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

Authors: S. K. Q. Al-Omari
Journal: ISRN Mathematical Analysis
 (2014)

---

## 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.


---

## Body

## 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 Section1. 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 off
(
x
) is given by [1]
(1)
(
μ
f
)
(
ζ
)
=
f
~
(
ζ
)
=
:
1
ζ
∫
ℝ
+
f
(
t
)
exp
⁡
(
-
t
ζ
)
d
t
,
over a set of functions given as
(2)
𝔸
=
{
f
(
t
)
:
∃
m
,
τ
1
,
τ
2
>
0
,
|
f
(
t
)
|
<
m
e
t
/
τ
j
,
(
-
1
)
j
×
(
0
,
∞
]
t
∈
(
-
1
)
j
×
(
0
,
∞
]
}
,
where f
(
t
) is of convergent series type and ζ
∈
(
-
τ
1
,
τ
2
).Denote by(3)
(
f
⊛
g
)
(
x
)
=
∫
ℝ
+
f
(
x
-
t
)
g
(
t
)
d
t
,
the usual convolution product of f and g.Then the Sumudu transform of the convolution product is given as(4)
μ
(
f
⊛
g
)
(
ζ
)
=
ζ
f
~
(
ζ
)
g
~
(
ζ
)
,
where f
~ and g
~ denote the Sumudu transform of f and g, respectively.Following are general properties of Sumudu transforms.(i)
Ifk
1, k
2
∈
ℝ
+ and f
~, g
~ are the Sumudu transforms of f and g, respectively, then
(5)
μ
(
k
1
f
(
t
)
+
k
2
g
(
t
)
)
(
ζ
)
=
k
1
f
~
(
ζ
)
+
k
2
g
~
(
ζ
)
;
(ii)
(6)
μ
f
(
k
t
)
(
ζ
)
=
μ
(
k
ζ
)
,
k
∈
ℝ
+
;
(iii)
(7)
lim
⁡
t
→
0
f
(
t
)
=
lim
⁡
ζ
→
0
μ
(
ζ
)
=
f
(
0
)
.For details, see [8].For strong Boehmians, see [9]. For usual Boehmians, see [10–19].


## 2. Strong Boehmians

Denote byℝ
+ 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].Betweenf
∈
𝔢
(
Ω
) and υ
∈
𝔡
(
ℝ
+
), the convolution product ⊗ of f and υ is given by
(8)
(
f
⊗
υ
)
(
x
)
=
∫
ℝ
+
f
(
α
,
t
)
υ
(
x
-
t
)
d
t
,
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.Let𝔲
(
ℝ
+
) be the subset of 𝔡
(
ℝ
+
) of test functions such that
(9)
∫
ℝ
+
υ
(
x
)
d
x
=
1
.
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
(10)
f
(
α
,
x
)
⊗
d
β
υ
(
x
)
=
f
(
β
,
x
)
⊗
d
α
υ
(
x
)
,
where
(11)
d
r
υ
(
x
)
=
r
υ
(
r
x
)
,
α
,
β
∈
ℝ
+
1
.
Two quotients f
/
υ and g
/
ψ are said to be equivalent, f
/
υ
~
g
/
ψ, if and only if
(12)
f
(
α
,
x
)
⊗
d
β
ψ
(
x
)
=
g
(
β
,
x
)
⊗
d
α
υ
(
x
)
,
iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii
α
,
β
∈
ℝ
+
1
.Let the setA be given as
(13)
A
=
{
f
υ
:
∀
f
∈
𝔢
(
Ω
)
,
υ
∈
𝔲
(
ℝ
+
)
}
.
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].(𝔠
1)
Letυ
,
ψ
∈
𝔲
(
ℝ
+
); then υ
⊛
ψ
∈
𝔲
(
ℝ
+
).
(𝔠
2)
Letf
∈
𝔢
(
Ω
) and υ
∈
𝔲
(
ℝ
+
); then f
⊗
υ
∈
𝔢
(
Ω
).
(𝔠
3)
Let(
f
/
υ
)
∈
A and ψ
∈
𝔲
(
ℝ
+
); then we have that (
f
⊗
ψ
/
υ
⊛
ψ
)
∈
A and
(14)
f
υ
is
equivalent
to
f
⊗
ψ
υ
⊛
ψ
.
(𝔠
4)
Letυ
∈
𝔲
(
ℝ
+
); then d
r
υ
∈
𝔲
(
ℝ
+
)
,
r
≥
1.
(𝔠
5)
Let(
f
/
υ
)
∈
A, w
>
0 and g
(
α
,
x
)
=
f
(
α
+
w
,
x
); then we have
(15)
g
ψ
∈
A
,
g
ψ
~
f
υ
,
whereψ
=
d
w
υ.Addition and scalar multiplications in£
st
(
𝔢
,
𝔲
,
⊗
) are defined in the usual way as
(16)
[
f
υ
]
+
[
g
ψ
]
=
[
f
⊗
ψ
+
g
⊗
υ
υ
⊛
ψ
]
,
λ
[
f
υ
]
=
[
λ
f
υ
]
.
Differentiation and the operation ⊗ in the space in £
st
(
𝔢
,
𝔲
,
⊗
) are given as
(17)
𝒟
k
[
f
υ
]
=
[
𝒟
k
f
υ
]
,
[
f
υ
]
⊗
ψ
=
[
f
⊗
ψ
υ
]
.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 υ
1, υ
2, a collection Δ
⊂
ℜ, x
⨀
(
υ
1
*
υ
2
)
=
(
x
⨀
υ
1
)
⨀
υ
2; (
v
i
) 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
(18)
𝒜
=
{
(
x
n
,
υ
n
)
:
x
n
∈
ℑ
,
(
υ
n
)
∈
Δ
,
x
n
⨀
υ
m
=
x
m
⨀
υ
n
,
∀
m
,
n
∈
ℕ
}
.
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
→
δ
h, 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
→
Δ
h, 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 ofδ convergence: h
n
→
δ
h  (
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).
Letf
∈
𝔢
(
Ω
) and υ
∈
𝔲
(
ℝ
+
). Then one has
(19)
μ
(
f
⊗
d
β
υ
)
(
ζ
)
=
ζ
f
~
(
α
,
ζ
)
d
β
υ
~
(
ζ
)
,
where d
β
υ
~
(
ζ
)
=
β
υ
~
(
β
ζ
), ζ
∈
ℝ
+.Proof.
By using the definition of Sumudu transform, Fubini’s theorem then gives(20)
μ
(
f
(
α
,
x
)
⊗
d
β
υ
(
x
)
)
(
ζ
)
=
∫
ℝ
+
f
(
α
,
t
)
d
t
∫
ℝ
+
d
β
υ
(
x
-
t
)
e
-
x
/
ζ
ζ
d
x
.
The substitutionβ
x
-
β
t
=
y implies x
=
(
y
+
β
t
)
/
β and, hence
(21)
μ
(
f
(
α
,
x
)
⊗
d
β
υ
(
x
)
)
(
ζ
)
=
∫
ℝ
+
f
(
α
,
t
)
d
t
∫
ℝ
+
d
β
υ
(
y
)
e
(
-
y
+
β
t
)
/
β
ζ
β
ζ
d
y
.
Hence(22)
μ
(
f
(
α
,
x
)
⊗
d
β
υ
(
x
)
)
(
ζ
)
=
ζ
f
~
(
α
,
ζ
)
d
β
υ
~
(
ζ
)
.
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.
LetΔ
1
(
ℝ
+
), or simply Δ
1, be a set of delta sequences such that (
υ
n
)
∈
𝔲
(
ℝ
+
) and supp
⁡
υ
n
⊂
(
0
,
ε
n
),  ε
n
>
0,  ε
n
→
0 as n
→
∞.
Let𝔫
(
ℝ
+
) be the set of images of Sumudu transforms of all 𝔲
(
ℝ
+
) elements and, Δ
2
(
ℝ
+
) be the set of Sumudu transforms of all delta sequences from Δ
1.
Forf
∈
𝔢
(
Ω
) and υ
~
∈
𝔫
(
ℝ
+
), we introduce the operation • as
(23)
f
(
α
,
ζ
)
•
υ
~
(
ζ
)
=
ζ
f
(
α
,
ζ
)
d
β
υ
~
(
ζ
)
,
iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii
β
∈
ℝ
+
1
.Lemma 3.
Letf
∈
𝔢
(
Ω
) and υ
~
∈
𝔫
(
ℝ
+
); then f
•
υ
~
∈
𝔢
(
Ω
), ∀
n
∈
ℕ.
Proof is immediate sincef
∈
𝔢
(
Ω
) and d
β
υ
~
∈
𝔫
(
ℝ
+
).Lemma 4.
The following hold true.(i)
If(
υ
~
n
)
,
(
ψ
~
n
)
∈
Δ
2
(
ℝ
+
), then υ
~
n
•
ψ
~
n
∈
Δ
2
(
ℝ
+
),  ∀
n
∈
ℕ.
(ii)
Letf
,
g
∈
𝔢
(
Ω
) and (
υ
~
n
)
∈
Δ
2
(
ℝ
+
) be such that f
(
α
,
ζ
)
•
υ
~
n
(
ζ
)
=
g
(
β
,
ζ
)
•
υ
~
n
(
ζ
); then
(24)
f
(
α
,
ζ
)
=
g
(
β
,
ζ
)
,
α
,
β
∈
ℝ
+
1
.Proof.
Proof of (i). For all(
υ
~
n
)
,
(
ψ
~
n
)
∈
Δ
2
(
ℝ
+
) we can find their related sequences (
υ
n
)
,
(
ψ
n
)
∈
Δ
1
(
ℝ
+
). Since υ
n
⊛
ψ
n
∈
Δ
1
(
ℝ
+
),  ∀
n, it follows
(25)
μ
(
υ
n
⊛
ψ
n
)
=
ζ
υ
~
n
(
ζ
)
ψ
~
n
(
ζ
)
=
υ
~
n
•
ψ
~
n
∈
Δ
2
(
ℝ
+
)
,
∀
n
∈
ℕ, by (23).
This proves Part (i) of the lemma.
Proof of (ii). For each delta sequence(
υ
~
n
)
∈
Δ
2
(
ℝ
+
) we see that
(26)
d
β
υ
~
n
⟶
1
ζ
,
υ
~
n
⟶
1
ζ
as
n
⟶
∞
.
Hence
(27)
f
(
α
,
ζ
)
•
υ
~
n
(
ζ
)
=
ζ
f
(
α
,
ζ
)
d
β
υ
~
n
(
ζ
)
⟶
f
(
α
,
ζ
)
iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii
as
n
⟶
∞
,
g
(
β
,
ζ
)
•
υ
~
n
(
ζ
)
=
ζ
g
(
β
,
ζ
)
d
α
υ
~
n
(
ζ
)
⟶
g
(
β
,
ζ
)
iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii
as
n
⟶
∞
.
This implies(28)
f
(
α
,
ζ
)
=
g
(
β
,
ζ
)
,
∀
ζ
∈
ℝ
+
,
α
,
β
∈
ℝ
+
1
.
This completes the proof of the lemma.Hence, from Lemma4, (
Δ
2
(
ℝ
+
)
,
⊛
) can be regarded as delta sequence in 𝔫
(
ℝ
+
).Lemma 5.
The mapping𝔢
(
Ω
)
×
𝔫
(
ℝ
+
)
→
𝔫
(
ℝ
+
) defined by
(29)
f
(
α
,
ζ
)
•
υ
~
(
ζ
)
=
ζ
f
(
α
,
ζ
)
d
β
υ
~
(
ζ
)
.
Satisfies the following properties.(i)
υ
~
n
•
ψ
~
n
=
ψ
~
n
•
υ
~
n, for every (
υ
n
)
,
(
ψ
n
)
∈
Δ
1, n
∈
ℕ.
(ii)
Iff
,
g
∈
𝔢
(
Ω
) and (
υ
~
n
)
∈
Δ
2
(
ℝ
+
), then (
f
+
g
)
•
υ
~
n
=
f
•
υ
~
n
+
g
•
υ
~
n.
(iii)
Iff
∈
𝔢
(
Ω
), (
υ
~
n
)
,
(
ψ
~
n
)
∈
Δ
2
(
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)
Iff
n
→
f
∈
𝔢
(
Ω
) and υ
~
∈
𝔫
(
ℝ
+
) then f
n
•
υ
~
→
f
•
υ
~ as n
→
∞.
(ii)
Iff
n
→
f
∈
𝔢
(
Ω
) and (
υ
~
n
)
∈
Δ
2
(
ℝ
+
) then f
n
•
υ
~
n
→
f as n
→
∞.Proof.
Proof of (i). Letυ
~
∈
𝔫
(
ℝ
+
), f
n
,
f
∈
𝔢
(
Ω
), ∀
n. The hypothesis of the lemma implies
(30)
|
𝒟
ζ
k
(
f
n
(
α
,
ζ
)
•
υ
~
(
ζ
)
-
f
(
α
,
ζ
)
•
υ
~
(
ζ
)
)
|
=
|
𝒟
ζ
k
ζ
d
β
υ
~
(
ζ
)
(
f
n
-
f
)
(
α
,
ζ
)
|
⟶
0
asn
→
∞ in 𝔢
(
Ω
).
Proof of (ii). Let(
υ
~
n
)
∈
Δ
2
(
ℝ
+
); then the fact that d
β
υ
~
n
(
ζ
)
→
1
/
ζ as n
→
∞ implies
(31)
|
𝒟
ζ
k
(
f
n
•
υ
~
n
(
ζ
)
-
f
(
α
,
ζ
)
)
|
⟶
|
𝒟
ζ
k
(
f
n
(
α
,
ζ
)
-
f
(
α
,
ζ
)
)
|
⟶
0
asn
→
∞ in 𝔢
(
Ω
).
This completes the proof of the lemma.The general Boehmian space£
ge
(
𝔢
,
𝔫
,
Δ
2
,
•
), or £
ge, is, therefore, constructed.The sum and multiplication by a scalar of two Boehmians in£
ge are defined in a natural way as
(32)
[
f
n
υ
~
n
]
+
[
g
n
ψ
~
n
]
=
[
f
n
•
ψ
~
n
+
g
n
•
υ
~
n
υ
~
n
•
ψ
~
n
]
,
α
[
f
n
υ
~
n
]
=
[
α
f
n
υ
~
n
]
,
α
∈
ℂ
.
The operation • and differentiation in £
ge are, respectively, defined by
(33)
[
f
n
υ
~
n
]
•
[
g
n
ψ
~
n
]
=
[
f
n
•
g
n
υ
~
n
•
ψ
~
n
]
,
𝒟
α
[
f
n
υ
~
n
]
=
[
𝒟
α
f
n
υ
~
n
]
.
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 Δ
1 of delta sequences. Such a space is denoted by £
st
(
𝔢
,
(
𝔲
,
⊛
)
,
Δ
1
,
⊗
), 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.
Letf
∈
𝔢
(
Ω
) and υ
∈
𝔲
(
ℝ
+
); then we define the Sumudu transform of the strong Boehmian [
f
n
/
υ
n
] in £
st by
(34)
ð
[
f
n
υ
n
]
=
[
f
n
~
d
α
υ
~
n
]
∈
£
ge
,
where d
α has its usual meaning.Theorem 8.
The Sumudu transformð
:
£
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
(35)
f
n
(
α
,
x
)
⊗
d
β
ψ
n
(
x
)
=
g
n
(
β
,
x
)
⊗
d
α
υ
n
(
x
)
.
Applying the convolution theorem on both sides of (35) yields
(36)
g
~
n
d
β
ψ
~
n
=
f
n
~
d
α
υ
~
n
.
Hence
(37)
ð
[
f
n
υ
n
]
=
ð
[
g
n
ψ
n
]
.
This completes the proof of the theorem.Theorem 9.
Let(
ψ
n
)
,
(
υ
n
)
∈
Δ
1
(
ℝ
+
) and f
,
g
∈
𝔢
(
Ω
); then the mapping ð
:
£
st
→
£
ge is one-one.Proof.
Assumeð
[
f
n
/
υ
n
]
=
ð
[
g
n
/
ψ
n
] in £
ge. Using (34) we get
(38)
f
n
~
(
α
,
ζ
)
•
ψ
~
n
(
ζ
)
=
g
~
n
(
β
,
ζ
)
•
υ
~
n
(
ζ
)
.
By (23) we get
(39)
ζ
f
n
~
(
α
,
ζ
)
d
β
ψ
~
n
(
ζ
)
=
ζ
g
~
n
(
β
,
ζ
)
d
α
υ
~
n
(
ζ
)
.
Using Theorem1, (39) becomes
(40)
μ
(
f
n
(
α
,
x
)
⊗
d
β
ψ
n
(
x
)
)
=
μ
(
g
n
(
β
,
x
)
⊗
d
α
υ
n
(
x
)
)
.
Sinceμ is one-one we get
(41)
f
n
(
α
,
x
)
⊗
d
β
ψ
n
(
x
)
=
g
n
(
β
,
x
)
⊗
d
α
υ
n
(
x
)
.
The property of equivalence classes in£
st, therefore, implies
(42)
f
n
(
α
,
x
)
υ
n
(
x
)
~
g
n
(
β
,
x
)
ψ
n
(
x
)
.
Hence[
f
n
/
υ
n
]
=
[
g
n
/
ψ
n
].
This completes the proof of the theoremTheorem 10.
ð
:
£
st
→
£
ge is continuous with respect to 𝔲 convergence.Proof.
Letx
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
(43)
f
n
~
d
α
υ
~
⟶
f
~
d
α
υ
~
as
n
⟶
∞
.
Hence,ð
x
n
→
ð
x as n
→
∞ in £
ge.
This completes the proof of the theorem.Definition 11.
Lety
=
[
f
n
~
/
d
α
υ
~
n
] in £
ge; then we define the inverse ð
-
1 of ð by the formula
(44)
ð
-
1
y
=
[
f
n
υ
n
]
,
in£
st.Theorem 12.
The mappingð
-
1
:
£
ge
→
£
st is well-defined.Proof.
Let[
f
n
~
/
d
α
υ
~
n
]
=
[
g
~
n
/
d
β
ψ
~
n
]; then f
n
~
(
α
,
ζ
)
•
ψ
~
m
(
ζ
)
=
g
m
~
(
β
,
ζ
)
•
υ
~
n
(
ζ
). Using (23) we get
(45)
ζ
f
n
~
(
α
,
ζ
)
d
β
ψ
~
m
(
ζ
)
=
ζ
g
m
~
(
β
,
ζ
)
d
α
υ
~
n
(
ζ
)
.
Theorem1 implies μ
(
f
n
(
α
,
x
)
⊗
d
β
ψ
m
n
(
x
)
)
=
μ
(
g
m
(
β
,
x
)
⊗
d
α
υ
n
(
x
)
). Hence
(46)
f
n
(
α
,
x
)
⊗
d
β
ψ
m
(
x
)
=
g
m
(
β
,
x
)
⊗
d
α
υ
n
(
x
)
.
This completes the proof of the Theorem.Theorem 13.
The mappingð
-
1
:
£
ge
→
£
st is linear.Proof.
Let[
f
n
~
/
d
α
υ
~
n
]
,
[
g
~
n
/
d
β
ψ
~
n
]
∈
£
ge and k
∈
ℝ
+; then using (23) we write
(47)
ð
-
1
[
f
n
~
d
α
υ
~
n
]
+
[
g
~
n
d
β
ψ
~
n
]
=
ð
-
1
[
ζ
f
n
~
(
α
,
ζ
)
d
β
ψ
~
n
(
ζ
)
+
ζ
g
~
n
(
β
,
ζ
)
d
α
υ
~
n
(
ζ
)
d
α
υ
~
n
(
ζ
)
⊛
d
β
ψ
~
n
(
ζ
)
]
=
[
f
n
(
α
,
x
)
⊗
d
β
ψ
n
(
x
)
+
g
n
(
β
,
x
)
⊗
d
α
υ
n
(
x
)
d
α
υ
n
(
x
)
⊛
d
β
ψ
n
(
x
)
]
=
[
f
n
υ
n
]
+
[
g
n
ψ
n
]
.
Also,k
ð
-
1
(
[
f
n
~
/
d
α
υ
~
n
]
)
=
ð
-
1
(
[
k
f
n
~
/
d
α
υ
~
n
]
) is obvious.
The proof is completed.Theorem 14.
The mappingð
-
1
:
£
ge
→
£
st is continuous with respect to δ convergence.Proof.
Lety
n
→
y
∈
£
ge, y
n
=
[
f
n
,
k
~
/
d
α
υ
~
k
], y
=
[
f
k
~
/
d
α
υ
~
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 familyΔ
1 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
(48)
𝔦
(
f
)
=
[
f
⊗
d
α
υ
υ
]
,is continuous and independent of the choice ofυ.(ii) The injection𝔧
:
£
st
→
£
ge defined by
(49)
𝔧
[
f
υ
]
=
[
f
(
n
,
·
)
d
n
υ
]
,is continuous.Proof.
Proof of (i) is same as that of [9, Theorem 3.1].
Detailed proof of (ii) is as follows.
Supposey
n
=
[
f
n
/
υ
], y
=
[
f
/
υ
], and f
n
(
n
,
·
)
→
f
(
n
,
·
) as n
→
∞ uniformly on compact subsets of Ω. Then
(50)
[
f
n
υ
]
⊗
d
k
υ
=
[
f
n
(
k
,
·
)
d
k
υ
]
=
[
f
n
(
k
,
·
)
⊗
d
k
υ
d
k
υ
]
~
f
n
(
k
,
·
)
⟶
f
(
k
,
·
)
,
asn
→
∞.
Employing (i) we get(51)
[
f
n
υ
]
⊗
d
k
υ
⟶
f
(
k
,
·
)
=
[
f
(
k
,
·
)
⊗
d
k
υ
υ
]
=
[
f
(
k
,
·
)
υ
]
⊗
d
k
υ
.
That is(52)
[
f
n
υ
]
⊗
d
k
υ
⟶
y
⊗
d
k
υ
,
asn
→
∞. Therefore
(53)
y
n
⊗
d
k
υ
⟶
y
⊗
d
k
υ
,
asn
→
∞.
This completes the proof of the remark.



---

*Source: 463901-2014-04-24.xml*