On the Generalized Krätzel Transform and Its Extension to Bohemian Spaces

Abstract

We investigate the Krätzel transform on certain class of generalized functions. We propose operations that lead to the construction of desired spaces of generalized functions. The Krätzel transform is extended and some of its properties are obtained.

1. Introduction

In recent years, integral transforms of Bohemian have comprised an active area of research. Several integral transforms are extended to various spaces of Bohemian, especially, that permit a factorization property of Fourier convolution type.

On the other hand, several integral transforms that have not permitted a factorization property of Fourier convolution type are also extended to various spaces of Bohemian. In the sequence of these integral transforms, the Krätzel transform [1] 𝒦^𝜌_𝑣𝑓(𝑥)=_ℝ+𝒵^𝑣_𝜌𝑥𝑦𝑓𝑦𝑑𝑦, (1) where 𝒵^𝑣_𝜌𝑥𝑦=_ℝ+𝑡^𝑣−1𝑒^{−𝑡𝜌−𝑥𝑦/𝑡}𝑑𝑡, (2)𝜌>0(∈ℕ), 𝑣∈ℂ, 𝑥>0 is extended to certain space of ultra-Bohemian, denoted by 𝒮₊(𝑙^𝑟,𝛼,(𝛼_𝑖),𝑎) and 𝒮₊(𝑙^𝑟,𝛼,{𝛼_𝑖},𝑎), 1≤𝑟≤∞, respectively.

Another form of the Krätzel transform was initially introduced in [2], and defined as a generalization of the Laplace transform in [3] as 𝑙_𝑣^(𝜌)𝑓(𝑥)=_ℝ+𝜆_𝑣^(𝜌)𝑥𝑦𝑓𝑦𝑑𝑦, (3) where 𝜆_𝑣^(𝜌)(𝑥)=(2𝜋)^{((𝜌−1)/2)𝜌(1/2)}𝑇×𝑥𝑣+1−1/𝜌𝜌^𝜌𝑣∞₁𝑡^𝜌−1^{𝑣−(1/𝜌)}𝑒^−𝑥𝑡𝑑𝑡, (4) for 𝜌∈ℕ and Re𝑣>−1+(1/𝜌),𝑥>0. The 𝑙_𝑣^(𝜌) transform was extended to generalized functions in [3] and to distributions in [4]. By 𝑙^𝜌(𝑋), we denote the space of equivalence classes of measurable functions 𝑓∶𝑋→ℝ such that _𝑋𝑓^𝜌𝑑𝜇<∞, (5) where two measurable functions are equivalent when they are equal to 𝜇 a.e.

For 𝑙^𝜌(𝑋), the 𝑙^𝜌-norm is defined as 𝑓_𝑙^𝜌=_𝑋𝑓^𝜌𝑑𝜇^1/𝜌. (6) The Banach space 𝑙_𝜌,𝜇 of Lebesgue measurable functions is defined by [3, page 446] 𝑓_𝜌,𝜇=𝑥^−𝜇𝑓_𝜌<∞. (7) Due to [3, Proposition 2.1], we state the following theorem.

Theorem 1. Let 1≤𝜌≤∞, 𝜇,𝑣∈ℂ, 𝜌>0, (1/𝜌)+(1/𝜌^′)=1 and Re𝜇>−(1/𝜌^′)−min{0,𝜌Re𝑣} and then 𝑙_𝑣^(𝜌) is a continuous linear mapping from 𝑙_𝜌,𝜇 into 𝑙_{𝜌,2/(𝜌−𝜇−1)}.

Note that we always assume that the hypothesis of Theorem 1 is satisfied. By ⋎ we denote the Mellin-type convolution product of first order defined by [5] 𝑓⋎𝜑𝑦=_ℝ+𝑓𝑦𝑡⁻¹𝑡⁻¹𝜑(𝑡)𝑑𝑡. (8) By 𝒟(ℝ₊), or simply 𝒟, denote the Schwartz space of test functions of compact support defined on ℝ₊. Then we have the following definition.

Definition 2. Let 𝑓,𝑔∈𝑙_𝜌,𝜇. For 𝑓 and 𝑔, we define the operation ⊗ given by 𝑓⊗𝑔(𝑥)=_ℝ+𝑓(𝑥𝑡)𝑔(𝑡)𝑑𝑡. (9)

Then we easily see that the operations in (8) and (9) are very basic for the next construction of the desired Bohemian spaces.

Theorem 3. Let 𝑓∈𝑙_𝜌,𝜇 and 𝜑∈𝒟, then 𝑙_𝑣^(𝜌)(𝑓⋎𝜑)(𝑥)=((𝑙_𝑣^(𝜌)𝑓)⊗𝜑)(𝑥).

Proof. Let 𝑥>0. For 𝑓∈𝑙_𝜌,𝜇 and 𝜑∈𝒟 we, by (3) and (8), get that 𝑙_𝑣^(𝜌)𝑥)=𝑓⋎𝜑(_ℝ+ℝ+𝑓𝑦𝑡⁻¹𝑡⁻¹𝜑(𝑡)𝑑𝑡𝜆_𝑣^(𝜌)=𝑥𝑦𝑑𝑦_ℝ+ℝ+𝑓𝑦𝑡⁻¹𝑡⁻¹𝜑(𝑡)𝑑𝑡. (10) A change of variables 𝑦=𝑡𝑧 in (10) implies 𝑙_𝑣^(𝜌)𝑓⋎𝜑(𝑥)=_ℝ+ℝ+𝑓(𝑧)𝜆_𝑣^(𝜌)(𝑥𝑡𝑧)𝑑𝑧𝜑(𝑡)𝑑𝑡=_ℝ+𝑙_𝑣^(𝜌)𝑓(𝑥𝑡)𝜑(𝑡)𝑑𝑡. (11) Hence, by (9), we have 𝑙_𝑣^(𝜌)𝑓⋎𝜑(𝑥)=𝑙_𝑣^(𝜌)𝑓⊗𝜑(𝑥). (12) This completes the proof of theorem.

2. General Bohemian

The structure necessary for the construction of Bohemian consists of the following:(1)a group 𝑋 and a commutative semigroup (𝑌,∗);(2)a operation ⊙∶𝑋×𝑌→𝑋 such that 𝑥⊙(𝜐₁∗𝜐₂)=(𝑥⊙𝜐₁)⊙𝜐₂, for all 𝑥∈𝑋 and 𝜐₁,𝜐₂,∈𝑌;(3)a collection Δ⊂𝑌^ℕ such that(i)for all (𝜐_𝑛),(𝜎_𝑛)∈Δ, we have (𝜐_𝑛∗𝜎_𝑛)∈Δ,(ii)If 𝑥⊙𝜐_𝑛=𝑦⊙𝜐_𝑛, then 𝑥=𝑦 where 𝑥,𝑦∈𝑋,(𝜐_𝑛)∈Δ, and 𝑛∈ℕ.

Then the set Δ that satisfies (i) and (ii) is called the set of all delta sequences.

Let 𝐴={(𝑥_𝑛,𝜐_𝑛)∶𝑥_𝑛∈𝑋,(𝜐_𝑛)∈Δ,𝑥_𝑛⊙𝜐_𝑚=𝑥_𝑚⊙𝜐_𝑛,forall𝑚,𝑛∈ℕ}. Then we say (𝑥_𝑛,𝜐_𝑛)∼(𝑦_𝑛,𝜎_𝑛) if there are (𝑥_𝑛,𝜐_𝑛),(𝑦_𝑛,𝜎_𝑛)∈𝐴, such that 𝑥_𝑛⊙𝜎_𝑚=𝑦_𝑚⊙𝜐_𝑛, for all 𝑚,𝑛∈ℕ. The relation ∼ is an equivalence relation in 𝐴. The space of equivalence classes in 𝐴 is called the space of Bohemian and denoted by 𝔹ℍ. Each element of 𝔹ℍ is called Bohemian. Then the convergence in 𝔹ℍ is defined as follows.(1)(ℎ_𝑛)∈𝔹ℍ is said to be 𝛿-convergent to ℎ∈𝔹ℍ, denoted by ℎ→−_𝑛^𝛿ℎ as 𝑛→∞, if ∃ a delta sequence (𝜐_𝑛) such that (ℎ_𝑛⊙𝜐_𝑛),(ℎ⊙𝜐_𝑛)∈𝑋, and (ℎ_𝑛⊙𝜐_𝑘)→(ℎ⊙𝜐_𝑘)∈𝑋 as 𝑛→∞, for all 𝑘,𝑛∈ℕ.(2)(ℎ_𝑛)∈𝔹ℍ is said to be Δ-convergent to ℎ∈𝔹ℍ, denoted by ℎ−→_𝑛^Δℎ as 𝑛→∞, if ∃ a (𝜐_𝑛)∈Δ such that (ℎ_𝑛−ℎ)⊙𝜐_𝑛∈𝑋, for all 𝑛∈ℕ, and (ℎ_𝑛−ℎ)⊙𝜐_𝑛→0∈𝑋 as 𝑛→∞.The following theorem is equivalent to the statement of 𝛿-convergence.

Theorem 4. ℎ→−_𝑛^𝛿ℎ∈𝔹ℍ as 𝑛→∞ if and only if there are 𝑓_𝑛,𝑘,𝑓_𝑘∈𝑋 and 𝜐_𝑘∈Δ such that ℎ_𝑛=[𝑓_𝑛,𝑘/𝜐_𝑘],ℎ=[𝑓_𝑘/𝜐_𝑘] and, for all 𝑘∈ℕ, 𝑓_𝑛,𝑘→𝑓_𝑘∈𝑋 as 𝑛→∞. See [6–17] for more details.

3. The Spaces 𝔹ℍ₁(𝑙_𝜌,𝜇) and 𝔹ℍ₂(𝑙_{𝜌,2/(𝜌−𝜇−1)})

In this section we construct the space 𝔹ℍ(𝑙_𝜌,𝜇,(𝒟,⋎),⋎,Δ)(or𝔹ℍ₁(𝑙_𝜌,𝜇)) and the space 𝔹ℍ(𝑙_{𝜌,2/(𝜌−𝜇−1)},(𝒟,⋎),⊗,Δ)(or𝔹ℍ₂(𝑙_{𝜌,2/(𝜌−𝜇−1)})) of Bohemian and give their properties.

At the first step, we prove the following connecting theorem.

Theorem 5. Let 𝑓∈𝑙_𝜌,𝜇 and 𝜑,𝜓∈𝒟; then, 𝑓⊗(𝜑⋎𝜓)=(𝑓⊗𝜑)⊗𝜓.

Proof. Let 𝑓∈𝑙_𝜌,𝜇 and 𝜑,𝜓∈𝒟; then, applying the Fubinitz theorem yields (𝑥)=𝑓⊗𝜑⋎𝜓_ℝ+𝑓()(𝑡)=𝑥𝑡𝜑⋎𝜓𝑑𝑡_ℝ+𝑓(𝑥𝑡)_ℝ+𝜑𝑡𝑦𝜓𝑦𝑦⁻¹=𝑑𝑦𝑑𝑡_ℝ+ℝ+𝑓()𝜑𝑡𝑥𝑡𝑦𝑦⁻¹𝑑𝑦𝜓𝑦𝑑𝑦. (13) The change of variables 𝑡=𝑦𝑧 implies 𝑑𝑡=𝑦𝑑𝑧 and further 𝑓⊗𝜑⋎𝜓(𝑥)=_ℝ+ℝ+𝑓𝑥𝑦𝑧𝜑(𝑧)𝑑𝑧𝜓𝑦𝑑𝑦. (14) Therefore, 𝑓⊗𝜑⋎𝜓(𝑥)=_ℝ+𝑓⊗𝜑𝑥𝑦𝜓𝑦𝑑𝑦. (15) Hence, (15) implies 𝑓⊗𝜑⋎𝜓(𝑥)=𝑓⊗𝜑⊗𝜓(𝑥). (16) This completes the proof of the theorem.

Next, forthcoming theorems prove the existence of the space 𝔹ℍ₁(𝑙_𝜌,𝜇).

Theorem 6. Let 𝑓∈𝑙_𝜌,𝜇 and 𝜑∈𝒟 and then 𝑓⋎𝜑∈𝑙_𝜌,𝜇.

Proof. For each 𝑓∈𝑙_𝜌,𝜇 and 𝜑∈𝒟, we have 𝑥^𝜇𝑓⋎𝜑(𝑥)^𝜌_𝜌,𝜇=_ℝ+𝑥^𝜇_ℝ+𝑓(𝑥𝑡)𝜑(𝑡)𝑑𝑡^𝜌𝑑𝑥. (17) By Jensen’s theorem, we write 𝑥^𝜇𝑓⋎𝜑(𝑥)^𝜌_𝜌,𝜇≤_ℝ+ℝ+𝑥^𝜇_ℝ+𝑓(𝑥𝑡)𝜑(𝑡)𝑑𝑡^𝜌≤𝑑𝑥_ℝ+ℝ+𝑥^𝜇𝑓(𝑥𝑡)^𝜌𝜑(𝑡≤𝑓)𝑑𝑡𝑑𝑥_{𝜌𝜌,𝜇𝐾}𝜑(𝑡)𝑑𝑡, (18) where 𝐾=[𝑎,𝑏], 𝑏>𝑎>0 is a compact subset containing the support of 𝜑. Hence, from (18), we get 𝑥^𝜇𝑓⋎𝜑(𝑥)^𝜌_𝜌,𝜇≤𝑓^𝜌_𝜌,𝜇𝑀(𝑏−𝑎), (19) where 𝑀 is certain positive real number. This completes the proof of the theorem.

Theorem 7. Let 𝑓∈𝑙_𝜌,𝜇 and 𝜑,𝜓∈𝒟 and then(i)𝑓⋎(𝜑+𝜓)=𝑓⋎𝜑+𝑓⋎𝜓;(ii)(𝛼𝑓)⋎𝜑=𝛼(𝑓⋎𝜑);(iii)let (𝑓_𝑛)∈𝑙_𝜌,𝜇 and 𝑓_𝑛→𝑓 as 𝑛→∞ then for 𝜑∈𝒟, and we have 𝑓_𝑛⋎𝜑→𝑓⋎𝜑 as 𝑛→∞;(iv)𝑓⋎(𝜑⋎𝜓)=(𝑓⋎𝜑)⋎𝜓.

Proof of Part (i), (ii), and (iii) follows from properties of integral operators ∫. Similarly, the proof of Part (iv) is straightforward from the properties of ⋎ proved in [5].

Following theorem is straightforward, and detailed proof is omitted.

Theorem 8. Let 𝑓∈𝑙_𝜌,𝜇 and (𝜇_𝑛)∈Δ then 𝑓⋎𝜇_𝑛→𝑓 as 𝑛→∞.

Proof. Since 𝒟 is a dense subspace of 𝑙_𝜌 then it is a dense subspace of 𝑙_𝜌,𝜇. Hence, there can be found 𝜓∈𝒟 such that 𝑓−𝜓_𝜌,𝜇<𝜀 (20) for 𝜀>0. Also, from (18), we have 𝑓−𝜓⋎𝜇_𝑛^𝜌_𝜌,𝜇≤𝑓^𝜌_𝜌,𝜇𝜇_𝑛𝑙1. (21) Let 𝑔(𝑡)=𝑦^𝜇𝜓(𝑦𝑡⁻¹)𝑡⁻¹ then; 𝑔(𝑡)∈𝒟 and hence uniformly continuous on ℝ₊.
Therefore, there is 𝛿>0 such that 𝑔𝑦−𝑔(𝑥)<𝜀whenever𝑦−𝑥<𝛿. (22) Thus, using (22), we get 𝜓×𝜇_𝑛−𝜓𝑦^𝜌_𝜌,𝜇=_ℝ+𝑦^𝜇𝜓×𝜇_𝑛−𝜓𝑦^𝜌≤𝑑𝑦_ℝ+ℝ+𝑦^𝜇𝜓𝑦𝑡⁻¹𝑡⁻¹𝜇_𝑛(𝑡)−𝜓𝑦𝜇_𝑛(𝑡)^𝜌≤𝑑𝑡𝑑𝑦_ℝ+ℝ+𝑦^𝜇𝜓𝑦𝑡⁻¹𝑡⁻¹−𝜓𝑦^𝜌×𝜇_𝑛(𝑡=)𝑑𝑡𝑑𝑦_ℝ+ℝ+𝑔𝑦−𝑔(1)^𝜌𝜇_𝑛(𝑡=)𝑑𝑡𝑑𝑦_ℝ+ℝ+𝜀^𝜌𝜇_𝑛(𝑡)𝑑𝑡𝑑𝑦. (23) By (22), supp𝜇_𝑛(𝑡)→0 as 𝑛→∞ implies that there can be found 𝑁∈ℕ, such that supp𝜇_𝑛⊆[0,𝛿], for all 𝑛≥𝑁. Further, the fact that the function 𝜓 is of compact support thus this implies that supp𝜓(𝑦)⊆𝐾=[𝑎,𝑏], where 𝐾 is a compact subset of ℝ₊. Thus, from (23), we write 𝜓⋎𝜇_𝑛−𝜓^𝜌_𝜌,𝜇≤𝜀^𝜌𝑏_𝑎^𝛿₀𝑀𝑑𝑡=𝜀^𝜌(𝑏−𝑎)(2𝛿)𝑀. (24) Now, we have 𝑓×𝜇_𝑛−𝑓_𝜌,𝜇≤𝑓−𝜓⋎𝜇_{𝑛𝜌,𝜇}+𝜓×𝜇_𝑛−𝜓_𝜌,𝜇+𝑓−𝜓_𝜌,𝜇. (25) On using (20), (21) and (24) prove that 𝑓×𝜇_𝑛−𝑓_𝜌,𝜇⟶0as𝜖⟶0. (26) Thus the theorem is proved. Then the Bohemian space 𝔹ℍ₁(𝑙_𝜌,𝜇) is therefore constructed. The operations such as sum and multiplication by a scalar of two Bohemian in 𝔹ℍ₁(𝑙_𝜌,𝜇) are defined in a natural way 𝑓_𝑛𝜇_𝑛+𝑔_𝑛𝜏_𝑛=𝑓_𝑛⋎𝜏_𝑛+𝑔_𝑛⋎𝜇_𝑛𝜇_𝑛⋎𝜏_𝑛,𝛼𝑓_𝑛𝜇_𝑛=𝑎𝑓_𝑛𝜇_𝑛, (27) where α  is a complex number.
Similarly, the operation ⋎ and differentiation are defined by 𝑓_𝑛𝜇_𝑛⋎𝑔_𝑛𝜏_𝑛=𝑓_𝑛⋎𝑔_𝑛𝜇_𝑛⋎𝜏_𝑛,𝒟^𝛼𝑓_𝑛𝜇_𝑛=𝒟^𝛼𝑓_𝑛𝜇_𝑛. (28) Now constructing the space 𝔹ℍ₂(𝑙_{𝜌,2/(𝜌−𝜇−1)}) follows from theorems which were used for constructing the space 𝔹ℍ₁(𝑙_𝜌,𝜇). Therefore, the corresponding proofs of Theorems 10 and 11 are omitted.

Theorem 9. Let 𝑓∈𝑙_{𝜌,2/(𝜌−𝜇−1)} and 𝜑∈𝒟 and then 𝑓⊗𝜑∈𝑙_{𝜌,2/(𝜌−𝜇−1)}.

Theorem 10. Let 𝑓∈𝑙_{𝜌,2/(𝜌−𝜇−1)}, 𝜑,𝜓∈𝒟 and then the following hold:(i)𝑓⊗(𝜑+𝜓)=𝑓⊗𝜑+𝑓⊗𝜓; (ii)(𝛼𝑓)⊗𝜑=𝛼(𝑓⊗𝜑); (iii)if 𝑓_𝑛→𝑓 in 𝑙_{𝜌,2/(𝜌−𝜇−1)}, as 𝑛→∞ then 𝑓_𝑛⊗𝜑→𝑓⊗𝜑 as 𝑛→∞;(iv)if (𝜇_𝑛)∈Δ, then 𝑓⊗𝜇_𝑛→𝑓 as 𝑛→∞.

Theorem 11. Let 𝑓∈𝑙_{𝜌,2/(𝜌−𝜇−1)} and 𝜑,𝜓∈𝒟 and then 𝑓⊗(𝜑⋎𝜓)=(𝑓⊗𝜑)⊗𝜓.

Proof of this theorem is similar to that of Theorem 6. Thus the space 𝔹ℍ₂(𝑙_{𝜌,2/(𝜌−𝜇−1)}) can be regarded as Bohemian space.

4. Krätzel Transform of Bohemian

By aid of Theorem 4, we have the right to define the Krätzel transform of [(𝑓_𝑛)/(𝜇_𝑛)]∈𝔹ℍ₁(𝑙_𝜌,𝜇) as the Bohemian 𝑙_𝑣^(𝜌)𝑓_𝑛𝜇_𝑛=𝑙_𝑣^(𝜌)𝑓_𝑛𝜇_𝑛 (29) is embedded in the space 𝔹ℍ₂(𝑙_{𝜌,2/(𝜌−𝜇−1)}).

Theorem 12. The mapping 𝑙_𝑣^(𝜌)∶𝔹ℍ₁(𝑙_𝜌,𝜇)→𝔹ℍ₂(𝑙_{𝜌,2/(𝜌−𝜇−1)}) is(i)well defined,(ii)linear.

Proof. Let [(𝑓_𝑛)/(𝜇_𝑛)],[(𝑔_𝑛)/(𝜓_𝑛)]∈𝔹ℍ₁(𝑙_𝜌,𝜇) be such that [(𝑓_𝑛)/(𝜇_𝑛)]=[(𝑔_𝑛)/(𝜓_𝑛)], and then 𝑓_𝑛⋎𝜓_𝑚=𝑔_𝑚⋎𝜇_𝑛=𝑔_𝑛⋎𝜇_𝑚. Using Theorem 4 implies 𝑙_𝑣^(𝜌)𝑓_𝑛⊗𝜓_𝑚=𝑙_𝑣^(𝜌)𝑔_𝑛⊗𝜇_𝑚, for all 𝑛,𝑚. The idea of quotient of sequences in 𝔹ℍ₂(𝑙_{𝜌,2/(𝜌−𝜇−1)}) implies that 𝑙_𝑣^(𝜌)𝑓_𝑛𝜇_𝑛𝑙isequivalentto^𝑣(𝜌)𝑔_𝑛𝜓_𝑛. (30) That is, 𝑙_𝑣^(𝜌)𝑓_𝑛𝜇_𝑛=𝑙_𝑣^(𝜌)𝑔_𝑛𝜓_𝑛. (31) To prove part (ii) of the theorem, if [(𝑓_𝑛)/(𝜇_𝑛)],[(𝑔_𝑛)/(𝜓_𝑛)]∈𝔹ℍ₁(𝑙_𝜌,𝜇), then 𝑙_𝑣^(𝜌)𝑓_𝑛𝜇_𝑛+𝑔_𝑛𝜓_𝑛=𝑙_𝑣^(𝜌)𝑓_𝑛⊗𝜓_𝑛+𝑙_𝑣^(𝜌)𝑔_𝑛⊗𝜇_𝑛𝜇_𝑛⊗𝜓_𝑛. (32) Hence 𝑙_𝑣^(𝜌)𝑓_𝑛𝜇_𝑛+𝑔_𝑛𝜓_𝑛=𝑙_𝑣^(𝜌)𝑓_𝑛𝜇_𝑛+𝑙_𝑣^(𝜌)𝑔_𝑛𝜓_𝑛. (33) Also, if 𝛼∈ℂ, then 𝛼𝑙_𝑣^(𝜌)𝑓_𝑛𝜇_𝑛𝑙=𝛼^𝑣(𝜌)𝑓_𝑛𝜇_𝑛=𝑙_𝑣^(𝜌)𝛼𝑓_𝑛𝜇_𝑛. (34) Hence 𝛼𝑙_𝑣^(𝜌)𝑓_𝑛𝜇_𝑛=𝑙_𝑣^(𝜌)𝛼𝑓_𝑛𝜇_𝑛. (35) This completes the proof.

Definition 13. Let [(𝑙_𝑣^(𝜌)𝑓_𝑛)/(𝜇_𝑛)]∈𝔹ℍ₂(𝑙_{𝜌,2/(𝜌−𝜇−1)}). We define the inverse of 𝑙_𝑣^(𝜌) transform of the Bohemian [(𝑙_𝑣^(𝜌)𝑓_𝑛)/(𝜇_𝑛)] as 𝑙_𝑣^(𝜌)−1𝑙_𝑣^(𝜌)𝑓_𝑛𝜇_𝑛=𝑙_𝑣^(𝜌)−1𝑙_𝑣^(𝜌)𝑓_𝑛𝜇_𝑛=𝑓_𝑛𝜇_𝑛, (36) for each (𝜇_𝑛)∈Δ.

Theorem 14. 𝑙_𝑣^(𝜌)∶𝔹ℍ₁(𝑙_𝜌,𝜇)→𝔹ℍ₂(𝑙_{𝜌,2/(𝜌−𝜇−1)}) is an isomorphism.

Proof. Let 𝑙_𝑣^(𝜌)[(𝑓_𝑛)/(𝜇_𝑛)]=𝑙_𝑣^(𝜌)[(𝑔_𝑛)/(𝜓_𝑛)]∈𝔹ℍ₂(𝑙_{𝜌,2/(𝜌−𝜇−1)}) and then by (29) we get 𝑙_𝑣^(𝜌)𝑓_𝑛⊗𝜓_𝑚=𝑙_𝑣^(𝜌)𝑔_𝑚⊗𝜇_𝑛. Therefore, Theorem 4 implies 𝑙_𝑣^(𝜌)𝑓_𝑛⋎𝜓_𝑚=𝑙_𝑣^(𝜌)𝑔_𝑚⋎𝜇_𝑛. (37) Thus 𝑓_𝑛⋎𝜓_𝑚=𝑔_𝑚⋎𝜇_𝑛, for all 𝑚,𝑛∈ℕ. The concept of quotients of equivalent classes of 𝔹ℍ₁(𝑙_𝜌,𝜇) then gives 𝑓_𝑛𝜇_𝑛=𝑔_𝑛𝜓_𝑛. (38) This proves that 𝑙_𝑣^(𝜌) is injective.
To show that 𝑙_𝑣^(𝜌) is surjective, let [(𝑙_𝑣^(𝜌)𝑓_𝑛)/(𝜇_𝑛)]∈𝔹ℍ₂(𝑙_{𝜌,2/(𝜌−𝜇−1)}). Then (𝑙_𝑣^(𝜌)𝑓_𝑛)/(𝜇_𝑛) is a quotient of sequences in 𝔹ℍ₂(𝑙_{𝜌,2/(𝜌−𝜇−1)}).Hence, 𝑙_𝑣^(𝜌)𝑓_𝑛⊗𝜇_𝑚=𝑙_𝑣^(𝜌)𝑓_𝑚⊗𝜇_𝑛, for all 𝑚,𝑛∈ℕ. Once again, Theorem 4 implies that 𝑙_𝑣^(𝜌)(𝑓_𝑛⋎𝜇_𝑚)=𝑙_𝑣^(𝜌)(𝑓_𝑚⋎𝜇_𝑛). Hence [(𝑓_𝑛)/(𝜇_𝑛)]∈𝔹ℍ₁(𝑙_𝜌,𝜇) satisfies 𝑙_𝑣^(𝜌)𝑓_𝑛𝜇_𝑛=𝑙_𝑣^(𝜌)𝑓_𝑛𝜇_𝑛. (39) This completes the proof of the theorem.

Theorem 15. Let 𝜓∈𝒟 and[(𝑙_𝑣^(𝜌)𝑓_𝑛)/(𝜇_𝑛)]∈𝔹ℍ₂(𝑙_{𝜌,2/(𝜌−𝜇−1)}) then one has(i)𝑙_𝑣^(𝜌)−1([(𝑙_𝑣^(𝜌)𝑓_𝑛)/(𝜇_𝑛)]⊗𝜓)=[(𝑓_𝑛)/(𝜇_𝑛)]⋎𝜓;(ii)𝑙_𝑣^(𝜌)([(𝑓_𝑛)/(𝜇_𝑛)]⋎𝜓)=[(𝑙_𝑣^(𝜌)𝑓_𝑛)/(𝜇_𝑛)]⊗𝜓.

Proof. By using (29), we have 𝑙_𝑣^(𝜌)−1𝑙_𝑣^(𝜌)𝑓_𝑛𝜇_𝑛=𝑙⊗𝜓^{𝑣(𝜌)−1}𝑙_𝑣^(𝜌)𝑓_𝑛⊗𝜓𝜇_𝑛. (40) Theorem 4 then gives 𝑙_𝑣^(𝜌)−1𝑙_𝑣^(𝜌)𝑓_𝑛𝜇_𝑛=𝑓⊗𝜓^𝑛⋎𝜓𝜇_𝑛=𝑓_𝑛𝜇_𝑛⋎𝜓. (41) Hence the part (i) of the theorem is proved. The proof of part (ii) is similar thus we omit the details. This completes the proof of the theorem.

Theorem 16. The mappings(i)𝑙_𝑣^(𝜌)∶𝔹ℍ₁(𝑙_𝜌,𝜇)→𝔹ℍ₂(𝑙_{𝜌,2/(𝜌−𝜇−1)}) are continuous with respect to 𝛿 and Δ-convergence.(ii)𝑙_𝑣^(𝜌)−1∶𝔹ℍ₂(𝑙_{𝜌,2/(𝜌−𝜇−1)})→𝔹ℍ₁(𝑙_𝜌,𝜇) are continuous with respect to 𝛿 and Δ-convergence.

Proof. At first, let us show that 𝑙_𝑣^(𝜌) and 𝑙_𝑣^(𝜌)−1 are continuous with respect to 𝛿-convergence.
Let 𝛽→−_𝑛^𝛿𝛽 in 𝔹ℍ₁(𝑙_𝜌,𝜇) as 𝑛→∞ and then we show that 𝑙_𝑣^(𝜌)𝛽_𝑛→𝑙_𝑣^(𝜌)𝛽 as 𝑛→∞. By virtue of Theorem 5, we can find 𝑓_𝑛,𝑘 and 𝑓_𝑘 in 𝑙_𝜌,𝜇 such that 𝛽_𝑛=𝑓_𝑛,𝑘𝜇_𝑘𝑓,𝛽=^𝑘𝜇_𝑘 (42) such that 𝑓_𝑛,𝑘→𝑓_𝑘 as 𝑛→∞ for every 𝑘∈ℕ. Hence, 𝑙_𝑣^(𝜌)𝑓_𝑛,𝑘→𝑙_𝑣^(𝜌)𝑓_𝑘∈𝑙_{𝜌,2/(𝜌−𝜇−1)} as 𝑛→∞. Thus, 𝑙_𝑣^(𝜌)𝑓_𝑛,𝑘𝜇_𝑘⟶𝑙_𝑣^(𝜌)𝑓_𝑘𝜇_𝑘∈𝔹ℍ₂𝑙_{𝜌,2/(𝜌−𝜇−1)} (43) as 𝑛→∞.
To prove the Part (ii), Let 𝑔_𝑛,𝑔∈𝔹ℍ₂(𝑙_{𝜌,2/(𝜌−𝜇−1)}) be such that 𝑔→−_𝑛^𝛿𝑔 as 𝑛→∞. Then, once again, by Theorem 5, 𝑔_𝑛=[𝑙_𝑣^(𝜌)𝑓_𝑛,𝑘/𝜇_𝑘] and 𝑔=[𝑙_𝑣^(𝜌)𝑓_𝑘/𝜇_𝑘] for some 𝑓_𝑛,𝑘,𝑓_𝑘∈𝑙_𝜌,𝜇 and 𝑙_𝑣^(𝜌)𝑓_𝑛,𝑘→𝑙_𝑣^(𝜌)𝑓_𝑘 as 𝑛→∞. Hence [𝑓_𝑛,𝑘/𝜇_𝑘]→[𝑓_𝑘/𝜇_𝑘] as 𝑛→∞.
Using (36), we get 𝑙_𝑣^(𝜌)−1𝑙_𝑣^(𝜌)𝑓_𝑛,𝑘𝜇_𝑘⟶𝑙_𝑣^(𝜌)−1𝑙_𝑣^(𝜌)𝑓_𝑘𝜇_𝑘as𝑛→∞. (44) Now, we establish that 𝑙_𝑣^(𝜌) and 𝑙_𝑣^(𝜌)−1 are continuous with respect to Δ-convergence.
Let 𝛽−→_𝑛^Δ𝛽 in 𝔹ℍ₁(𝑙_𝜌,𝜇) as 𝑛→∞. Then, there exist 𝑓_𝑛∈𝑙_𝜌,𝜇 and (𝜇_𝑛)∈Δ such that (𝛽_𝑛−𝛽)⋎𝜇_𝑛=[((𝑓_𝑛)⋎𝜇_𝑘)/𝜇_𝑘] and 𝑓_𝑛→0 as 𝑛→∞. By applying (29) we get 𝑙_𝑣^(𝜌)𝛽_𝑛−𝛽⋎𝜇_𝑛=𝑙_𝑣^(𝜌)𝑓_𝑛⋎𝜇_𝑘𝜇_𝑘. (45) Hence, we have 𝑙_𝑣^(𝜌)((𝛽_𝑛−𝛽)⋎𝜇_𝑛)=[((𝑙_𝑣^(𝜌)𝑓_𝑛)⊗𝜇_𝑘)/𝜇_𝑘]=𝑙_𝑣^(𝜌)𝑓_𝑛→0 as 𝑛→∞ in 𝑙_{𝜌,2/(𝜌−𝜇−1)}.
Therefore 𝑙_𝑣^(𝜌)𝛽_𝑛−𝛽⋎𝜇_𝑛=𝑙_𝑣^(𝜌)𝛽_𝑛−𝑙_𝑣^(𝜌)𝛽⊗𝜇_𝑛⟶0as𝑛⟶∞. (46) Hence, 𝑙_𝑣^(𝜌)𝛽−→_𝑛^Δ𝑙_𝑣^(𝜌)𝛽 as 𝑛→∞.
Finally, let 𝑔−→_𝑛^Δ𝑔∈𝔹ℍ₂(𝑙_{𝜌,2/(𝜌−𝜇−1)}) as 𝑛→∞ and then we find 𝑙_𝑣^(𝜌)𝑓_𝑘∈𝑙_{𝜌,2/(𝜌−𝜇−1)} such that (𝑔_𝑛−𝑔)⊗𝜇_𝑛=[(𝑙_𝑣^(𝜌)𝑓_𝑘⊗𝜇_𝑘)/𝜇_𝑘] and 𝑙_𝑣^(𝜌)𝑓_𝑘→0 as 𝑛→∞ for some (𝜇_𝑛)∈Δ.
Now, using (36), we obtain 𝑙_𝑣^(𝜌)−1𝑔_𝑛−𝑔⊗𝜇_𝑛=𝑙_𝑣^(𝜌)−1𝑙_𝑣^(𝜌)𝑓_𝑘⊗𝜇_𝑘𝜇_𝑘. (47) Theorem 4 implies 𝑙_𝑣^(𝜌)−1𝑔_𝑛−𝑔⊗𝜇_𝑛=𝑓_𝑛⋎𝜇_𝑘𝜇_𝑘=𝑓_𝑛⟶0as𝑛⟶∞in𝑙_𝜌,𝜇. (48) Thus 𝑙_𝑣^(𝜌)−1𝑔_𝑛−𝑔⊗𝜇_𝑛=𝑙_𝑣^(𝜌)−1𝑔_𝑛−𝑙_𝑣^(𝜌)−1𝑔⋎𝜇_𝑛⟶0as𝑛⟶∞. (49) From this, we find that 𝑙_𝑣^(𝜌)−1𝑔−→_𝑛^Δ𝑙_𝑣^(𝜌)−1𝑔 as 𝑛→∞ in 𝔹ℍ₁(𝑙_𝜌,𝜇).
This completes the proof of the theorem.

Theorem 17. The 𝑙_𝑣^(𝜌) transform is consistent with the classical transform 𝑙_𝑣^(𝜌).

Proof. For every 𝑓∈𝑙_𝜌,𝜇, let 𝛽 be its representative in 𝔹ℍ₁(𝑙_𝜌,𝜇) and then 𝛽=[(𝑓⋎(𝜇_𝑛))/(𝜇_𝑛)] where (𝜇_𝑛)∈Δ, for all 𝑛∈ℕ. Since (𝜇_𝑛) is independent from the representative, for all 𝑛∈ℕ, therefore 𝑙_𝑣^(𝜌)𝛽=𝑙_𝑣^(𝜌)𝑓⋎𝜇_𝑛𝜇_𝑛=𝑙_𝑣^(𝜌)𝑓⊗𝜇_𝑛𝜇_𝑛, (50) which is the representative of 𝑙_𝑣^(𝜌)𝑓∈𝑙_{𝜌,2/(𝜌−𝜇−1)}. Thus this completes the proof.

Theorem 18. Let [(𝑔_𝑛)/(𝜓_𝑛)]∈𝔹ℍ₂(𝑙_{𝜌,2/(𝜌−𝜇−1)}) and then a necessary and sufficient condition for [(𝑔_𝑛)/(𝜓_𝑛)] to be in the range of 𝑙_𝑣^(𝜌) is that 𝑔_𝑛 belongs to range of 𝑙_𝑣^(𝜌), for all 𝑛∈ℕ.

Proof. Let [(𝑔_𝑛)/(𝜓_𝑛)] be in the range of 𝑙_𝑣^(𝜌) and then it is clear that 𝑔_𝑛 belongs to the range of 𝑙_𝑣^(𝜌), for all 𝑛∈ℕ.
To establish the converse, let 𝑔_𝑛 be in the range of 𝑙_𝑣^(𝜌), for all 𝑛∈ℕ. Then there is 𝑓_𝑛∈𝑙_𝜌,𝜇 such that 𝑙_𝑣^(𝜌)𝑓_𝑛=𝑔_𝑛, 𝑛∈ℕ. Since [(𝑔_𝑛)/(𝜓_𝑛)]∈𝔹ℍ₂(𝑙_{𝜌,2/(𝜌−𝜇−1)}), 𝑔_𝑛⊗𝜓_𝑚=𝑔_𝑚⊗𝜓_𝑛, (51) for all 𝑚,𝑛∈ℕ, therefore, 𝑙_𝑣^(𝜌)𝑓_𝑛⋎𝜑_𝑛=𝑙_𝑣^(𝜌)𝑓_𝑚⋎𝜑_𝑛,∀𝑚,𝑛∈ℕ, (52) where 𝑓_𝑛∈𝑙_𝜌,𝜇 and 𝜑_𝑛∈Δ, for all 𝑛∈ℕ. Then it follows that we get 𝑓_𝑛⋎𝜑_𝑚=𝑓_𝑚⋎𝜑_𝑛, for all 𝑚,𝑛∈ℕ. Thus 𝑓_𝑛/𝜑_𝑛 is a quotient of sequences in 𝔹ℍ₁(𝑙_𝜌,𝜇). Therefore, we have [(𝑓_𝑛)/(𝜑_𝑛)]∈𝔹ℍ₁(𝑙_𝜌,𝜇) and 𝑙_𝑣^(𝜌)𝑓_𝑛𝜑_𝑛=𝑔_𝑛𝜓_𝑛. (53) Hence the theorem is proved.

Theorem 19. If 𝛽=[(𝑓_𝑛)/(𝜑_𝑛)]∈𝔹ℍ₁(𝑙_𝜌,𝜇) and 𝛾=[(𝜅_𝑛)/(𝜇_𝑛)]∈𝔹ℍ₁(𝑙_𝜌,𝜇), then one has 𝑙_𝑣^(𝜌)𝑓_𝑛𝜑_𝑛⋎𝜅_𝑛𝜇_𝑛=𝑙_𝑣^(𝜌)𝑓_𝑛𝜑_𝑛⊗𝜅_𝑛𝜇_𝑛. (54)

Proof. Let the hypothesis of the theorem be satisfied for some [(𝑓_𝑛)/(𝜑_𝑛)] and [(𝜅_𝑛)/(𝜇_𝑛)]. Therefore, 𝑙_𝑣^(𝜌)𝑓_𝑛𝜑_𝑛⋎𝜅_𝑛𝜇_𝑛=𝑙_𝑣^(𝜌)𝑓_𝑛⋎𝜅_𝑛𝜑_𝑛⋎𝜇_𝑛=𝑙_𝑣^(𝜌)𝑓_𝑛⊗𝜅_𝑛𝜑_𝑛⊗𝜇_𝑛=𝑙_𝑣^(𝜌)𝑓_𝑛𝜑_𝑛⊗𝜅_𝑛𝜇_𝑛. (55) This completes the proof.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors gratefully acknowledge that this research was partially supported by the University Putra Malaysia under the ERGS Grant Scheme having project no. 5527068.