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.
1. Introduction
The Mellin transform μ 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 function f 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 functions f 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).
Let ℒ1 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
Let μa,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+.
Then μa,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 by μ´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: let f∈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).
Let f∈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 that μfm^ is injective and linear from μ´a,b into μ´a,b.
The Mellin-type convolution product of f,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 by ϑ(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
Let 𝒢 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
if ω,ψ∈𝒮, then ω*ψ∈𝒮 and ω*ψ=ψ*ω;
if f∈𝒢 and ω,ψ∈𝒮, then (f*ω)*ψ=f*(ω⋆ψ);
if f,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
if f,g∈𝒢,(δn)∈Δ, and f*δn=g*δn(n=1,2,…), then f=g, for all n;
if (ωn),(δn)∈Δ, then (ωn*ψn)∈Δ.
Elements of Δ will be called delta sequences.
Consider the class 𝒜 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 sequences fn/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 of δ-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 group μ´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]
∫ℝ+φn(t)dt=1, for all n∈ℕ;
∫ℝ+|φn(t)|dt≤𝔪, for all n∈ℕ, for some 𝔪>0;
supp φn⊂(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, say β2, 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.
Let g∈ℓm and ψ∈ϑe, then g⋏ψ∈ℓm.
Proof.
Let g 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.
Let ψ1,ψ2∈ϑe; then ψ1⋏ψ2∈ϑe.
Proof.
By definition of ϑe 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.
Let g1,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.
Let gn→g in ℓm,ψ∈ϑe; then gn⋏ψ→g⋏ψ as n→∞ in ℓm.
Proof.
Can easily be checked.
Theorem 7.
Let g∈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.
Since μφn(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.
Let ψn=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 space β2 is therefore constructed.
In addition, scalar multiplication, differentiation, and the operation ⋏ in β2 are defined similar to that of usual Boehmian spaces.
Each g∈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 transform μm↷: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 transform μm↷ is consistent with the distributional μfm^(μfm^:μ´a,b→μ´a,b).
Proof.
For every f∈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, let gn 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 that μfm^ is injective, μfm^:μ´a,b→μ´a,b, implies that fn⋏φm=fm⋏φn,m,n∈ℕ.
Thus fn/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).
Let β=[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 some β 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 transform μm↷:β1→β2 is bijective.
Proof.
Assume μm↷[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.
μm↷:β1→β2 and (μm↷)-1:β2→β1 are continuous with respect to δ-convergence.
Proof.
Let βn→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 of μfn,km^ implies μfn,km^→μfkm^ as n→∞. Thus,
(42)[μfn,km^μφk]⟶[μfkm^μφk]
as n→∞ in β2. This proves continuity of μm↷.
Next, let gn→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.
μm↷:β1→β2 and (μm↷)-1:β2→β1 are continuous with respect to Δ-convergence.
Proof.
Let βn→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, let gn→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.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The authors acknowledge that this research was partially supported by Universiti Putra Malaysia under the ERGS 1-2013/5527179.
ZemanianA. H.1987New York, NY, USADoverMR896486Al-OmariS. K. Q.On the distributional Mellin transformation and its extension to Boehmian spaces2011617–20801810MR2837986ZBL1245.46032YangJ.SarkarT. K.AntonikP.Applying the Fourier-modified Mellin transform to Doppler-distorted waveforms200717610301039Al-OmariS. K. Q.A Mellin transform for a space of Lebesgue integrable Boehmians2011629–3215971606MR2837944ZBL1253.46047Al-OmariS. K. Q.Distributional and tempered distributional diffraction Fresnel transforms and their extension to Boehmian spaces201330179194MR3066507BanerjiP. K.Al-OmariS. K.DebnathL.Tempered distributional Fourier sine (cosine) transform2006171175976810.1080/10652460600856534MR2263952ZBL1131.42003MikusińskiP.Fourier transform for integrable Boehmians198717357758210.1216/RMJ-1987-17-3-577MR908263ZBL0629.44005MikusińskiP.Tempered Boehmians and ultradistributions1995123381381710.2307/2160805MR1223517ZBL0821.46053MikusińskiP.Convergence of Boehmians198391159179MR722539ZBL0524.44005Al-OmariS. K. Q.LoonkerD.BanerjiP. K.KallaS. L.Fourier sine (cosine) transform for ultradistributions and their extensions to tempered and ultraBoehmian spaces2008195-645346210.1080/10652460801936721MR2426735ZBL1215.42007Al-OmariS. K. Q.Hartley transforms on a certain space of generalized functions201320341542610.1515/gmj-2013-0034MR3100963ZBL06216217Al-OmariS. K. Q.KılıçmanA.Note on Boehmians for class of optical Fresnel wavelet transforms201220121410.1155/2012/405368405368MR2980402ZBL1266.46030Al-OmariS. K. Q.KılıçmanA.On generalized hartley-Hilbert and Fourier-Hilbert transforms20122012, article 2321210.1186/1687-1847-2012-232BoehmeT. K.The support of Mikusiński operators1973176319334MR0313727ZBL0268.44005Al-OmariS. K. Q.KılıçmanA.On diffraction Fresnel transforms for Boehmians201120111171274610.1155/2011/712746MR2861514ZBL1243.46032