On Modified Mellin Transform of Generalized Functions

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.

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] 𝜇^𝑚_𝑓𝑦=∶𝜇^𝑚𝑓𝑦=∶_ℝ+𝑓(𝑥)𝑦𝑥^𝑦−1𝑑𝑥 (1) 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 𝑓⋎𝑔(𝑥)=_ℝ+𝑓(𝜏)𝑔𝑥𝜏𝜏⁻¹𝑑𝜏. (2) From [1] it has been noted that 𝜇𝑓⋎𝑔𝑦=𝜇_𝑓𝑦𝜇_𝑔𝑦. (3) 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 𝜇^𝑚𝑓⋎𝑔𝑦=𝜇^𝑚_𝑓𝑦𝜇_𝑔𝑦, (4) where 𝜇^𝑚_𝑓 and 𝜇_𝑔 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 𝜇^𝑚𝑓⋎𝑔𝑦=_ℝ+𝑓⋎𝑔(𝑥)𝑦𝑥^𝑦−1𝑑𝑥=_ℝ+ℝ+𝑓(𝜏)𝑔𝑥𝜏𝜏⁻¹𝑑𝜏𝑦𝑥^𝑦−1=𝑑𝑥_ℝ+𝑦𝑓(𝜏)𝜏⁻¹_ℝ+𝑔𝑥𝜏𝑥^𝑦−1𝑑𝑥𝑑𝜏. (5) Employing Fubinis theorem, then the substitution 𝑧=𝑥/𝜏 together with simple computations establishes that 𝜇^𝑚𝑓⋎𝑔𝑦=𝜇^𝑚_𝑓𝑦𝜇_𝑔𝑦. (6) Hence the theorem is proved.

2. Modified Mellin Transform of Distribution

Let 𝜇_𝑎,𝑏 be the space of smooth functions 𝜑 over ℝ₊ such that [1] 𝛾_𝑘𝜑=sup_𝑥∈ℝ+𝜉_𝑎,𝑏(𝑥)𝑥^𝑘−1𝑑^𝑘𝑑𝑥^𝑘𝜑(𝑥) (7) is finite, 𝑘=0,1,2,…, where 𝜉_𝑎,𝑏(𝑥)≜𝑥^−𝑎𝑥,for0<𝑥≤1,_−𝑏,for1<𝑥<∞, (8) and 𝑎,𝑏∈ℝ₊.

Then 𝜇_𝑎,𝑏 is linear space under addition and multiplication by complex numbers. 𝜇_𝑎,𝑏 can also be generated by the multinorms (𝛾_𝑘)^∞₀ which turns to be a countably multinormed space.

Denote by 𝜇́_𝑎,𝑏 the complete strong dual space of 𝜇_𝑎,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, 𝜇_𝑎,𝑏 is dense in 𝜉(ℝ₊) and the topology assigned to 𝜇_𝑎,𝑏 is stronger than that induced on 𝜇_𝑎,𝑏 by 𝜉(ℝ₊) and is identified with a subspace of 𝜇́_𝑎,𝑏.

The straightforward conclusion is that the kernel function (𝑦𝑥^𝑦−1) of 𝜇^𝑚_𝑓 is a member of 𝜇_𝑎,𝑏 for 𝑎≤Re𝑦≤𝑏.

This usually leads to the following definition: let 𝑓∈𝜇́_𝑎,𝑏; then the distributional transform 𝜇^𝑚_𝑓 of 𝑓 is defined as𝜇^𝑚_𝑓𝑦≜𝑓(𝑥),𝑦𝑥^𝑦−1,𝑦∈Ω_𝑓, (9) where 𝑓∈𝜇́_𝑎,𝑏 and Ω_𝑓={𝑦∈ℂ∶𝑎<Re𝑦<𝑏}.

Theorem 2 (analyticity theorem). Let 𝑓∈𝜇́_𝑎,𝑏; then 𝜇^𝑚_𝑓 is analytic and d^𝑘d𝑦^𝑘𝜇^𝑚_𝑓𝑦=𝑓(𝑥),𝑦𝑥^𝑦−1ln^𝑘𝑥+𝑘𝑥^𝑦−1ln𝑥^𝑘−1, (10) where 𝑘 is nonnegative integer and 𝑦∈Ω_𝑓.

It is easy for reader to see that 𝜇^𝑚_𝑓 is injective and linear from 𝜇́_𝑎,𝑏 into 𝜇́_𝑎,𝑏.

The Mellin-type convolution product of 𝑓,𝑔∈𝜇́_𝑎,𝑏 is given as 𝑓⋎𝑔,𝜑≜𝑓(𝑥),𝑔(𝑡),𝜑(𝑡𝑥), (11) where 𝜑∈𝜇_𝑎,𝑏.

From (11) it is clear that 𝑓⋎𝑔 is a member of 𝜇́_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∈𝜇́_𝑎,𝑏 and 𝑔∈𝜗(ℝ₊) can be given as (𝑥)=𝑓(𝑡),1𝑓⋎𝑔𝑡𝑔𝑥𝑡, (12) 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(1)if 𝜔,𝜓∈𝒮, then 𝜔∗𝜓∈𝒮 and 𝜔∗𝜓=𝜓∗𝜔;(2)if 𝑓∈𝒢 and 𝜔,𝜓∈𝒮, then (𝑓∗𝜔)∗𝜓=𝑓∗(𝜔⋆𝜓);(3)if 𝑓,𝑔∈𝒢,𝜔∈𝒮, and 𝜆∈ℝ, then 𝑓+𝑔∗𝜔=𝑓∗𝜔+𝑔∗𝜔,𝜆𝑓∗𝜔=𝜆𝑓∗𝜔. (13)Let Δ be a family of sequences from 𝒮 such that(1)if 𝑓,𝑔∈𝒢,(𝛿_𝑛)∈Δ, and 𝑓∗𝛿_𝑛=𝑔∗𝛿_𝑛(𝑛=1,2,…), then 𝑓=𝑔, for all 𝑛;(2)if (𝜔_𝑛),(𝛿_𝑛)∈Δ, then (𝜔_𝑛∗𝜓_𝑛)∈Δ. Elements of Δ will be called delta sequences.

Consider the class 𝒜 of pair of sequences defined by 𝒜=𝑓_𝑛,𝜔_𝑛∶𝑓_𝑛⊆𝒢^ℕ,𝜔_𝑛∈Δ, (14) for each 𝑛∈ℕ.

An element ((𝑓_𝑛),(𝜔_𝑛))∈𝒜 is called a quotient of sequences, denoted by [𝑓_𝜐/𝜔_𝑛], if 𝑓_𝑛∗𝜔_𝑚=𝑓_𝑚∗𝜔_𝑛, for all 𝑛,𝑚∈ℕ.

Two quotients of sequences 𝑓_𝑛/𝜔_𝑛 and 𝑔_𝑛/𝜓_𝑛 are said to be equivalent, 𝑓_𝑛/𝜔_𝑛∼𝑔_𝑛/𝜓_𝑛, if 𝑓_𝑛∗𝜓_𝑚=𝑔_𝑚∗𝜔_𝑛, for all 𝑛,𝑚∈ℕ.

The relation ∼ is an equivalent relation on 𝒜. The equivalence class containing 𝑓_𝑛/𝜔_𝑛 is denoted by [𝑓_𝑛/𝜔_𝑛]. 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 𝑓_𝑛𝜔_𝑛+𝑔_𝑛𝜓_𝑛=𝑓_𝑛∗𝜓_𝑛+𝑔_𝑛∗𝜔_𝑛𝜔_𝑛∗𝜓_𝑛,𝛼𝑓_𝑛𝜔_𝑛=𝛼𝑓_𝑛𝜔_𝑛=𝛼𝑓_𝑛𝜔_𝑛, (15)𝛼∈ℂ, space of complex numbers.

The operation ∗ and the differentiation are defined by [𝑓_𝑛/𝜔_𝑛]∗[𝑔_𝑛/𝜓_𝑛]=[(𝑓_𝑛∗𝑔_𝑛)/(𝜔_𝑛∗𝜓_𝑛)] and 𝒟^𝛼[𝑓_𝑛/𝜔_𝑛]=[𝒟^𝛼𝑓_𝑛/𝜔_𝑛].

The operation ∗ can be extended to 𝛽×𝒮 as follows. If [𝑓_𝑛/𝜔_𝑛]∈𝛽₁ and 𝜔∈𝒮, then 𝑓_𝑛𝜔_𝑛𝑓∗𝜔=^𝑛∗𝜔𝜔_𝑛. (16)

δ-Convergence. A sequence of Boehmians (𝛽_𝑛) in 𝛽₁ is said to be 𝛿 convergent to a Boehmian 𝛽 in 𝛽₁, denoted by 𝛽→−_𝑛^𝛿𝛽, if there exists a delta sequence (𝜔_𝑛) such that 𝛽_𝑛∗𝜔_𝑛,𝛽∗𝜔_𝑛𝛽∈𝒢,∀𝑘,𝑛∈ℕ,_𝑛∗𝜔_𝑘⟶𝛽∗𝜔_𝑘as𝑛⟶∞,in𝒢,forevery𝑘∈ℕ. (17)

The following is equivalent for the statement of 𝛿-convergence: 𝛽→−_𝑛^𝛿𝛽(𝑛→∞) in 𝛽₁ if and only if there is 𝑓_𝑛,𝑘,𝑓_𝑘∈𝒢 and (𝜔_𝑘)∈Δ such that 𝛽_𝑛=[𝑓_𝑛,𝑘/𝜔_𝑘], 𝛽=[𝑓_𝑘/𝜔_𝑘] and for each 𝑘∈ℕ, 𝑓_𝑛,𝑘→𝑓_𝑘 as 𝑛→∞ in 𝒢.

A sequence of Boehmians (𝛽_𝑛) in 𝛽₁ is said to be Δ convergent to a Boehmian 𝛽 in 𝛽₁, denoted by 𝛽−→_𝑛^Δ𝛽, if there exists a (𝜔_𝑛)∈Δ such that (𝛽_𝑛−𝛽)∗𝜔_𝑛∈𝛽₁, for all 𝑛∈ℕ, and (𝛽_𝑛−𝛽)∗𝜔_𝑛→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 𝜇́_𝑎,𝑏(ℝ₊) and 𝜗(ℝ₊) as a subgroup of 𝜇́_𝑎,𝑏(ℝ₊). Let ⋎ be the operation between 𝜇́_𝑎,𝑏(ℝ₊) and 𝜗(ℝ₊) and Δ the set of delta sequences given by [2](1)∫_ℝ+𝜑_𝑛(𝑡)𝑑𝑡=1, for all 𝑛∈ℕ;(2)∫_ℝ+|𝜑_𝑛(𝑡)|𝑑𝑡≤𝔪, for all 𝑛∈ℕ, for some 𝔪>0;(3)supp 𝜑_𝑛⊂(𝑎_𝑛,𝑏_𝑛), for all 𝑛∈ℕ for some 0<𝑎_𝑛<𝑏_𝑛<∞ with 𝑎_𝑛→1, 𝑏_𝑛→1 as 𝑛→∞. Let 𝛽₁ be the Boehmian space obtained from 𝜇́_𝑎,𝑏,𝜗(ℝ₊) and Δ; then 𝛽₁ will serve as the domain space of 𝜇^↷𝑚.

Our next objective is to construct a range space, say 𝛽₂, for 𝜇^↷𝑚.

Let Δ^𝑒_ℝ+=𝜇_𝜑𝑛∶𝜑_𝑛∈Δ,𝜗^𝑒=𝜇_𝜑,ℓ∶𝜑∈𝜗_𝑚=𝜇^𝑚_𝑓∶𝑓∈𝜇́_𝑎,𝑏. (18) For 𝑔∈ℓ^𝑚,𝜓∈𝜗^𝑒, define 𝑔⋏𝜓𝑦=𝑔𝑦𝜓𝑦. (19) We have the following theorem.

Theorem 3. Let 𝑔∈ℓ^𝑚 and 𝜓∈𝜗^𝑒, then 𝑔⋏𝜓∈ℓ^𝑚.

Proof. Let 𝑔 and 𝜓 belong to ℓ^𝑚 and 𝜗^𝑒; respectively. Then there are 𝑓∈𝜇́_𝑎,𝑏 and 𝜑∈𝜗 such that 𝑔=𝜇^𝑚_𝑓 and 𝜓=𝜇_𝜑, respectively. Therefore, by the convolution theorem and (19) we get that 𝑦𝑦𝜓𝑦=𝜇𝑔⋏𝜓=𝑔_𝑚𝑓𝜇_𝜑𝑦=𝜇^𝑚𝑓⋎𝜑𝑦. (20) Since 𝑓⋎𝜑∈𝜇́_𝑎,𝑏, it follows 𝑔⋏𝜓∈ℓ^𝑚.
Hence we have proved the theorem.

Theorem 4. Let 𝜓₁,𝜓₂∈𝜗^𝑒; then 𝜓₁⋏𝜓₂∈𝜗^𝑒.

Proof. By definition of 𝜗^𝑒 we can find 𝜑₁,𝜑₂∈𝜗 such that 𝜓₁=𝜇_𝜑1 and 𝜓₂=𝜇_𝜑2.
Therefore, by [1], 𝜓₁⋏𝜓₂=𝜓₁𝑦𝜓₂𝑦=𝜇_𝜑1𝑦𝜇_𝜑2𝑦=𝜇𝜑₁⋎𝜑₂𝑦. (21) But since 𝜑₁⋎𝜑₂∈𝜗, we get 𝜓₁⋏𝜓₂∈𝜗^𝑒. Thus we have the theorem.

Theorem 5. Let 𝑔₁,𝑔₂∈ℓ^𝑚 and 𝜓∈𝜗^𝑒; then (𝑔₁+𝑔₂)⋏𝜓=𝑔₁⋏𝜓+𝑔₂⋏𝜓 and (𝛼𝑔)⋏𝜓=𝑔⋏(𝛼𝜓)=𝛼(𝑔₁⋏𝜓).

Proof. Is straightforward.

Theorem 6. Let 𝑔_𝑛→𝑔 in ℓ^𝑚,𝜓∈𝜗^𝑒; then 𝑔_𝑛⋏𝜓→𝑔⋏𝜓 as 𝑛→∞ in ℓ^𝑚.

Proof. Can easily be checked.

Theorem 7. Let 𝑔∈ℓ^𝑚 and (𝜓_𝑛)∈Δ^𝑒_ℝ+; then 𝑔⋏𝜓_𝑛→𝑔 as 𝑛→∞.

Proof. By (19) we have 𝑔⋏𝜓_𝑛𝑦=𝜇^𝑚_𝑓𝜇_𝜑𝑛𝑦, (22) where 𝑓∈𝜇́_𝑎,𝑏 and (𝜑_𝑛)∈Δ are such that 𝜇^𝑚_𝑓=𝑔 and 𝜇_𝜑𝑛=𝜓_𝑛, for all 𝑛∈ℕ.
Since 𝜇_𝜑𝑛(𝑦)→1 as 𝑛→∞ on compact subsets of ℝ₊, (22) implies that (𝑔⋏𝜓_𝑛)(𝑦)→𝜇^𝑚_𝑓(𝑦)=𝑔(𝑦), for all 𝑦, as 𝑛→∞. Hence we obtain the theorem.

Theorem 8. Let (𝜓_𝑛),(𝜃_𝑛)∈Δ^𝑒_ℝ+; then 𝜓_𝑛⋏𝜃_𝑛∈Δ^𝑒_ℝ+.

Let 𝜓_𝑛=𝜇_𝛼𝑛,𝜃_𝑛=𝜇_𝜎𝑛; then taking into account the fact that (𝜓_𝑛⋏𝜃_𝑛)(𝑦)=(𝜇_𝛼𝑛⋏𝜇_𝜎𝑛)(𝑦)=𝜇(𝛼_𝑛⋎𝜎_𝑛)(𝑦), since 𝛼_𝑛⋎𝜎_𝑛∈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∈ℓ^𝑚 can be identified by a member of 𝛽₂ given as 𝑔⟶𝑔⋏𝜓_𝑛𝜓_𝑛as𝑛⟶∞, (23) where 𝜓_𝑛∈Δ^𝑒_ℝ+.

Definition 9. The extended modified Mellin transform 𝜇^↷𝑚∶𝛽₁→𝛽₂ is defined by 𝜇^↷𝑚𝛽=𝜇^𝑚_𝑓𝑛𝜇_𝜑𝑛𝑓,∀𝛽=^𝑛𝜑_𝑛∈𝛽₁. (24)

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 𝜇^↷𝑚 is consistent with the distributional 𝜇^𝑚_𝑓(𝜇^𝑚_𝑓∶𝜇́_𝑎,𝑏→𝜇́_𝑎,𝑏).

Proof. For every 𝑓∈𝜇́_𝑎,𝑏, let 𝛽 be its representative in 𝛽₁; then 𝛽=[(𝑓⋎𝜑_𝑛)/𝜑_𝑛], where 𝜑_𝑛∈Δ, for all 𝑛. Then it is clear that 𝜑_𝑛 is independent of the representative, for all 𝑛.
Therefore 𝜇^↷𝑚𝛽=𝜇^↷𝑚𝑓⋎𝜑_𝑛𝜑_𝑛𝜇i.e.=^𝑚𝑓𝜇_𝜑𝑛𝜇_𝜑𝑛𝜇i.e.=^𝑚𝑓𝜇_𝜑𝑛𝜇_𝜑𝑛 (25) which is the representative of 𝜇^𝑚_𝑓 in 𝜇́_𝑎,𝑏.
Hence we have the proof.

Theorem 12 (necessity theorem). Let [𝑔_𝑛/𝜓_𝑛]∈𝛽₂; then the necessary and sufficient condition that [𝑔_𝑛/𝜓_𝑛]is to be in the range of 𝜇^↷𝑚 is that 𝑔_𝑛 belongs to range of 𝜇^𝑚_𝑓 for every 𝑛∈ℕ.

Proof. Let [𝑔_𝑛/𝜓_𝑛] be in the range of 𝜇^↷𝑚; then of course 𝑔_𝑛 belongs to the range of (𝜇^𝑚_𝑓), for all 𝑛∈ℕ.
To establish the converse, let 𝑔_𝑛 be in the range of 𝜇^𝑚_𝑓, for all 𝑛∈ℕ. Then there is 𝑓_𝑛∈𝜇́_𝑎,𝑏 such that 𝜇^𝑚_𝑓𝑛=𝑔_𝑛,𝑛∈ℕ.
Since [𝑔_𝑛/𝜓_𝑛]∈𝛽₂, 𝑔_𝑛⋎𝜓_𝑚=𝑔_𝑚⋎𝜓_𝑛, (26) for all 𝑚,𝑛∈ℕ. Therefore, 𝜇^𝑚𝑓_𝑛⋏𝜑_𝑛=𝜇^𝑚𝑓_𝑚⋏𝜑_𝑛,∀𝑚,𝑛∈ℕ, (27) where 𝑓_𝑛∈𝜇́_𝑎,𝑏 and 𝜑_𝑛∈Δ.
The fact that 𝜇^𝑚_𝑓 is injective, 𝜇^𝑚_𝑓∶𝜇́_𝑎,𝑏→𝜇́_𝑎,𝑏, implies that 𝑓_𝑛⋏𝜑_𝑚=𝑓_𝑚⋏𝜑_𝑛,𝑚,𝑛∈ℕ.
Thus 𝑓_𝑛/𝜑_𝑛 is quotient of sequences in 𝛽₁. Hence, [𝑓_𝑛/𝜑_𝑛]∈𝛽₁ and 𝜇^↷𝑚𝑓_𝑛𝜑_𝑛=𝑔_𝑛𝜓_𝑛. (28) Hence the theorem is proved.

Theorem 13 (generalized convolution theorem). Let 𝛽=[𝑓_𝑛/𝜑_𝑛]∈𝛽₁ and 𝛾=[𝜅_𝑛/𝜙_𝑛]∈𝛽₁; then 𝜇^↷𝑚𝛽⋎𝛾=𝜇^↷𝑚𝑓_𝑛𝜑_𝑛⋏𝜇^↷𝑚𝜅_𝑛𝜙_𝑛. (29)

Proof. Assume that the requirements of the theorem satisfy for some 𝛽 and 𝛾∈𝛽₁; then using Definition 9 and the operation ⋏ yields 𝜇^↷𝑚𝛽⋎𝛾=𝜇^↷𝑚𝑓_𝑛⋎𝜅_𝑛𝜑_𝑛⋎𝜙_𝑛=𝜇^𝑚𝑓_𝑛⋎𝜅_𝑛𝜇𝜑_𝑛⋎𝜙_𝑛=𝜇^𝑚_𝑓𝑛⋏𝜇^𝑚_𝜅𝑛𝜇_𝜑𝑛⋏𝜇_𝜙𝑛=𝜇^𝑚_𝑓𝑛𝜇_𝜑𝑛⋏𝜇^𝑚_𝜅𝑛𝜇_𝜙𝑛. (30) Therefore 𝜇^↷𝑚𝛽⋎𝛾=𝜇^↷𝑚𝑓_𝑛𝜑_𝑛⋏𝜇^↷𝑚𝜅_𝑛𝜙_𝑛. (31) This completes the proof.

Theorem 14. The extended modified Mellin transform 𝜇^↷𝑚∶𝛽₁→𝛽₂ is bijective.

Proof. Assume 𝜇^↷𝑚[𝑓_𝑛/𝜑_𝑛]=𝜇^↷𝑚[𝜅_𝑛/𝜙_𝑛]; then it follows from the concept of quotients of sequences that 𝜇^𝑚_𝑓𝑛⋏𝜇_𝜙𝑚=𝜇^𝑚_𝜅𝑚⋏𝜇_𝜑𝑛. Therefore, 𝜇^𝑚(𝑓_𝑛⋎𝜙_𝑚)=𝜇^𝑚(𝜅_𝑚⋎𝜑_𝑛). The property that 𝜇^𝑚 is one to one implies 𝑓_𝑛⋎𝜙_𝑚=𝜅_𝑚⋎𝜑_𝑛. Therefore, 𝑓_𝑛𝜑_𝑛=𝜅_𝑛𝜙_𝑛. (32) Next to establish that 𝜇^↷𝑚 is onto, let [𝜇^𝑚_𝑓𝑛/𝜇_𝜑𝑛](∈𝛽₂) be arbitrary; then 𝜇^𝑚_𝑓𝑛⋏𝜇_𝜑𝑚=𝜇^𝑚_𝑓𝑚⋏𝜇_𝜑𝑛 for every 𝑚,𝑛∈ℕ. Hence 𝑓_𝑛,𝑓_𝑚∈𝜇́_𝑎,𝑏 are such that 𝜇^𝑚(𝑓_𝑛⋎𝜑_𝑚)=𝜇^𝑚(𝑓_𝑚⋎𝜑_𝑛), for all 𝑚,𝑛∈ℕ.
Hence, the Boehmian [𝑓_𝑛/𝜑_𝑛] belongs to 𝛽₁ and satisfies 𝜇^↷𝑚𝑓_𝑛𝜑_𝑛=𝜇^𝑚_𝑓𝑛𝜇_𝜑𝑛. (33) This completes the proof of the theorem.

Now we introduce (𝜇^↷𝑚)⁻¹ as the inverse transform of 𝜇^↷𝑚, where 𝜇^↷𝑚−1𝜇^𝑚_𝑓𝑛𝜇_𝜑𝑛=𝜇^𝑚_𝑓𝑛⁻¹𝜇^𝑚_𝑓𝑛𝜇_𝜑𝑛⁻¹𝜇_𝜑𝑛, (34) for every [𝑓_𝑛/𝜑_𝑛]∈𝛽₁.

Theorem 15. Let [𝜇^𝑚_𝑓𝑛/𝜇_𝜑𝑛]∈𝛽₂ and 𝜙∈𝜗^𝑒, then 𝜇^↷𝑚−1𝜇^𝑚_𝑓𝑛𝜇_𝜑𝑛=𝑓⋏𝜙^𝑛𝜑_𝑛⋎𝜙,_↷𝜇^𝑚𝑓_𝑛𝜑_𝑛=𝜇⋎𝜙^𝑚𝑓𝑛𝜇_𝜑𝑛⋏𝜙. (35)(36)

Proof. We prove (35) and omit the proof of (36) due to its similarity. Given [𝜇^𝑚_𝑓𝑛/𝜇_𝜑𝑛]∈𝛽₂ and 𝜓∈𝜗 such that 𝜙=𝜇^𝑚_𝜓 then employing (34) yields 𝜇^↷𝑚−1𝜇^𝑚_𝑓𝑛𝜇_𝜑𝑛⋏𝜙=𝜇^↷𝑚−1𝜇^𝑚_𝑓𝑛⋏𝜙𝜇_𝜑𝑛=𝜇^𝑚_𝑓𝑛⁻¹𝜇^𝑚_𝑓𝑛⋏𝜇^𝑚_𝜓𝜇_𝜑𝑛⁻¹𝜇_𝜑𝑛. (37) Using (19) gives 𝜇^↷𝑚−1𝜇^𝑚_𝑓𝑛𝜇_𝜑𝑛=𝜇⋏𝜙^{𝑚𝑓𝑛−1}𝜇^𝑚_𝑓𝑛𝜇^𝑚_𝜓𝜇_𝜑𝑛⁻¹𝜇_𝜑𝑛. (38) Hence the convolution theorem gives 𝜇^↷𝑚−1𝜇^𝑚_𝑓𝑛𝜇_𝜑𝑛=𝜇⋏𝜙^{𝑚𝑓𝑛−1}𝜇^𝑚_𝑓𝑛𝑓_𝑛⋎𝜙𝜇_𝜑𝑛⁻¹𝜇_𝜑𝑛. (39) Thus 𝜇^↷𝑚−1𝜇^𝑚_𝑓𝑛𝜇_𝜑𝑛=𝑓⋏𝜙^𝑛𝜑_𝑛⋎𝜙. (40) Proof of the second part is similar.
This completes the proof of the theorem.

Theorem 16. 𝜇^↷𝑚∶𝛽₁→𝛽₂ and (𝜇^↷𝑚)⁻¹∶𝛽₂→𝛽₁ are continuous with respect to 𝛿-convergence.

Proof. Let 𝛽→−_𝑛^𝛿𝛽 in 𝛽₁ as 𝑛→∞; then we establish that 𝜇^↷𝑚𝛽_𝑛→𝜇^↷𝑚𝛽 as 𝑛→∞. Let 𝑓_𝑛,𝑘 and 𝑓_𝑘 be in 𝜇́_𝑎,𝑏 such that 𝛽_𝑛=𝑓_𝑛,𝑘𝜑_𝑘𝑓,𝛽=^𝑘𝜑_𝑘 (41) and 𝑓_𝑛,𝑘→𝑓_𝑘 as 𝑛→∞ for every 𝑘∈ℕ.
The continuity of 𝜇^𝑚_{𝑓𝑛,𝑘} implies 𝜇^𝑚_{𝑓𝑛,𝑘}→𝜇^𝑚_𝑓𝑘 as 𝑛→∞. Thus, 𝜇^𝑚_{𝑓𝑛,𝑘}𝜇_𝜑𝑘⟶𝜇^𝑚_𝑓𝑘𝜇_𝜑𝑘 (42) as 𝑛→∞ in 𝛽₂. This proves continuity of 𝜇^↷𝑚.
Next, let 𝑔→−_𝑛^𝛿𝑔∈𝛽₂ as 𝑛→∞; then we have 𝑔_𝑛=[𝜇^𝑚_{𝑓𝑛,𝑘}/𝜇_𝜑𝑘] and 𝑔=[𝜇^𝑚_𝑓𝑘/𝜇_𝜑𝑘] for some 𝜇^𝑚_{𝑓𝑛,𝑘},𝜇^𝑚_𝑓𝑘∈ℓ^𝑚, where 𝜇^𝑚_{𝑓𝑛,𝑘}→𝜇^𝑚_𝑓𝑘 as 𝑛→∞. Hence 𝜇^𝑚_𝑓𝑛,k⁻¹𝜇^𝑚_{𝑓𝑛,𝑘}⟶𝜇^𝑚_𝑓𝑘⁻¹𝜇^𝑚_𝑓𝑘 (43) as 𝑛→∞ in 𝛽₁. That is, 𝜇^𝑚_{𝑓𝑛,𝑘}⁻¹𝜇^𝑚_{𝑓𝑛,𝑘}𝜑_𝑘=𝜇^𝑚_{𝑓𝑛,𝑘}⁻¹𝜇^𝑚_{𝑓𝑛,𝑘}𝜇_𝜑𝑘⁻¹𝜇_𝜑𝑘⟶𝜇^𝑚_𝑓𝑘⁻¹𝜇^𝑚_𝑓𝑘𝜑_𝑘 (44) as 𝑛→∞.
Hence 𝜇^𝑚_{𝑓𝑛,𝑘}⁻¹𝜇^𝑚_{𝑓𝑛,𝑘}𝜑_𝑘⟶𝜇^𝑚_𝑓𝑘⁻¹𝜇^𝑚_𝑓𝑘𝜇_𝜑𝑘⁻¹𝜇_𝜑𝑘 (45) as 𝑛→∞.
That is, 𝜇^↷𝑚−1𝑔_𝑛⟶𝜇^↷𝑚−1𝑔 (46) as 𝑛→∞. This completes the proof.

Theorem 17. 𝜇^↷𝑚∶𝛽₁→𝛽₂ and (𝜇^↷𝑚)⁻¹∶𝛽₂→𝛽₁ are continuous with respect to Δ-convergence.

Proof. Let 𝛽−→_𝑛^Δ𝛽 in 𝛽₁ as 𝑛→∞. Then, we find 𝑓_𝑛∈𝜇́_𝑎,𝑏 and (𝜑_𝑘)∈Δ such that (𝛽_𝑛−𝛽)⋏𝜑_𝑘=[(𝑓_𝑛⋏𝜑_𝑘)/𝜑_𝑘] and 𝑓_𝑛→0 as 𝑛→∞. Therefore 𝜇^↷𝑚𝛽_𝑛−𝛽⋎𝜑_𝑘=𝜇^𝑚𝑓_𝑛⋎𝜑_𝑘𝜇_𝜑𝑘. (47) Hence, 𝜇^↷𝑚((𝛽_𝑛−𝛽)⋎𝜑_𝑘)=[(𝜇^𝑚_𝑓𝑛⋏𝜇_𝜑𝑘)/𝜇_𝜑𝑘]=𝜇^𝑚_𝑓𝑛→0 as 𝑛→∞ in ℓ^𝑚.
Therefore 𝜇^↷𝑚𝛽_𝑛−𝛽⋎𝜑_𝑛=𝜇^↷𝑚𝛽_𝑛−𝜇^↷𝑚𝛽⋏𝜇_𝜑𝑘⟶as𝑛⟶∞. (48) Hence,𝜇^↷𝑚𝛽_𝑛−→^Δ𝜇^↷𝑚𝛽 as 𝑛→∞.
Proof of the second part is analogous. Detailed proof is as follows.
Finally, let 𝑔−→_𝑛^Δ𝑔 in 𝛽₂ as 𝑛→∞; then we can find 𝜇^𝑚_𝑓𝑘∈ℓ^𝑚 such that (𝑔_𝑛−𝑔)⋏𝜇_𝜑𝑘=[(𝜇^𝑚_𝑓𝑘⋏𝜇_𝜑𝑘)/𝜇_𝜑𝑘] and 𝜇^𝑚_𝑓𝑘→0 as 𝑛→∞ for some (𝜇_𝜑𝑘)∈Δ^𝑒_ℝ+.
Next, we have 𝜇^↷𝑚−1𝑔_𝑛−𝑔⋏𝜇_𝜑𝑘=𝜇^𝑚_𝑓𝑘⁻¹𝜇^𝑚_𝑓𝑘⋏𝜇_𝜑𝑘𝜇_𝜑𝑘⁻¹𝜇_𝜑𝑘. (49) Thus, by (34) we get 𝜇^↷𝑚−1𝑔_𝑛−𝑔⋏𝜇_𝜑𝑘=𝑓_𝑛⋎𝜑_𝑘𝜑_𝑘=𝑓_𝑛⟶0 (50) as 𝑛→∞ in 𝜇́_𝑎,𝑏.
Therefore 𝜇^↷𝑚−1𝑔_𝑛−𝑔⋏𝜇_𝜑𝑘=𝜇^↷𝑚−1𝑔_𝑛−𝜇^↷𝑚−1𝑔⋎𝜑_𝑘⟶0 (51) as 𝑛→∞.
Thus, we have 𝜇^↷𝑚−1𝑔−→_𝑛^Δ𝜇^↷𝑚−1𝑔 (52) 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.