∞. Hence
(43)(μfn,km^)-1(μfn,km^)⟶(μfkm^)-1(μfkm^)
as n→∞ in β1. That is,
(44)[(μfn,km^)-1(μfn,km^)φk]=[(μfn,km^)-1(μfn,km^)(μφk)-1(μφk)]⟶[(μfkm^)-1(μfkm^)φk]
as n→∞.
Hence(45)[(μfn,km^)-1(μfn,km^)φk]⟶[(μfkm^)-1(μfkm^)(μφk)-1(μφk)]
as n→∞.
That is,(46)(μm↷)-1gn⟶(μm↷)-1g
as n→∞. This completes the proof.Theorem 17.
μ
m
↷
:
β
1
→
β
2 and (μm↷)-1:β2→β1 are continuous with respect to Δ-convergence.Proof.
Letβn→Δβ in β1 as n→∞. Then, we find fn∈μ´a,b and (φk)∈Δ such that (βn-β)⋏φk=[(fn⋏φk)/φk] and fn→0 as n→∞. Therefore
(47)μm↷((βn-β)⋎φk)=[μm(fn⋎φk)μφk].
Hence, μm↷((βn-β)⋎φk)=[(μfnm^⋏μφk)/μφk]=μfnm^→0 as n→∞ in ℓm.
Therefore(48)μm↷((βn-β)⋎φn)=(μm↷βn-μm↷β)⋏μφk⟶asn⟶∞.
Hence,μm↷βn→Δμm↷β as n→∞.
Proof of the second part is analogous. Detailed proof is as follows.
Finally, letgn→Δg in β2 as n→∞; then we can find μfkm^∈ℓm such that (gn-g)⋏μφk=[(μfkm^⋏μφk)/μφk] and μfkm^→0 as n→∞ for some (μφk)∈Δℝ+e.
Next, we have(49)(μm↷)-1((gn-g)⋏μφk)=[(μfkm)-1(μfkm^⋏μφk)(μφk)-1(μφk)].
Thus, by (34) we get
(50)(μm↷)-1((gn-g)⋏μφk)=[fn⋎φkφk]=fn⟶0
as n→∞ in μ´a,b.
Therefore(51)(μm↷)-1((gn-g)⋏μφk)=((μm↷)-1gn-(μm↷)-1g)⋎φk⟶0
as n→∞.
Thus, we have(52)(μm↷)-1gn→Δ(μm↷)-1g
as n→∞ in β1.
This completes the proof of the theorem.



---

*Source: 539240-2013-12-11.xml*