On the Exponential Radon Transform and Its Extension to Certain Functions Spaces

Abstract

We investigate the exponential Radon transform on a certain function space of generalized functions. We establish certain space of generalized functions for the cited transform. The transform that is obtained is well defined. More properties of consistency, convolution, analyticity, continuity, and sufficient theorems have been established.

1. Introduction

The Radon transform of a sufficiently nice function 𝑓 defined on ℝ^𝑛 is given by ℛ_𝜗𝑓𝜂≡ℛ𝑓𝜗,𝜂≡_𝜗⟂𝑓𝜂𝜗+𝑢𝐝𝑢, (1) where (𝜗,𝜂)∈ℝ^𝑛=∑_𝑛−1𝑥𝑝,∑_𝑛−1 is the unit sphere in ℝ^𝑛, and 𝐝𝑢 is the Euclidean measure on the subspace 𝜗^⟂ orthogonal to 𝜗.

Applications of the Radon transform occur in a number of areas, such as seismic signal processing, remote sensing, and system identification from output data [1, 2]. The Radon transform is extended to various spaces of distributions, rapidly decreasing and integrable Boehmians [3, 4]. More about the Radon transform is given in [5–9].

The discrete Radon transform is defined by [10, 11]. The attenuated Radon transform is defined in Mikusiński et al. [12, 13]. For a uniform attenuation coefficient 𝜇∈ℂ, the exponential Radon transform of a compactly supported real valued function 𝑓, defined on ℝ², is given by Kurusa and Hertle [7, 8]: 𝐓^𝑒_𝜇𝑓(𝜗,𝑡)=_ℝ2𝑓(𝐱)𝛿(𝐱⋅𝜗−𝑡)𝑒^{𝜇𝐱⋅𝜗⟂}𝐝𝐱,𝑡∈ℝ, (2) where 𝜗=(cos𝜑,sin𝜑)^𝜏 is a unit vector on 𝒮¹, 𝜑∈[0,2𝜋),𝜗^⟂=(−sin𝜑,cos𝜑).

The exponential Radon transform constitutes a mathematical model for imaging modalities such as X-ray tomography for 𝜇=0, single photon emission tomography for 𝜇∈ℝ, and optical polarization tomography of trass tensor field [14]. However, if in addition 𝜇 is unknown, then one first must find 𝜇 and then find 𝑓. This is the identification problem.

The exponential Radon transform, as a generalization of the Radon transform, is defined as a mapping of function spaces and is also represented in terms of Fourier transforms of its domain and range, and this leads to a characterization of the range of the transform. For more information about the exponential Radon transform, we refer to [15, 16].

2. General Construction of Boehmians

The minimal structure necessary for the construction of Boehmians consists of the following elements:(i)a set 𝐚 and a commutative semigroup (𝐠,∗);(ii) an operation ⊙∶𝐚×𝐠→𝐚 such that for each 𝑥∈𝐚 and 𝜐₁,𝜐₂,∈𝐠, 𝑥⊙𝜐₁∗𝜐₂=𝑥⊙𝜐₁⊙𝜐₂; (3)(iii) a collection Δ⊂𝐠^ℕ satisfying the following:(1)if 𝑥,𝑦∈𝐚, (𝜐_𝑛)∈Δ, 𝑥⊙𝜐_𝑛=𝑦⊙𝜐_𝑛 for all 𝑛, then 𝑥=𝑦;(2)if (𝜐_𝑛),(𝜎_𝑛)∈Δ, then (𝜐_𝑛∗𝜎_𝑛)∈Δ,Δ being the set of all delta sequences.

Consider 𝒜=𝑥_𝑛,𝜐_𝑛∶𝑥_𝑛∈𝐚,𝜐_𝑛∈Δ,𝑥_𝑛⊙𝜐_𝑚=𝑥_𝑚⊙𝜐_𝑛,∀𝑚,𝑛∈ℕ. (4) If (𝑥_𝑛,𝜐_𝑛),(𝑦_𝑛,𝜎_𝑛)∈𝒜, 𝑥_𝑛⊙𝜎_𝑚=𝑦_𝑚⊙𝜐_𝑛, ∀𝑚,𝑛∈ℕ, then we say (𝑥_𝑛,𝜐_𝑛)∼(𝑦_𝑛,𝜎_𝑛). The relation ∼ is an equivalence relation in 𝒜. The space of equivalence classes in 𝒜 is denoted by 𝜹(𝐚,(𝐠,∗),⊙,Δ). Elements of 𝜹(𝐚,(𝐠,∗),⊙,Δ) are called Boehmians.

Between 𝐚 and 𝜹(𝐚,(𝐠,∗),⊙,Δ) there is a canonical embedding expressed as 𝑥⟶𝑥⊙𝑠_𝑛𝑠_𝑛as𝑛⟶∞. (5)

The operation ⊙ can be extended to 𝜹(𝐚,(𝐠,∗),⊙,Δ)×𝐚 by 𝑥_𝑛𝜐_𝑛𝑥⊙𝑡=^𝑛⊙𝑡𝜐_𝑛. (6) In 𝜹(𝐚,(𝐠,∗),Δ), two types of convergence exist:(1)a sequence (ℎ_𝑛) in 𝜹(𝐚,(𝐠,∗),⊙,Δ) is said to be 𝛿-convergent to ℎ in 𝜹(𝐚,(𝐠,∗),⊙,Δ), denoted by ℎ→−_𝑛^𝛿ℎ as 𝑛→∞, if there exists a delta sequence (𝜐_𝑛) such that (ℎ_𝑛⊙𝜐_𝑛),(ℎ⊙𝜐_𝑛)∈𝐚, ∀𝑘,𝑛∈ℕ, and (ℎ_𝑛⊙𝜐_𝑘)→(ℎ⊙𝜐_𝑘) as 𝑛→∞, in 𝐚, for every 𝑘∈ℕ;(2)a sequence (ℎ_𝑛) in 𝜹(𝐚,(𝐠,∗),⊙,Δ) is said to be Δ-convergent to ℎ in 𝜹(𝐚,(𝐠,∗),⊙,Δ), denoted by ℎ−→_𝑛^Δℎ as 𝑛→∞, if there exists a (𝜐_𝑛)∈Δ such that (ℎ_𝑛−ℎ)⊙𝜐_𝑛∈𝐚, ∀𝑛∈ℕ, and (ℎ_𝑛−ℎ)⊙𝜐_𝑛→0 as 𝑛→∞ in 𝐚.

The following theorem is equivalent to the statement of 𝛿-convergence.

Theorem 1. ℎ→−_𝑛^𝛿ℎ(𝑛→∞) in 𝜹(𝐚,(𝐠,∗),⊙,Δ) if and only if there is 𝑓_𝑛,𝑘,𝑓_𝑘∈𝐚 and 𝜐_𝑘∈Δ such that ℎ_𝑛=[𝑓_𝑛,𝑘/𝜐_𝑘], ℎ=[𝑓_𝑘/𝜐_𝑘] and for each 𝑘∈ℕ, 𝑓_𝑛,𝑘→𝑓_𝑘 as 𝑛→∞ in 𝐚.

For further discussion see [3, 17–21].

3. Necessary and Sufficient Conditions

Denote by 𝐥¹(𝒮¹×ℝ) the space of Lebesgue complex-valued measurable functions of bounded support defined on 𝒮¹×ℝ and satisfying 𝜓((𝜗,𝑡))=_𝒮1ℝ𝜓((𝜗,𝑡))²𝑒^{𝜇𝐲⋅𝜗⟂}𝐝𝜗𝐝𝑡<∞ (7) then 𝐓^𝑒_𝜇𝜓∈𝐥¹(𝒮¹×ℝ),𝜇∈ℝ,𝐲∈ℝ² being arbitrary but fixed.

By 𝜿(ℝ²) denote the space of test functions of bounded support defined on ℝ².

Let Δ be the set of sequences (𝜇_𝑛(𝐱))∈𝜿(ℝ²) such that [3, (2.6)–(2.8)] 𝜇𝑛(_ℝ2𝐱)𝐝𝐱=1,_ℝ2(𝐱𝜇𝜇𝑛)𝐝𝐱<𝑀,0<𝑀∈ℝ,supp_𝑛(𝐱)⟶0as𝑛⟶∞. (8)

The convolution product between two functions is defined by the integral equation 𝑓∗𝑔(𝐱)=_ℝ2𝑓(𝐱−𝐲)𝑔(𝐲)𝐝𝐲, (9) where 𝐱∈ℝ².

Now we construct the space 𝜹(𝐥¹,(𝜿,∗),×,Δ) of Boehmians.

We have the following definition.

Definition 2. Let 𝜓∈𝐥¹(𝒮¹×ℝ) and 𝜉∈𝜿(ℝ²); then we define the mapping × as 𝜓×𝜉(𝜗,𝑡)=_ℝ2𝜓(𝜗,𝑡−𝐲⋅𝜗)𝜉(𝐲)𝑒^{𝜇𝐲⋅𝜗⟂}()𝐝𝐲𝜗,𝑡∈𝒮¹×ℝ. (10)

Theorem 3. Let 𝜓∈𝐥¹(𝒮¹×ℝ) and 𝜉∈𝜿(ℝ²); then 𝜓×𝜉∈𝐥¹(𝒮¹×ℝ).

Proof. Let 𝜓∈𝐥¹(𝒮¹×ℝ) and 𝜉∈𝜿(ℝ²). By using (10), Fubini’s theorem, and Jensen’s inequality we get 𝜓×𝜉(𝜗,𝑡)²=_𝒮1ℝ𝜓×𝜉(𝜗,𝑡)²=𝐝𝜗𝐝𝑡_𝒮1ℝℝ2𝜓(𝜗,𝑡−𝐲⋅𝜃)𝜉(𝐲)𝑒^{𝜇𝐲⋅𝜗⟂}𝐝𝐲²𝐝𝜗𝐝𝑡=_ℝ2𝜉(𝐲)_𝒮1ℝ𝜓(𝜗,𝑡−𝐲⋅𝜗)²𝑒^{𝜇𝐲⋅𝜗⟂}𝜓𝐝𝜗𝐝𝑡𝐝𝐲<𝑀², (11) where 𝑀 is a positive constant.
The proof is therefore completed.

Theorem 4. Let 𝜓_𝑛→𝜓 in 𝐥¹(𝒮¹×ℝ) and 𝜉∈𝜿(ℝ²); then 𝜓_𝑛×𝜉→𝜓×𝜉 as 𝑛→∞.

Proof of this theorem follows from Theorem 3.

Theorem 5. Let 𝜓∈𝐥¹(𝒮¹×ℝ) and 𝜉₁,𝜉₂∈𝜿(ℝ²); then one has 𝜓×𝜉₁∗𝜉₂=𝜓×𝜉₁×𝜉₂. (12)

Proof. Let (𝜗,𝑡)∈𝒮¹×ℝ. Using (10) and (9) we write 𝜓×𝜉₁∗𝜉₂()=𝜗,𝑡_ℝ2𝜓(𝜗,𝑡−𝐲⋅𝜗)𝜉₁∗𝜉₂(𝐲)𝑒^{𝜇𝐲⋅𝜗⟂}=𝐝𝐲_ℝ2𝜓(𝜗,𝑡−𝐲⋅𝜗)_ℝ2𝜉₁(𝐲−𝐱)𝜉₂(𝐱)𝐝𝐱𝑒^{𝜇𝐲⋅𝜗⟂}=𝐝𝐲_ℝ2ℝ2𝜓(𝜗,𝑡−𝐲⋅𝜗)𝜉₁(𝐲−𝐱)𝑒^{𝜇𝐲⋅𝜗⟂}𝐝𝐲𝜉₂(𝐱)𝐝𝐱. (13)
The substitution 𝐲−𝐱=𝐳, 𝐲,𝐱,𝐳∈ℝ², implies 𝜓×𝜉₁∗𝜉₂()=𝜗,𝑡_ℝ2ℝ2𝜓(𝜗,𝑡−(𝐳+𝐱)⋅𝜗)𝜉₁(𝐳)𝑒^{𝜇(𝐳+𝐱)𝜗⟂}𝐝𝐳×𝜉₂(𝐱)=𝐝𝐱_ℝ2ℝ2𝜓(𝜗,(𝑡−𝐱⋅𝜗)−𝐳⋅𝜗)𝜉₁(𝐳)𝑒^{𝜇𝐱⋅𝜗⟂}𝐝𝐳×𝜉₂(𝐱)𝑒^{𝜇𝐱⋅𝜗⟂}=𝐝𝐱_ℝ2𝜓×𝜉₁(𝜗,𝑡−𝐱⋅𝜗)𝜉₂(𝐱)𝑒^{𝜇𝐱⋅𝜗⟂}=𝐝𝐱𝜓×𝜉₁×𝜉₂(𝜗,𝑡). (14)
This completes the proof of the theorem.

Theorem 6. Let 𝜓₁,𝜓₂∈𝐥¹(𝒮¹×ℝ) and 𝜉∈𝜿(ℝ²); then 𝜓₁+𝜓₂×𝜉=𝜓₁×𝜉+𝜓₂×𝜉₂. (15)

Theorem 7. Let 𝛼∈ℂ, 𝜓∈𝐥¹(𝒮¹×ℝ) and 𝜉∈𝜿(ℝ²); then 𝛼𝜓×𝜉=𝛼𝜓×𝜉. (16)

Proof of Theorems 6 and 7 follows from simple integration. Detailed proof is thus avoided.

Theorem 8. Let 𝜓∈𝐥¹(𝒮¹×ℝ) and (𝜇_𝑛)∈Δ; then 𝜓×𝜇_𝑛→𝜓 as 𝑛→∞.

Proof. Let 𝜓∈𝐥¹(𝒮¹×ℝ). Since 𝜿(ℝ×ℝ) is dense in 𝐥¹(𝒮¹×ℝ), we can choose 𝛼∈𝜿(ℝ×ℝ) such that 𝜓−𝛼<𝜖,𝜖>0. (17)
From the analysis applied for proving Theorem 3 and by applying (17) we get 𝜓−𝛼×𝜇_𝑛≤𝑀𝜓−𝛼<𝑀𝜖. (18) Also, for each fixed (𝜗,𝑡)∈𝒮¹×ℝ define 𝑔(𝐲)=𝛼(𝜗,𝑡−𝐲⋅𝜗)𝑒^{𝜇𝐲⋅𝜗⟂},𝐲∈ℝ²; (19) then 𝑔(𝐲)∈𝜿(ℝ²) and hence 𝑔(𝐲) uniformly continuous on ℝ². Thus, there is 𝛿>0 such that |𝑔(𝐲)−𝑔(𝐱)|<𝜖 whenever |𝐲−𝐱|≤𝛿.
Moreover supp𝛼(𝜗,𝑡)⊆[𝑎,𝑏]×𝕜,𝕜⊆ℝ−{0} implies 𝛼(𝜗,𝑡)=0, ∀(𝜗,𝑡)∉[𝑎−𝛿,𝑏+𝛿]×𝕜.
Hence, by (8) and the fact that |(𝑔(𝑦)−𝑔(0))|<𝜖, by Jensen’s inequality, we have 𝛼×𝜇_𝑛−𝛼(𝜗,𝑡)²=_𝒮1ℝ𝛼×𝜇_𝑛−𝛼(𝜗,𝑡)²=𝐝𝜗𝐝𝑡_𝒮1ℝℝ2𝛼(𝜗,𝑡−𝐲⋅𝜗)𝜇_𝑛(𝐲)𝑒^{𝜇𝐲⋅𝜗⟂}−𝛼(𝜗,𝑡)𝜇_𝑛(𝐲)𝐝𝐲²=𝐝𝜗𝐝𝑡_𝒮1ℝℝ2𝑔(𝐲)−𝑔(0)𝜇_𝑛(𝐲)𝐝𝐲²≤𝐝𝜗𝐝𝑡_{𝕜𝑏+𝛿𝑎−𝛿ℝ2}𝑔(𝐲)−𝑔(0)²𝜇_𝑛(𝐲)𝐝𝐲𝐝𝜗𝐝𝑡≤𝜖²_𝕜^𝑏+𝛿_{𝑎−𝛿ℝ2}𝜇_𝑛(𝐲)𝐝𝐲𝐝𝜗𝐝𝑡=𝜖²𝑀𝑚(𝕜)(𝑏−𝑎+2𝛿), (20) where 𝑚(𝕜) is the Lebesgue measure of 𝕜.
Hence, using (17), (18), and (20) we, for large values of 𝑛, get 𝜓×𝜇_𝑛−𝜓≤𝜓−𝛼×𝜇_𝑛+𝛼×𝜇_𝑛−𝛼+𝛼−𝜓<𝑀𝜖+𝜖²(𝕜)=𝑀𝑚)(𝑏−𝑎+2𝛿+𝜖𝑀+𝜖²𝑀𝑚(𝕜)(𝑏−𝑎+2𝛿)+1𝜖. (21) Hence 𝜓×𝜇_𝑛→𝜓 as 𝑛→∞.
The Boehmian space 𝜹(𝐥¹,(𝜿,∗),×,Δ) is constructed.
The sum and multiplication by a scalar of two Boehmians are naturally defined in the respective ways: 𝑓_𝑛𝜙_𝑛+𝑔_𝑛𝜏_𝑛=𝑓_𝑛×𝜏_𝑛+𝑔_𝑛×𝜙_𝑛𝜙_𝑛∗𝜏_𝑛,𝜂𝑓_𝑛𝜙_𝑛=𝜂𝑓_𝑛𝜙_𝑛, (22)𝜂 being complex number.
The operations × and the derivative are defined by 𝑓_𝑛𝜙_𝑛×𝑔_𝑛𝜏_𝑛=𝑓_𝑛×𝑔_𝑛𝜖_𝑛∗𝜏_𝑛,𝒟^𝛼𝑓_𝑛𝜙_𝑛=𝒟^𝛼𝑓_𝑛𝜙_𝑛. (23)
Between 𝐥¹(𝒮¹×ℝ) and 𝜹(𝐥¹,(𝜿,∗),×,Δ) the canonical embedding admits 𝑓⟶𝑓×𝜙_𝑛𝜙_𝑛as𝑛⟶∞. (24)
The operation × can be extended to 𝜹(𝐥¹,(𝜿,∗),×,Δ)×𝐥¹(𝒮¹×ℝ) by 𝑓_𝑛𝜙_𝑛𝑓×𝑓=^𝑛×𝑓𝜙_𝑛. (25)
By 𝜹(𝐥¹,(𝜿,∗),∗,Δ) denote the corresponding Boehmian space obtained from 𝐥¹(𝒮¹×ℝ), 𝜿(ℝ²), Δ and the product ∗.

Theorem 9. Let 𝜓∈𝐥¹(𝒮¹×ℝ) and 𝜉∈𝜿(ℝ²); then 𝐓^𝑒_𝜇𝜓∗𝜉(𝜗,𝑡)=𝐓^𝑒_𝜇𝜓×𝜉(𝜗,𝑡),(𝜗,𝑡)∈𝒮¹×ℝ. (26)

Proof. Let (𝜗,𝑡)∈(𝒮¹×ℝ). By employing (2) for (9) we get 𝐓^𝑒_𝜇𝜓∗𝜉(𝜗,𝑡)=_ℝ2𝜓∗𝜉(𝐱)𝛿(𝐱⋅𝜗−𝑡)𝑒^{𝜇𝐱⋅𝜗⟂}=𝐝𝐱_ℝ2ℝ2𝜓(𝐱−𝐲)𝜉(𝐲)𝐝𝐲𝛿(𝐱⋅𝜗−𝑡)𝑒^{𝜇𝐱⋅𝜗⟂}=𝐝𝐱_ℝ2ℝ2𝜓(𝐱−𝐲)𝛿(𝐱⋅𝜗−𝑡)𝑒^{𝜇𝐱⋅𝜗⟂}𝐝𝐱𝜉(𝐲)𝐝𝐲. (27)
The substitution 𝐱−𝐲=𝐳 implies 𝐱=𝐲+𝐳 and 𝐝𝐱=𝐝𝐳.
Thus we get 𝐓^𝑒_𝜇𝜓∗𝜉(𝜗,𝑡)=_ℝ2ℝ2𝜓(𝐳)𝛿(𝐳⋅𝜗−(𝑡−𝐲𝜗))𝑒^{𝜇𝐳⋅𝜗⟂}𝑒^{𝜇𝐲⋅𝜗⟂}(𝐲)=𝐝𝐳𝜉𝐝𝐲_ℝ2ℝ2𝐓^𝑒_𝜇𝜓(𝑡−𝐲⋅𝜗)𝑒^{𝜇𝐲⋅𝜗⟂}𝜉(𝐲)=𝐓𝐝𝐲_𝑒𝜇𝜓×𝜉(𝜗,𝑡). (28)
This completes the proof of the theorem.

4. The Exponential Radon Transform of Boehmians

Definition 10. Let 𝛽_𝑛=[𝜓_𝑛/𝜇_𝑛]∈𝜹(𝐥¹,(𝜿,∗),∗,Δ); then we define its exponential Radon transform as the mapping 𝐓^𝑒_𝜇𝜓_𝑛𝜇_𝑛=𝐓^𝑒_𝜇𝜓_𝑛𝜇_𝑛 (29) in the space 𝜹(𝐥¹,(𝜿,∗),×,Δ).
Definition 10 is well defined by Theorem 9.
To show that (29) is well defined, let [𝜓_𝑛/𝜇_𝑛],[𝜉_𝑛/𝜖_𝑛]∈𝜹(𝐥¹,(𝜿,∗),∗,Δ) and [𝜓_𝑛/𝜇_𝑛]=[𝜉_𝑛/𝜖_𝑛]; then 𝜓_𝑛∗𝜖_𝑚=𝜉_𝑚∗𝜇_𝑛. (30) Employing 𝐓^𝑒_𝜇 for (30) and using Theorem 9 imply that 𝐓^𝑒_𝜇𝜓_𝑛×𝜖_𝑚=𝐓^𝑒_𝜇𝜉_𝑚×𝜇_𝑛. (31) From (31) we see that 𝐓^𝑒_𝜇𝜓_𝑛/𝜇_𝑛∼𝐓^𝑒_𝜇𝜉_𝑛/𝜖_𝑛 in the sense of 𝜹(𝐥¹,(𝜿,∗),×,Δ).
This completes the proof of the theorem.

Theorem 11. Let 𝛽₁,𝛽₂∈𝜹(𝐥¹,(𝜿,∗),∗,Δ); then 𝐓^𝑒_𝜇(𝛽₁∗𝛽₁)=𝐓^𝑒_𝜇𝛽₁×𝛽₂.

Proof. Assume the requirements of the theorem are satisfied for some 𝛽₁,𝛽₂∈𝜹(𝐥¹,(𝜿,∗),∗,Δ); then there are (𝑓_𝑛),(𝜅_𝑛)∈𝐥¹(𝒮¹×ℝ) and (𝜑_𝑛),(𝜙_𝑛)∈Δ such that 𝛽₁=[(𝑓_𝑛)/(𝜑_𝑛)] and 𝛽₂=[(𝜅_𝑛)/(𝜙_𝑛)]. Therefore, we write 𝐓^𝑒_𝜇𝛽₁∗𝛽₂=𝐓^𝑒_𝜇𝑓_𝑛∗𝜅_𝑛𝜑_𝑛∗𝜙_𝑛=𝐓^𝑒_𝜇𝑓_𝑛∗𝜅_𝑛𝜑_𝑛∗𝜙_𝑛=𝐓^𝑒_𝜇𝑓_𝑛×𝜅_𝑛𝜑_𝑛×𝜙_𝑛=𝐓^𝑒_𝜇𝑓_𝑛𝜑_𝑛×𝜅_𝑛𝜙_𝑛. (32)
Thus we get that 𝐓^𝑒_𝜇(𝛽₁∗𝛽₂)=𝐓^𝑒_𝜇(𝛽₁)×𝛽₂.
This completes the proof.

Theorem 12. 𝐓^𝑒_𝜇 defines a linear mapping from 𝜹(𝐥¹,(𝜿,∗),∗,Δ) into 𝜹(𝐥¹,(𝜿,∗),×,Δ).

The proof is straightforward.

Definition 13. Let 𝛽∈𝜹(𝐥¹,(𝜿,∗),×,Δ) be such that 𝛽=[(𝐓^𝑒_𝜇𝑓_𝑛)/(𝜙_𝑛)]. Then we define the inverse transform of 𝒯^𝑒_𝜇 as 𝐓^𝑒_𝜇⁻¹𝐓^𝑒_𝜇𝑓_𝑛𝜙_𝑛=𝑓_𝑛𝜙_𝑛 (33) for each (𝜙_𝑛)∈Δ.

Theorem 14. 𝐓^𝑒_𝜇 defines an isomorphism from 𝜹(𝐥¹,(𝜿,∗),∗,Δ) onto 𝜹(𝐥¹,(𝜿,∗),×,Δ).

Proof. Assume that [(𝐓^𝑒_𝜇𝑓_𝑛)/(𝜙_𝑛)]=[(𝐓^𝑒_𝜇𝑔_𝑛)/(𝜓_𝑛)] in 𝜹(𝐥¹,(𝜿,∗),×,Δ). Using (29) and Theorem 9 we get 𝐓^𝑒_𝜇𝑓_𝑛×𝜓_𝑚=𝐓^𝑒_𝜇𝑔_𝑚×𝜙_𝑛. Once again, Theorem 9 implies 𝐓^𝑒_𝜇(𝑓_𝑛∗𝜓_𝑚)=𝐓^𝑒_𝜇(𝑔_𝑚∗𝜙_𝑛). Hence 𝑓_𝑛∗𝜓_𝑚=𝑔_𝑚∗𝜙_𝑛. Therefore, [(𝑓_𝑛)/(𝜙_𝑛)]=[(𝑔_𝑛)/(𝜓_𝑛)].
Now, let [(𝐓^𝑒_𝜇𝑓_𝑛)/(𝜙_𝑛)]∈𝜹(𝐥¹,(𝜿,∗),×,Δ); then 𝐓^𝑒_𝜇𝑓_𝑛×𝜙_𝑚=𝐓^𝑒_𝜇𝑓_𝑚×𝜙_𝑛, ∀𝑚,𝑛∈ℕ. Theorem 9 leads to 𝐓^𝑒_𝜇(𝑓_𝑛∗𝜙_𝑚)=𝐓^𝑒_𝜇(𝑓_𝑚∗𝜙_𝑛). Hence [(𝑓_𝑛)/(𝜙_𝑛)]∈𝜹(𝐥¹,(𝜿,∗),∗,Δ) is the Boehmian that satisfies 𝐓^𝑒_𝜇[(𝑓_𝑛)/(𝜙_𝑛)]=[(𝐓^𝑒_𝜇𝑓_𝑛)/(𝜙_𝑛)].
This completes the proof of the theorem.

Theorem 15. Let [(𝑓_𝑛)/(𝜙_𝑛)]∈𝜹(𝐥¹,(𝜿,∗),∗,Δ) and 𝜙∈𝜿(ℝ²); then 𝐓^𝑒_𝜇𝑓_𝑛𝜙_𝑛=𝐓∗𝜙^𝑒𝜇𝑓_𝑛𝜙_𝑛×𝜙. (34)

Proof. Applying Definition 10 for each [(𝑓_𝑛)/(𝜙_𝑛)]∈𝜹(𝐥¹,(𝜿,∗),×,Δ) and 𝜙∈𝜿(ℝ²) yields 𝐓^𝑒_𝜇𝑓_𝑛𝜙_𝑛=𝐓∗𝜙^𝑒𝜇𝑓_𝑛∗𝜙𝜙_𝑛. (35)
By Theorem 9 we get 𝐓^𝑒_𝜇𝑓_𝑛𝜙_𝑛=𝑓∗𝜙^𝑛×𝜙𝜙_𝑛=𝑓_𝑛𝜙_𝑛×𝜙. (36)
This completes the proof of the theorem.

Theorem 16. The mappings 𝐓^𝑒_𝜇 and (𝐓^𝑒_𝜇)⁻¹ are continuous with respect to 𝛿 and Δ convergence.

Proof. First of all, we show that 𝐓^𝑒_𝜇 and (𝐓^𝑒_𝜇)⁻¹ are continuous with respect to 𝛿 convergence.
Let 𝛽→−_𝑛^𝛿𝛽 in 𝜹(𝐥¹,(𝜿,∗),∗,Δ) as 𝑛→∞; then we show that 𝐓^𝑒_𝜇𝛽_𝑛→𝐓^𝑒_𝜇𝛽 as 𝑛→∞. By virtue of Theorem 1 we can find 𝑓_𝑛,𝑘 and 𝑓_𝑘 in 𝐥¹(ℝ²) such that 𝛽_𝑛=[𝑓_𝑛,𝑘/𝜙_𝑘] and 𝛽=[𝑓_𝑘/𝜙_𝑘] such that 𝑓_𝑛,𝑘→𝑓_𝑘 as 𝑛→∞ for every 𝑘∈ℕ.
Employing the continuity condition of 𝐓^𝑒_𝜇 transform implies 𝐓^𝑒_𝜇𝑓_𝑛,𝑘→𝐓^𝑒_𝜇𝑓_𝑘 as 𝑛→∞ in the space 𝐥¹(𝒮¹×ℝ).
Thus, 𝐓^𝑒_𝜇𝑓_𝑛,𝑘𝜙_𝑘⟶𝐓^𝑒_𝜇𝑓_𝑘𝜙_𝑘as𝑛⟶∞ (37) in 𝜹(𝐥¹,(𝜿,∗),×,Δ).
To prove the second part, let 𝑔→−_𝑛^𝛿𝑔 in 𝜹(𝐥¹,(𝜿,∗),×,Δ) as 𝑛→∞. Then, once again, by Theorem 1, 𝑔_𝑛=[𝐓^𝑒_𝜇𝑓_𝑛,𝑘/𝜙_𝑘] and 𝑔=[𝐓^𝑒_𝜇𝑓_𝑘/𝜙_𝑘] and 𝐓^𝑒_𝜇𝑓_𝑛,𝑘→𝐓^𝑒_𝜇𝑓_𝑘 as 𝑛→∞. Hence 𝑓_𝑛,𝑘→𝑓_𝑘 in 𝜹(𝐥¹,(𝜿,∗),∗,Δ) as 𝑛→∞. That is, [𝑓_𝑛,𝑘/𝜙_𝑘]→[𝑓_𝑘/𝜙_𝑘] as 𝑛→∞. Using (33) we get 𝐓^𝑒_𝜇⁻¹𝐓^𝑒_𝜇𝑓_𝑛,𝑘𝜙_𝑘⟶𝐓^𝑒_𝜇⁻¹𝐓^𝑒_𝜇𝑓_𝑘𝜙_𝑘as𝑛⟶∞. (38)
Now, we establish continuity of 𝐓^𝑒_𝜇 and (𝐓^𝑒_𝜇)⁻¹ with respect to Δ convergence.
Let 𝛽_𝑛,𝛽∈𝜹(𝐥¹,(𝜿,∗),∗,Δ) be such that 𝛽−→_𝑛^Δ𝛽 as 𝑛→∞. Then, by Theorem 1 we can find that 𝑓_𝑛∈𝐥¹(𝒮¹×ℝ) and (𝜙_𝑛)∈Δ such that (𝛽_𝑛−𝛽)∗𝜙_𝑛=[((𝑓_𝑛)∗𝜙_𝑘)/𝜙_𝑘] and 𝑓_𝑛→0 as 𝑛→∞. Employing (29) we get 𝐓^𝑒_𝜇𝛽_𝑛−𝛽∗𝜙_𝑛=𝐓^𝑒_𝜇𝑓_𝑛∗𝜙_𝑘𝜙_𝑘. (39) Hence, we have 𝐓^𝑒_𝜇((𝛽_𝑛−𝛽)∗𝜙_𝑛)=[((𝐓^𝑒_𝜇𝑓_𝑛)×𝜙_𝑘)/𝜙_𝑘]=𝐓^𝑒_𝜇𝑓_𝑛→0 as 𝑛→∞ in 𝐥¹(𝒮¹×ℝ).
Therefore 𝐓^𝑒_𝜇𝛽_𝑛−𝛽∗𝜙_𝑛=𝐓^𝑒_𝜇𝛽_𝑛−𝐓^𝑒_𝜇𝛽×𝜙_𝑛⟹as𝑛⟶∞. (40) Hence, 𝐓^𝑒_𝜇𝛽−→_𝑛^Δ𝐓^𝑒_𝜇𝛽 as 𝑛→∞.
Finally, let 𝑔−→_𝑛^Δ𝑔 in 𝜹(𝐥¹,(𝜿,∗),∗,Δ) as 𝑛→∞; then we find 𝐓^𝑒_𝜇𝑓_𝑘∈𝐥¹(𝒮¹×ℝ) such that (𝑔_𝑛−𝑔)×𝜙_𝑛=[(𝐓^𝑒_𝜇𝑓_𝑘×𝜙_𝑘)/𝜙_𝑘] and 𝐓^𝑒_𝜇𝑓_𝑘→0 as 𝑛→∞ for some (𝜙_𝑛)∈Δ.
Now, using (33), we obtain 𝐓^𝑒_𝜇⁻¹𝑔_𝑛−𝑔×𝜙_𝑛=𝐓^𝑒_𝜇⁻¹𝐓^𝑒_𝜇𝑓_𝑘×𝜙_𝑘𝜙_𝑘. (41) Theorem 9 implies 𝐓^𝑒_𝜇⁻¹𝑔_𝑛−𝑔×𝜙_𝑛=𝑓_𝑛∗𝜙_𝑘𝜙_𝑘=𝑓_𝑛⟶0∈𝐥¹𝒮¹×ℝas𝑛⟶∞. (42) Thus 𝐓^𝑒_𝜇⁻¹𝑔_𝑛−𝑔×𝜙_𝑛=𝐓^𝑒_𝜇⁻¹𝑔_𝑛−𝐓^𝑒_𝜇⁻¹𝑔∗𝜙_𝑛→0∈𝐥¹𝒮¹×ℝas𝑛⟶∞. (43)
From this we find that (𝐓^𝑒_𝜇)⁻¹𝑔−→_𝑛^Δ(𝐓^𝑒_𝜇)⁻¹𝑔∈𝜹(𝐥¹,(𝜿,∗),∗,Δ) as 𝑛→∞.
This completes the proof of the theorem.

Theorem 17. The transform 𝐓^𝑒_𝜇 is consistent with 𝐓^𝑒_𝜇∶𝐥¹(𝒮¹×ℝ)→𝐥¹(𝒮¹×ℝ).

Proof. For every 𝑓∈𝐥¹(𝒮¹×ℝ), let 𝛽∈𝜹(𝐥¹,(𝜿,∗),∗,Δ) be its representative; then ∀𝑛∈ℕ, (𝜑_𝑛)∈Δ, 𝛽=[(𝑓∗(𝜑_𝑛))/(𝜑_𝑛)]. For all 𝑛∈ℕ it is clear that (𝜑_𝑛) is independent of the representative.
We have 𝐓^𝑒_𝜇𝛽=𝐓^𝑒_𝜇𝑓∗𝜑_𝑛𝜑_𝑛=𝐓^𝑒_𝜇𝑓∗𝜑_𝑛𝜑_𝑛=𝐓^𝑒_𝜇𝑓×𝜑_𝑛𝜑_𝑛 (44) which is the representative of 𝐓^𝑒_𝜇𝑓 in the space 𝐥¹(𝒮¹×ℝ).
Hence the proof is completed.

Theorem 18. The necessary and sufficient condition for [(𝑔_𝑛)/(𝜓_𝑛)]∈𝜹(𝐥¹,(𝜿,∗),×,Δ) to be in the range of 𝐓^𝑒_𝜇 is that 𝑔_𝑛 belongs to range of 𝐓^𝑒_𝜇 for every 𝑛∈ℕ.

Proof. Let [(𝑔_𝑛)/(𝜓_𝑛)] be in the range of 𝐓^𝑒_𝜇; then of course 𝑔_𝑛 belongs to the range of 𝐓^𝑒_𝜇,∀𝑛∈ℕ.
To establish the converse, let 𝑔_𝑛 be in the range of  𝐓^𝑒_𝜇, ∀𝑛∈ℕ. Then there is 𝑓_𝑛∈𝐥¹(𝒮¹×ℝ) such that 𝐓^𝑒_𝜇𝑓_𝑛=𝑔_𝑛,𝑛∈ℕ.
Since [(𝑔_𝑛)/(𝜓_𝑛)]∈𝜹(𝐥¹,(𝜿,∗),×,Δ) we get 𝑔_𝑛×𝜓_𝑚=𝑔_𝑚×𝜓_𝑛, ∀𝑚,𝑛∈ℕ.
Therefore, Theorem 9 yields 𝐓^𝑒_𝜇𝑓_𝑛∗𝜑_𝑛=𝐓^𝑒_𝜇𝑓_𝑚∗𝜑_𝑛,∀𝑚,𝑛∈ℕ, (45) where 𝑓_𝑛∈𝐥¹(𝒮¹×ℝ) and 𝜑_𝑛∈Δ, ∀𝑛∈ℕ.
Thus 𝑓_𝑛∗𝜑_𝑚=𝑓_𝑚∗𝜑_𝑛, 𝑚,𝑛∈ℕ. Hence, 𝑓_𝑛𝜑_𝑛∈𝜹𝐥¹,(),𝐓𝜿,∗,∗,Δ_𝑒𝜇𝑓_𝑛𝜑_𝑛=𝑔_𝑛𝜓_𝑛. (46) The theorem is therefore completely proved.

Conflict of Interests

The authors declare that they have no conflict of interests regarding publication of this paper.

Acknowledgment

The second author gratefully acknowledges the partial support that this research received from the Universiti Putra Malaysia under GP-IBT Grant Scheme having Project no. GPIBT/2013/9420100.