On the Sumudu Transform and Its Extension to a Class of Boehmians

Abstract

Boehmians are used for all objects obtained by an algebraic construction similar to that of the field of quotients. In literature, several integral transforms have been extended to various Boehmian spaces but a few to the space of strong Boehmians. As shown in the work of Al-Omari (2013), this work describes certain spaces of Boehmians. The Sumudu transform is therefore established and it is one-one and continuous in the space of Boehmians. The inverse transform is given and some results are also discussed.

1. Introduction

Integral transforms are widely used in the literature, where some are often used for solving differential equations. The Sumudu transform was introduced by Watugala [1] and discussed by Weerakoon in [2]. Many properties are established in [3–5]. Having scale and unit-preserving properties, the Sumudu transform can be used to solve problems without resorting to new frequency domains. In [6] the classical Sumudu transform has been investigated over functions of two variables. In [7] the classical theorems in [6] are extended to distribution spaces and a space of Boehmians.

In this note we discuss the cited transform on certain space of strong Boehmians. Definitions and classical properties of Sumudu transforms are briefly given in Section 1. The general construction of usual and strong Boehmians is given in Sections 2 and 3, respectively. In Section 4, the Sumudu transform is extended to Boehmians and many properties are also obtained.

The Sumudu transform of 𝑓(𝑥) is given by [1] (𝜁)=𝑓(𝜁)1𝜇𝑓=∶𝜁_ℝ+𝑓(𝑡)exp−𝑡𝜁𝑑𝑡, (1) over a set of functions given as 𝔸=𝑓(𝑡)∶∃𝑚,𝜏₁,𝜏₂>0,𝑓(𝑡)<𝑚𝑒^{𝑡/𝜏𝑗},𝑡∈(−1)^𝑗×(0,∞], (2) where 𝑓(𝑡) is of convergent series type and 𝜁∈(−𝜏₁,𝜏₂).

Denote by 𝑓⊛𝑔(𝑥)=_ℝ+𝑓(𝑥−𝑡)𝑔(𝑡)𝑑𝑡, (3) the usual convolution product of 𝑓 and 𝑔.

Then the Sumudu transform of the convolution product is given as 𝜇𝑓⊛𝑔(𝜁)=𝜁𝑓(𝜁)𝑔(𝜁), (4) where 𝑓 and 𝑔 denote the Sumudu transform of 𝑓 and 𝑔, respectively.

Following are general properties of Sumudu transforms.(i)If 𝑘₁, 𝑘₂∈ℝ₊ and 𝑓, 𝑔 are the Sumudu transforms of 𝑓 and 𝑔, respectively, then 𝜇𝑘₁𝑓(𝑡)+𝑘₂𝑔(𝑡)(𝜁)=𝑘₁𝑓(𝜁)+𝑘₂𝑔(𝜁); (5)(ii)𝜇𝑓(𝑘𝑡)(𝜁)=𝜇(𝑘𝜁),𝑘∈ℝ₊; (6)(iii)lim_𝑡→0𝑓(𝑡)=lim_𝜁→0𝜇(𝜁)=𝑓(0). (7)

For details, see [8].

For strong Boehmians, see [9]. For usual Boehmians, see [10–19].

2. Strong Boehmians

Denote by ℝ₊ the set of positive real numbers. By 𝔡(ℝ₊) denote the Schwartz space of test functions of compact support over ℝ₊ and by 𝔢(Ω) the space of smooth functions over Ω, where Ω=ℝ₊₁×ℝ₊, (ℝ₊₁=[1,∞)). The dual space of 𝔢(Ω), namely 𝔢^′(Ω), consists of distributions of compact support [14, 20, 21].

Between 𝑓∈𝔢(Ω) and 𝜐∈𝔡(ℝ₊), the convolution product ⊗ of 𝑓 and 𝜐 is given by 𝑓⊗𝜐(𝑥)=_ℝ+𝑓(𝛼,𝑡)𝜐(𝑥−𝑡)𝑑𝑡, (8) where 𝛼∈ℝ₊₁.

Since notations ⊗ and ⊛ denote a same product, when 𝑓 is either defined over Ω or ℝ₊, the role of ⊗ and ⊛ may be interchanged in the text of this paper.

Let 𝔲(ℝ₊) be the subset of 𝔡(ℝ₊) of test functions such that _ℝ+𝜐(𝑥)𝑑𝑥=1. (9) The pair (𝑓,𝜐) of functions, 𝑓∈𝔢(Ω), 𝜐∈𝔲(ℝ₊), is said to be a quotient of functions, denoted as 𝑓/𝜐, if and only if 𝑓(𝛼,𝑥)⊗𝚍_𝛽𝜐(𝑥)=𝑓𝛽,𝑥⊗𝚍_𝛼𝜐(𝑥), (10) where 𝚍_𝑟𝜐(𝑥)=𝑟𝜐(𝑟𝑥),𝛼,𝛽∈ℝ₊₁. (11) Two quotients 𝑓/𝜐 and 𝑔/𝜓 are said to be equivalent, 𝑓/𝜐∼𝑔/𝜓, if and only if 𝑓(𝛼,𝑥)⊗𝚍_𝛽𝜓(𝑥)=𝑔𝛽,𝑥⊗𝚍_𝛼𝜐(𝑥),𝛼,𝛽∈ℝ₊₁. (12)

Let the set 𝙰 be given as 𝑓𝙰=𝜐∶∀𝑓∈𝔢(Ω),𝜐∈𝔲ℝ₊. (13) Then the equivalence class [𝑓/𝜐] containing 𝑓/𝜐 is said to be a strong Boehmian.

The space of all such Boehmians is denoted by £_st(𝔢,𝔲,⊗) and is the so-called the space of strong Boehmians.

Following conclusions are useful in the sequel [9, p.p. 886].(𝔠₁)Let 𝜐,𝜓∈𝔲(ℝ₊); then 𝜐⊛𝜓∈𝔲(ℝ₊).(𝔠₂)Let 𝑓∈𝔢(Ω) and 𝜐∈𝔲(ℝ₊); then 𝑓⊗𝜐∈𝔢(Ω).(𝔠₃)Let (𝑓/𝜐)∈𝙰 and 𝜓∈𝔲(ℝ₊); then we have that (𝑓⊗𝜓/𝜐⊛𝜓)∈𝙰 and 𝑓𝜐isequivalentto𝑓⊗𝜓.𝜐⊛𝜓 (14)(𝔠₄)Let 𝜐∈𝔲(ℝ₊); then 𝚍_𝑟𝜐∈𝔲(ℝ₊),𝑟≥1.(𝔠₅)Let (𝑓/𝜐)∈𝙰, 𝑤>0 and 𝑔(𝛼,𝑥)=𝑓(𝛼+𝑤,𝑥); then we have 𝑔𝜓𝑔∈𝙰,𝜓∼𝑓𝜐, (15) where 𝜓=𝚍_𝑤𝜐.

Addition and scalar multiplications in £_st(𝔢,𝔲,⊗) are defined in the usual way as 𝑓𝜐+𝑔𝜓=𝑓⊗𝜓+𝑔⊗𝜐,𝜆𝑓𝜐⊛𝜓𝜐=𝜆𝑓𝜐. (16) Differentiation and the operation ⊗ in the space in £_st(𝔢,𝔲,⊗) are given as 𝒟^𝑘𝑓𝜐=𝒟^𝑘𝑓𝜐,𝑓𝜐⊗𝜓=𝑓⊗𝜓𝜐. (17)

Convergence, in £_st(𝔢,𝔲,⊗), is defined as follows. A sequence (𝑥_𝑛) of strong Boehmians is said to be 𝔲 convergent to a Boehmian 𝑥 in £_st(𝔢,𝔲,⊗) if for some 𝑓_𝑛,𝑓∈𝔢(Ω), 𝑛∈ℕ, and 𝜐∈𝔲(ℝ₊) such that 𝑥_𝑛=[𝑓_𝑛/𝜐], 𝑥=[𝑓/𝜐], we have 𝑓_𝑛→𝑓 as 𝑛→∞ in £_st(𝔢,𝔲,⊗).

3. General Boehmians

Boehmians were first constructed as a generalization of regular Mikusinski operators [16]. The minimal structure necessary for the construction of Boehmians consists of the following elements: (i) A set ℑ; (ii) a commutative semigroup (ℜ,*); (iii) an operation ⨀∶ℑ×ℜ→ℑ such that for each 𝑥∈ℑ and 𝜐₁, 𝜐₂, a collection Δ⊂ℜ, 𝑥⨀(𝜐₁∗𝜐₂)=(𝑥⨀𝜐₁)⨀𝜐₂; (𝑣𝑖) a collection Δ⊂ℜ^ℕ such that (𝑎) if 𝑥,𝑦∈ℑ, (𝜐_𝑛)∈Δ, 𝑥⨀𝜐_𝑛=𝑦⨀𝜐_𝑛  ∀𝑛, then 𝑥=𝑦; (𝑏) if (𝜐_𝑛),(𝜎_𝑛)∈Δ, then (𝜐_𝑛∗𝜎_𝑛)∈Δ.  Δ is the set of all delta sequences. Consider 𝒜=𝑥_𝑛,𝜐_𝑛∶𝑥_𝑛∈ℑ,𝜐_𝑛∈Δ,𝑥_𝑛𝜐_𝑚=𝑥_𝑚𝜐_𝑛,∀𝑚,𝑛∈ℕ. (18) 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 £_ge(ℑ,ℜ,Δ). Elements of £_ge(ℑ,ℜ,Δ) are general Boehmians. Between ℑ and £_ge(ℑ,ℜ,Δ) there is a canonical embedding expressed as 𝑥→(𝑥⨀𝑠_𝑛)/𝑠_𝑛. The operation ⨀ can be extended to £_ge(ℑ,ℜ,Δ)×ℑ by 𝑥_𝑛/𝜐_𝑛⨀𝑡=(𝑥_𝑛⨀𝑡)/𝜐_𝑛. In £_ge(ℑ,ℜ,Δ), there are two types of convergence. (1) A sequence (ℎ_𝑛) in £_ge(ℑ,ℜ,Δ) is said to be 𝛿 convergent to ℎ in £_ge(ℑ,ℜ,Δ), denoted by ℎ→_𝑛^𝛿ℎ, if there exists a delta sequence (𝜐_𝑛) such that (ℎ_𝑛⨀𝜐_𝑛),   (ℎ⨀𝜐_𝑛)∈ℑ,  ∀𝑘,𝑛∈ℕ, and (ℎ_𝑛⨀𝜐_𝑘)→(ℎ⨀𝜐_𝑘) as 𝑛→∞, in ℑ, for every 𝑘∈ℕ. (2) A sequence (ℎ_𝑛) in £_ge(ℑ,ℜ,Δ) is said to be Δ convergent to ℎ in £_ge(ℑ,ℜ,Δ), denoted by ℎ→_𝑛^Δℎ, if there exists a (𝜐_𝑛)∈Δ such that (ℎ_𝑛−ℎ)⨀𝜐_𝑛∈ℑ, ∀𝑛∈ℕ, and (ℎ_𝑛−ℎ)⨀𝜐_𝑛→0 as 𝑛→∞ in ℑ.

The following is equivalent for the statement of 𝛿 convergence: ℎ→_𝑛^𝛿ℎ  (𝑛→∞) in £_ge(ℑ,ℜ,Δ)  if and only if there is 𝑓_𝑛,𝑘,𝑓_𝑘∈ℑ 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 ℑ.

For further discussion see [10–19].

4. Sumudu Transform of Strong Boehmians

We start investigation by establishing the following theorem.

Theorem 1 (the convolution theorem). Let 𝑓∈𝔢(Ω) and 𝜐∈𝔲(ℝ₊). Then one has 𝜇𝑓⊗𝚍_𝛽𝜐(𝜁)=𝜁𝑓(𝛼,𝜁)𝚍_𝛽𝜐(𝜁), (19) where 𝚍_𝛽𝜐(𝜁)=𝛽𝜐(𝛽𝜁), 𝜁∈ℝ₊.

Proof. By using the definition of Sumudu transform, Fubini’s theorem then gives 𝜇𝑓(𝛼,𝑥)⊗𝚍_𝛽𝜐(𝑥𝜁)=)(_ℝ+𝑓(𝛼,𝑡)𝑑𝑡_ℝ+𝚍_𝛽𝜐()𝑒𝑥−𝑡^−𝑥/𝜁𝜁𝑑𝑥. (20)
The substitution 𝛽𝑥−𝛽𝑡=𝑦 implies 𝑥=(𝑦+𝛽𝑡)/𝛽 and, hence 𝜇𝑓(𝛼,𝑥)⊗𝚍_𝛽𝜐(𝑥𝜁)=)(_ℝ+𝑓(𝛼,𝑡)𝑑𝑡_ℝ+𝚍_𝛽𝜐𝑦𝑒^{(−𝑦+𝛽𝑡)/𝛽𝜁}𝛽𝜁𝑑𝑦. (21)
Hence 𝜇𝑓(𝛼,𝑥)⊗𝚍_𝛽𝜐(𝑥)(𝜁)=𝜁𝑓(𝛼,𝜁)𝚍_𝛽𝜐(𝜁). (22)
This completes the proof of the theorem.

Next, we are describing a usual space of Boehmians by images of Sumudu transforms of strong Boehmians.

Consider the following definition.

Definition 2. Let Δ₁(ℝ₊), or simply Δ₁, be a set of delta sequences such that (𝜐_𝑛)∈𝔲(ℝ₊) and supp𝜐_𝑛⊂(0,𝜀_𝑛),  𝜀_𝑛>0,  𝜀_𝑛→0 as 𝑛→∞.
Let 𝔫(ℝ₊) be the set of images of Sumudu transforms of all 𝔲(ℝ₊) elements and, Δ₂(ℝ₊) be the set of Sumudu transforms of all delta sequences from Δ₁.
For 𝑓∈𝔢(Ω) and 𝜐∈𝔫(ℝ₊), we introduce the operation • as 𝑓(𝛼,𝜁)•𝜐(𝜁)=𝜁𝑓(𝛼,𝜁)𝚍_𝛽𝜐(𝜁),𝛽∈ℝ₊₁. (23)

Lemma 3. Let 𝑓∈𝔢(Ω) and 𝜐∈𝔫(ℝ₊); then 𝑓•𝜐∈𝔢(Ω), ∀𝑛∈ℕ.
Proof is immediate since 𝑓∈𝔢(Ω) and 𝚍_𝛽𝜐∈𝔫(ℝ₊).

Lemma 4. The following hold true. (i)If (𝜐_𝑛),(𝜓_𝑛)∈Δ₂(ℝ₊), then 𝜐_𝑛•𝜓_𝑛∈Δ₂(ℝ₊),  ∀𝑛∈ℕ.(ii)Let 𝑓,𝑔∈𝔢(Ω) and (𝜐_𝑛)∈Δ₂(ℝ₊) be such that 𝑓(𝛼,𝜁)•𝜐_𝑛(𝜁)=𝑔(𝛽,𝜁)•𝜐_𝑛(𝜁); then 𝑓(𝛼,𝜁)=𝑔𝛽,𝜁,𝛼,𝛽∈ℝ₊₁. (24)

Proof. Proof of (i). For all (𝜐_𝑛),(𝜓_𝑛)∈Δ₂(ℝ₊) we can find their related sequences (𝜐_𝑛),(𝜓_𝑛)∈Δ₁(ℝ₊). Since 𝜐_𝑛⊛𝜓_𝑛∈Δ₁(ℝ₊),  ∀𝑛, it follows 𝜇𝜐_𝑛⊛𝜓_𝑛=𝜁𝜐_𝑛(𝜁)𝜓_𝑛(𝜁)=𝜐_𝑛•𝜓_𝑛∈Δ₂ℝ₊, (25)∀𝑛∈ℕ, by (23).
This proves Part (i) of the lemma.
Proof of (ii). For each delta sequence (𝜐_𝑛)∈Δ₂(ℝ₊) we see that 𝚍_𝛽𝜐_𝑛⟶1𝜁,𝜐_𝑛⟶1𝜁as𝑛⟶∞. (26) Hence 𝑓(𝛼,𝜁)•𝜐_𝑛(𝜁)=𝜁𝑓(𝛼,𝜁)𝚍_𝛽𝜐_𝑛(𝜁)()𝑔•𝜐⟶𝑓𝛼,𝜁as𝑛⟶∞,𝛽,𝜁_𝑛(𝜁)=𝜁𝑔𝛽,𝜁𝚍_𝛼𝜐_𝑛(𝜁)⟶𝑔𝛽,𝜁as𝑛⟶∞. (27)
This implies 𝑓(𝛼,𝜁)=𝑔𝛽,𝜁,∀𝜁∈ℝ₊,𝛼,𝛽∈ℝ₊₁. (28)
This completes the proof of the lemma.

Hence, from Lemma 4, (Δ₂(ℝ₊),⊛) can be regarded as delta sequence in 𝔫(ℝ₊).

Lemma 5. The mapping 𝔢(Ω)×𝔫(ℝ₊)→𝔫(ℝ₊) defined by 𝑓(𝛼,𝜁)•𝜐(𝜁)=𝜁𝑓(𝛼,𝜁)𝚍_𝛽𝜐(𝜁). (29)
Satisfies the following properties. (i)𝜐_𝑛•𝜓_𝑛=𝜓_𝑛•𝜐_𝑛, for every (𝜐_𝑛),(𝜓_𝑛)∈Δ₁, 𝑛∈ℕ.(ii)If 𝑓,𝑔∈𝔢(Ω) and (𝜐_𝑛)∈Δ₂(ℝ₊), then (𝑓+𝑔)•𝜐_𝑛=𝑓•𝜐_𝑛+𝑔•𝜐_𝑛.(iii)If 𝑓∈𝔢(Ω), (𝜐_𝑛),(𝜓_𝑛)∈Δ₂(ℝ₊), then (𝑓•𝜐_𝑛)•𝜓_𝑛=𝑓•(𝜐_𝑛•𝜓_𝑛).
Proof is straightforward from definitions.
For similar proofs see [15] and other cited papers.

Lemma 6. The following hold true. (i)If 𝑓_𝑛→𝑓∈𝔢(Ω) and 𝜐∈𝔫(ℝ₊) then 𝑓_𝑛•𝜐→𝑓•𝜐 as 𝑛→∞.(ii)If 𝑓_𝑛→𝑓∈𝔢(Ω) and (𝜐_𝑛)∈Δ₂(ℝ₊) then 𝑓_𝑛•𝜐_𝑛→𝑓 as 𝑛→∞.

Proof. Proof of (i). Let 𝜐∈𝔫(ℝ₊), 𝑓_𝑛,𝑓∈𝔢(Ω), ∀𝑛. The hypothesis of the lemma implies 𝒟^𝑘_𝜁𝑓_𝑛()•𝜐(𝜁)()•𝜐(𝜁)=𝒟𝛼,𝜁−𝑓𝛼,𝜁_𝑘𝜁𝜁𝚍_𝛽𝜐(𝜁)𝑓_𝑛−𝑓(𝛼,𝜁)⟶0 (30)
as 𝑛→∞ in 𝔢(Ω).
Proof of (ii). Let (𝜐_𝑛)∈Δ₂(ℝ₊); then the fact that 𝚍_𝛽𝜐_𝑛(𝜁)→1/𝜁 as 𝑛→∞ implies 𝒟^𝑘_𝜁𝑓_𝑛•𝜐_𝑛(𝜁)()⟶𝒟−𝑓𝛼,𝜁_𝑘𝜁𝑓_𝑛(𝛼,𝜁)−𝑓(𝛼,𝜁)⟶0 (31)
as 𝑛→∞ in 𝔢(Ω).
This completes the proof of the lemma.

The general Boehmian space £_ge(𝔢,𝔫,Δ₂,•), or £_ge, is, therefore, constructed.

The sum and multiplication by a scalar of two Boehmians in £_ge are defined in a natural way as 𝑓_𝑛𝜐_𝑛+𝑔_𝑛𝜓_𝑛=𝑓_𝑛•𝜓_𝑛+𝑔_𝑛•𝜐_𝑛𝜐_𝑛•𝜓_𝑛,𝛼𝑓_𝑛𝜐_𝑛=𝛼𝑓_𝑛𝜐_𝑛,𝛼∈ℂ. (32) The operation • and differentiation in £_ge are, respectively, defined by 𝑓_𝑛𝜐_𝑛•𝑔_𝑛𝜓_𝑛=𝑓_𝑛•𝑔_𝑛𝜐_𝑛•𝜓_𝑛,𝒟^𝛼𝑓_𝑛𝜐_𝑛=𝒟^𝛼𝑓_𝑛𝜐_𝑛. (33) On the other hand, we are concerned with the strong space of Boehmians that can be described by the set (𝔢,⊗) and the subset (𝔲,⊛), injected with the family Δ₁ of delta sequences. Such a space is denoted by £_st(𝔢,(𝔲,⊛),Δ₁,⊗), or, simply, by £_st, for more convenience. With this injection, the space £_st preserves the operations of addition, scalar multiplication, differentiation, and the convolution given to general Boehmians. Theorem 1 then suggests the following definition.

Definition 7. Let 𝑓∈𝔢(Ω) and 𝜐∈𝔲(ℝ₊); then we define the Sumudu transform of the strong Boehmian [𝑓_𝑛/𝜐_𝑛] in £_st by ð𝑓_𝑛𝜐_𝑛=𝑓_𝑛𝚍_𝛼𝜐_𝑛∈£_ge, (34) where 𝚍_𝛼 has its usual meaning.

Theorem 8. The Sumudu transform ð∶£_st→£_ge is well-defined.

Proof. Let [𝑓_𝑛/𝜐_𝑛],[𝑔_𝑛/𝜓_𝑛]∈£_st be such that [𝑓_𝑛/𝜐_𝑛]=[𝑔_𝑛/𝜓_𝑛]; then using (12) we get 𝑓_𝑛(𝛼,𝑥)⊗𝚍_𝛽𝜓_𝑛(𝑥)=𝑔_𝑛𝛽,𝑥⊗𝚍_𝛼𝜐_𝑛(𝑥). (35) Applying the convolution theorem on both sides of (35) yields 𝑔_𝑛𝚍_𝛽𝜓_𝑛=𝑓_𝑛𝚍_𝛼𝜐_𝑛. (36) Hence ð𝑓_𝑛𝜐_𝑛𝑔=ð^𝑛𝜓_𝑛. (37) This completes the proof of the theorem.

Theorem 9. Let (𝜓_𝑛),(𝜐_𝑛)∈Δ₁(ℝ₊) and 𝑓,𝑔∈𝔢(Ω); then the mapping ð∶£_st→£_ge is one-one.

Proof. Assume ð[𝑓_𝑛/𝜐_𝑛]=ð[𝑔_𝑛/𝜓_𝑛] in £_ge. Using (34) we get 𝑓_𝑛(𝛼,𝜁)•𝜓_𝑛(𝜁)=𝑔_𝑛𝛽,𝜁•𝜐_𝑛(𝜁). (38)
By (23) we get 𝜁𝑓_𝑛(𝛼,𝜁)𝚍_𝛽𝜓_𝑛(𝜁)=𝜁𝑔_𝑛𝛽,𝜁𝚍_𝛼𝜐_𝑛(𝜁). (39)
Using Theorem 1, (39) becomes 𝜇𝑓_𝑛(𝛼,𝑥)⊗𝚍_𝛽𝜓_𝑛(𝑥)=𝜇𝑔_𝑛𝛽,𝑥⊗𝚍_𝛼𝜐_𝑛(𝑥). (40)
Since 𝜇 is one-one we get 𝑓_𝑛(𝛼,𝑥)⊗𝚍_𝛽𝜓_𝑛(𝑥)=𝑔_𝑛𝛽,𝑥⊗𝚍_𝛼𝜐_𝑛(𝑥). (41)
The property of equivalence classes in £_st, therefore, implies 𝑓_𝑛(𝛼,𝑥)𝜐_𝑛(𝑥)∼𝑔_𝑛𝛽,𝑥𝜓_𝑛(𝑥). (42)
Hence [𝑓_𝑛/𝜐_𝑛]=[𝑔_𝑛/𝜓_𝑛].
This completes the proof of the theorem

Theorem 10. ð∶£_st→£_ge is continuous with respect to 𝔲 convergence.

Proof. Let 𝑥_𝑛→𝑥∈£_st. By using 𝔲 convergence in £_st and [9, Theorem 2.6] we can find common 𝜐 for all 𝑥_𝑛 such that 𝑥_𝑛=[𝑓_𝑛/𝜐], 𝑥=[𝑓/𝜐] and 𝑓_𝑛→𝑓 as 𝑛→∞. Hence 𝑓_𝑛→𝑓 as 𝑛→∞. Therefore 𝑓_𝑛𝚍_𝛼𝜐⟶𝑓𝚍_𝛼𝜐as𝑛⟶∞. (43)
Hence, ð𝑥_𝑛→ð𝑥 as 𝑛→∞ in £_ge.
This completes the proof of the theorem.

Definition 11. Let 𝑦=[𝑓_𝑛/𝚍_𝛼𝜐_𝑛] in £_ge; then we define the inverse ð⁻¹ of ð by the formula ð⁻¹𝑓𝑦=^𝑛𝜐_𝑛, (44)
in £_st.

Theorem 12. The mapping ð⁻¹∶£_ge→£_st is well-defined.

Proof. Let [𝑓_𝑛/𝚍_𝛼𝜐_𝑛]=[𝑔_𝑛/𝚍_𝛽𝜓_𝑛]; then 𝑓_𝑛(𝛼,𝜁)•𝜓_𝑚(𝜁)=𝑔_𝑚(𝛽,𝜁)•𝜐_𝑛(𝜁). Using (23) we get 𝜁𝑓_𝑛(𝛼,𝜁)𝚍_𝛽𝜓_𝑚(𝜁)=𝜁𝑔_𝑚𝛽,𝜁𝚍_𝛼𝜐_𝑛(𝜁). (45)
Theorem 1 implies 𝜇(𝑓_𝑛(𝛼,𝑥)⊗𝚍_𝛽𝜓𝑚_𝑛(𝑥))=𝜇(𝑔_𝑚(𝛽,𝑥)⊗𝚍_𝛼𝜐_𝑛(𝑥)). Hence 𝑓_𝑛(𝛼,𝑥)⊗𝚍_𝛽𝜓_𝑚(𝑥)=𝑔_𝑚𝛽,𝑥⊗𝚍_𝛼𝜐_𝑛(𝑥). (46)
This completes the proof of the Theorem.

Theorem 13. The mapping ð⁻¹∶£_ge→£_st is linear.

Proof. Let [𝑓_𝑛/𝚍_𝛼𝜐_𝑛],[𝑔_𝑛/𝚍_𝛽𝜓_𝑛]∈£_ge and 𝑘∈ℝ₊; then using (23) we write ð⁻¹𝑓_𝑛𝚍_𝛼𝜐_𝑛+𝑔_𝑛𝚍_𝛽𝜓_𝑛=ð⁻¹𝜁𝑓_𝑛(𝛼,𝜁)𝚍_𝛽𝜓_𝑛(𝜁)+𝜁𝑔_𝑛𝛽,𝜁𝚍_𝛼𝜐_𝑛(𝜁)𝚍_𝛼𝜐_𝑛(𝜁)⊛𝚍_𝛽𝜓_𝑛(𝜁)=𝑓_𝑛(𝛼,𝑥)⊗𝚍_𝛽𝜓_𝑛(𝑥)+𝑔_𝑛𝛽,𝑥⊗𝚍_𝛼𝜐_𝑛(𝑥)𝚍_𝛼𝜐_𝑛(𝑥)⊛𝚍_𝛽𝜓_𝑛(𝑥)=𝑓_𝑛𝜐_𝑛+𝑔_𝑛𝜓_𝑛. (47)
Also, 𝑘ð⁻¹([𝑓_𝑛/𝚍_𝛼𝜐_𝑛])=ð⁻¹([𝑘𝑓_𝑛/𝚍_𝛼𝜐_𝑛]) is obvious.
The proof is completed.

Theorem 14. The mapping ð⁻¹∶£_ge→£_st is continuous with respect to 𝛿 convergence.

Proof. Let 𝑦_𝑛→𝑦∈£_ge, 𝑦_𝑛=[𝑓_𝑛,𝑘/𝚍_𝛼𝜐_𝑘], 𝑦=[𝑓_𝑘/𝚍_𝛼𝜐_𝑘], and 𝑓_𝑛,𝑘→𝑓_𝑘 as 𝑛→∞. Applying the inverse Sumudu transform yields 𝑓_𝑛,𝑘→𝑓_𝑘 as 𝑛→∞. Thus 𝑓_𝑛/𝜐_𝑛→𝑓/𝜐_𝑛 as 𝑛→∞.
This completes the proof of the theorem.

If the family Δ₁ of delta sequences is not described in the strong Boehmian space we have considered in this note, it will be interesting to pay attention to the following remarks.

Useful Remark. (i) The injection 𝔦∶(𝔢,𝔲,⊗)→£_st defined by 𝔦𝑓=𝑓⊗𝚍_𝛼𝜐𝜐, (48)

is continuous and independent of the choice of 𝜐.

(ii) The injection 𝔧∶£_st→£_ge defined by 𝔧𝑓𝜐=𝑓(𝑛,⋅)𝚍_𝑛𝜐, (49)

is continuous.

Proof. Proof of (i) is same as that of [9, Theorem 3.1].
Detailed proof of (ii) is as follows.
Suppose 𝑦_𝑛=[𝑓_𝑛/𝜐], 𝑦=[𝑓/𝜐], and 𝑓_𝑛(𝑛,⋅)→𝑓(𝑛,⋅) as 𝑛→∞ uniformly on compact subsets of Ω. Then 𝑓_𝑛𝜐⊗𝚍_𝑘𝑓𝜐=^𝑛(𝑘,⋅)𝚍_𝑘𝜐=𝑓_𝑛(𝑘,⋅)⊗𝚍_𝑘𝜐𝚍_𝑘𝜐∼𝑓_𝑛(𝑘,⋅)⟶𝑓(𝑘,⋅), (50)
as 𝑛→∞.
Employing (i) we get 𝑓_𝑛𝜐⊗𝚍_𝑘()=𝑓()𝜐⟶𝑓𝑘,⋅𝑘,⋅⊗𝚍_𝑘𝜐𝜐=𝑓(𝑘,⋅)𝜐⊗𝚍_𝑘𝜐. (51)
That is 𝑓_𝑛𝜐⊗𝚍_𝑘𝜐⟶𝑦⊗𝚍_𝑘𝜐, (52)
as 𝑛→∞. Therefore 𝑦_𝑛⊗𝚍_𝑘𝜐⟶𝑦⊗𝚍_𝑘𝜐, (53)
as 𝑛→∞.
This completes the proof of the remark.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.