# Some Definition of Hartley-Hilbert and Fourier-Hilbert Transforms in a Quotient Space of Boehmians

Authors: S. K. Q. Al-Omari
Journal: Journal of Mathematics
 (2014)

---

## Abstract

We investigate the Hartley-Hilbert and Fourier-Hilbert transforms on a quotient space of Boehmians. The investigated transforms are well-defined and linear mappings in the space of Boehmians. Further properties are also obtained.


---

## Body

## 1. Introduction

The Hilbert transform of a functionf
(
y
) via the Hartley transform is defined in [1, 2] as
(1)
f
h
h
y
=
h
h
f
y
=
1
π
∫
0
∞
f
h
o
x
cos
⁡
x
y
+
f
h
e
x
sin
x
y
d
x
,
where (
f
)
h
e and (
f
)
h
o are, respectively, the even and odd components of the Hartley transform (
f
)
h given as [3]
(2)
f
h
x
=
∫
-
∞
∞
f
y
cas
x
y
d
y
≔
f
h
e
x
+
f
h
o
x
,
where cas
(
x
y
)
=
cos
⁡
⁡
(
x
y
)
+
sin
⁡
(
x
y
).Letf
(
y
) be a casual function; that is, f
(
y
)
=
0, for y
>
0, and then (
f
)
h
o
(
x
) and (
f
)
h
e
(
x
) are related by a Hilbert transform pair as [3]
(3)
f
h
o
x
=
1
π
∫
-
∞
∞
f
h
e
y
x
-
y
d
y
,
f
h
e
x
=
1
π
∫
-
∞
∞
f
h
o
y
x
-
y
d
y
.The Hilbert transform off
(
y
) via the Fourier transform is defined by
(4)
f
f
h
y
=
f
h
f
y
=
1
π
∫
0
∞
f
f
i
x
cos
⁡
x
y
-
f
f
r
x
sin
x
y
d
x
,
where (
f
)
f
r
(
x
) and (
f
)
f
i
(
x
) are, respectively, the real and imaginary components of the Fourier transform given as
(5)
f
f
x
=
∫
-
∞
∞
f
y
e
-
i
x
y
d
y
≔
f
f
r
x
-
i
f
f
i
x
.The Hartley transform is extended to Boehmians in [4] and to strong Boehmians in [5]. The Hartley-Hilbert and Fourier-Hilbert transforms were discussed in various spaces of distributions and spaces of Boehmians in [1, 6].In this paper,i aim at investigating the Hartley-Hilbert transform on the context of Boehmians. Investigating the later transform is analogous.


## 2.  Spaces of Quotients (Spaces of Boehmians)

One of the most youngest generalizations of functions and more particularly of distributions is the theory of Boehmians. The idea of the construction of Boehmians was initiated by the concept of regular operators [7]. Regular operators form a subalgebra of the field of Mikusinski operators and they include only such functions whose support is bounded from the left. In a concrete case, the space of Boehmians contains all regular operators, all distributions, and some objects which are neither operators nor distributions.The construction of Boehmians is similar to the construction of the field of quotients and, in some cases, it gives just the field of quotients. On the other hand, the construction is possible where there are zero divisors, such as spaceC (the space of continuous functions) with the operations of pointwise additions and convolution.A number of integral transforms have been extended to Boehmian spaces in the recent past by many authors such as Roopkumar in [8, 9]; Mikusinski and Zayed in [10]; Al-Omari and Kilicman in [1, 6, 11]; Karunakaran and Vembu [12]; Karunakaran and Roopkumar [13]; Al-Omari et al. in [14]; and many others.For abstract construction of Boehmian spaces we refer to [14–16].By⊛ denote the Mellin-type convolution product of first kind defined by [8]
(6)
ϕ
⊛
ψ
t
=
∫
0
∞
z
-
1
ϕ
t
z
-
1
g
z
d
z
.
Properties of ⊛ are presented as follows [8]:(i)
ϕ
⊛
ψ
t
=
ψ
⊛
ϕ
t;
(ii)
ϕ
+
ψ
⊛
φ
t
=
ϕ
⊛
φ
t
+
ψ
⊛
φ
t;
(iii)
α
ϕ
⊛
ψ
t
=
α
ψ
⊛
ϕ
t; α is complex number;
(iv)
ϕ
⊛
ψ
⊛
φ
t
=
ϕ
⊛
ψ
⊛
φ
t.By*, we denote the convolution product defined by
(7)
ϕ
*
ψ
y
=
∫
0
∞
z
-
1
ϕ
y
z
-
1
ψ
z
d
z
.Byϑ
(
R
+
) denote the space of test functions of bounded supports defined on R
+ that vanish more rapidly than every power of x as x
→
∞. Denote by ϑ
1 the subset of ϑ
(
R
+
) defined by
(8)
ϑ
1
=
φ
∈
ϑ
R
+
:
∫
0
∞
φ
x
d
x
=
1
.
Let Δ be the set of delta sequences satisfying the following properties:(
1
):
δ
n
∈
ϑ
(
R
+
);
(
2
):
∫
0
∞
‍
δ
n
x
d
x
=
1, n
∈
N;
(
3
):
∫
0
∞
‍
δ
n
x
d
x
<
∞, n
∈
N;
(
4
):
supp
⁡
δ
n
x
⊆
a
,
b
n, 0
<
a
n
<
b
n and a
n
,
b
n
→
0 as n
→
∞, where
(9)
supp
⁡
δ
n
x
=
x
∈
R
+
:
δ
n
x
≠
0
,
∀
n
∈
N
.Now we establish the following theorem.Theorem 1.
Letϕ
,
ψ
∈
ϑ
(
R
+
); then one has ϕ
⊛
ψ
h
h
y
=
ϕ
h
h
*
ψ
y.Proof.
Letϕ and ψ be in ϑ
(
R
+
); then we have
(10)
ϕ
⊛
ψ
h
h
y
=
∫
0
∞
ϕ
⊛
ψ
h
o
x
cos
⁡
x
y
mmcm
+
ϕ
⊛
ψ
h
e
x
sin
x
y
d
x
,
‍
i
.
e
.
=
∫
0
∞
‍
∫
0
∞
ϕ
⊛
ψ
t
sin
x
t
d
t
cos
⁡
x
y
mmmm
+
∫
0
∞
ϕ
⊛
ψ
t
cos
⁡
x
t
d
t
mmmmmm
∫
0
∞
×
sin
x
y
d
x
,
i
.
e
.
=
∫
0
∞
∫
0
∞
∫
0
∞
ϕ
t
z
-
1
sin
x
t
cos
⁡
x
y
mmmmmmmmm
+
ϕ
t
z
-
1
cos
⁡
x
t
mmmmmmmmm
∫
0
∞
t
z
-
1
×
sin
x
y
d
t
d
x
m
m
m
c
×
ψ
z
z
-
1
d
z
.
By the change of variablest
z
-
1
=
w, we get that
(11)
ϕ
⊛
ψ
h
h
y
=
∫
0
∞
∫
0
∞
∫
0
∞
ϕ
w
sin
x
z
w
cos
⁡
x
y
mmmmmmmmm
+
ϕ
w
cos
⁡
x
z
w
mmmmmmmmm
∫
0
∞
×
sin
x
y
d
w
d
x
m
m
m
×
ψ
z
z
d
z
,
i
.
e
.
=
∫
0
∞
∫
0
∞
ϕ
h
o
x
z
cos
⁡
x
y
mc
mmmmm
∫
0
∞
+
ϕ
h
e
x
z
sin
x
y
d
x
m
m
m
×
ψ
z
z
d
z
.
Once again, by the change of variablesx
z
=
r, we get
(12)
ϕ
⊛
ψ
h
h
y
=
∫
0
∞
z
-
1
ϕ
h
h
y
z
-
1
ψ
z
d
z
.
This completes the proof of the theorem.Now, we establish the following theorem.Theorem 2.
Letϕ, ψ, and θ be test functions in ϑ
(
R
+
); then ϕ
*
ψ
⊛
θ
=
ϕ
*
ψ
*
θ.Proof.
Lety
>
0 be given. Then, from (6) and (7), we have
(13)
ϕ
*
ψ
⊛
θ
y
=
∫
0
∞
z
-
1
ϕ
y
z
-
1
ψ
⊛
θ
z
d
z
,
i
.
e
.
=
∫
0
∞
z
-
1
ϕ
y
z
-
1
∫
0
∞
t
-
1
ψ
z
t
-
1
θ
t
d
t
d
z
,
i
.
e
.
=
∫
0
∞
z
-
1
ϕ
y
z
-
1
ψ
*
θ
z
d
z
.
Hence, the theorem is completely proved.Theorem 3.
Letφ
,
ψ
∈
ϑ
1; then φ
⊛
ψ
=
ψ
⊛
φ.Proof.
Proof of this theorem follows from technique similar to that of Theorem2. Therefore, we omit the details.Theorem 4.
Letϕ
1
,
ϕ
2
∈
ϑ
(
R
+
) and φ
1
,
φ
2
∈
ϑ
1; then the following hold: (1)
ϕ
1
+
ϕ
2
*
φ
1
=
ϕ
1
*
φ
1
+
ϕ
2
*
φ
1;
(2)
ifϕ
n
→
ϕ
∈
ϑ
(
R
+
) as n
→
∞, then ϕ
n
*
φ
→
ϕ
*
φ as n
→
∞;
(3)
ϕ
1
*
φ
1
⊛
φ
2
=
ϕ
1
*
φ
1
*
φ
2;
(4)
ϕ
1
*
φ
1
∈
ϑ
(
R
+
).Proof.
Proof of (1) and (2) follows from elementary integral calculus. Proof of (3) is already obtained in Theorem2. Proof of (4) is obvious by the properties of the product (7). Further details are, thus, avoided.
This completes the proof of the theorem.Theorem 5.
Letϕ
∈
ϑ
(
R
+
) and δ
n
∈
Δ; then ϕ
*
δ
n
→
ϕ as n
→
∞.Proof.
LetK be a compact set containing the support of ϕ; then, by Property 2 of delta sequences, we have
(14)
D
y
k
ϕ
*
δ
n
-
ϕ
y
≤
∫
K
D
y
k
z
-
1
ϕ
y
z
-
1
-
ϕ
y
δ
n
y
d
z
.
Hence, (14) goes to zero as n
→
∞.
Hence the theorem is completely proved.The Boehmian spaceB
ϑ
,
ϑ
1
,
⊛
,
*
,
Δ is constructed.The sum of two Boehmians and multiplication by a scalar can be defined in a natural way:(15)
f
n
ω
n
+
g
n
ψ
n
=
f
n
*
ψ
n
+
g
n
*
ω
n
ω
n
⊛
ψ
n
,
α
f
n
ω
n
=
α
f
n
ω
n
=
α
f
n
ω
n
,
where α
∈
C, the field of complex numbers.The operation* and the differentiation are defined by
(16)
f
n
ω
n
*
g
n
ψ
n
=
f
n
*
g
n
ω
n
⊛
ψ
n
,
D
α
f
n
ω
n
=
D
α
f
n
ω
n
.Similarly, the spaceB
ϑ
,
ϑ
1
,
⊛
,
⊛
,
Δ can be proved.The sum of two Boehmians and multiplication by a scalar can be defined in a natural way:(17)
f
n
ω
n
+
g
n
ψ
n
=
f
n
⊛
ψ
n
+
g
n
⊛
ω
n
ω
n
⊛
ψ
n
,
α
f
n
ω
n
=
α
f
n
ω
n
=
α
f
n
ω
n
.The operation* and the differentiation are defined by
(18)
f
n
ω
n
⊛
g
n
ψ
n
=
f
n
⊛
g
n
ω
n
⊛
ψ
n
,
D
α
f
n
ω
n
=
D
α
f
n
ω
n
.As next, letϕ
∈
ϑ
(
R
+
); then ϕ
h
h
y
∈
ϑ
(
R
+
).For some details, we by definitions have that(19)
D
y
k
ϕ
h
h
y
≤
1
π
∫
0
∞
‍
D
y
k
ϕ
h
o
x
cos
⁡
x
y
+
ϕ
h
e
x
sin
x
y
d
x
≤
1
π
∫
0
∞
‍
x
k
ϕ
h
o
x
d
x
+
1
π
∫
0
∞
‍
x
k
ϕ
h
e
x
d
x
.But since(20)
1
π
∫
0
∞
x
k
ϕ
h
o
x
d
x
=
1
π
∫
0
∞
∫
a
b
x
k
ϕ
y
d
y
d
x
⟶
0
,for every power ofx as x
→
∞, where a
,
b is a compact set containing the support of ϕ, similarly we see that
(21)
1
π
∫
0
∞
x
k
ϕ
h
e
x
d
x
⟶
0
,for every power ofx as x
→
∞. Therefore, we get that ϕ
h
h
y
∈
ϑ
(
R
+
).


## 3. Hartley-Hilbert Transform of Quotients

Letϕ
n
/
δ
n
∈
B
ϑ
,
ϑ
1
,
⊛
,
⊛
,
Δ; then we define the extended Hartley-Hilbert transform of ϕ
n
/
δ
n as
(22)
ϕ
n
δ
n
h
h
=
ϕ
n
h
h
δ
n
,
in the quotient space B
ϑ
,
ϑ
1
,
⊛
,
*
,
Δ.Theorem 6.
The operatorϕ
n
/
δ
n
h
h is well defined and linear from B
ϑ
,
ϑ
1
,
⊛
,
⊛
,
Δ into B
ϑ
,
ϑ
1
,
⊛
,
*
,
Δ.Proof.
We show thatϕ
n
/
δ
n
h
h is well defined.
Letψ
n
/
μ
n
=
ξ
n
/
ϵ
n in B
ϑ
,
ϑ
1
,
⊛
,
⊛
,
Δ; then, by the concept of quotients of the space B
ϑ
,
ϑ
1
,
⊛
,
⊛
,
Δ we have
(23)
ψ
n
⊛
ϵ
m
=
ξ
m
⊛
μ
n
.
Hence, by Theorem1, we get from (23) that
(24)
ψ
n
h
h
*
ϵ
m
=
ξ
m
h
h
*
μ
n
.
Hence, from (24), it follows that
(25)
ψ
n
h
h
μ
n
~
ξ
n
h
h
ϵ
n
,
inB
ϑ
,
ϑ
1
,
⊛
,
*
,
Δ.
Therefore,(26)
ψ
n
h
h
μ
n
=
ξ
n
h
h
ϵ
n
,
inB
ϑ
,
ϑ
1
,
⊛
,
*
,
Δ.
That is,(27)
ψ
n
μ
n
h
h
=
ξ
n
ϵ
n
h
h
.
To show thatϕ
n
/
δ
n
h
h is linear, let f
n
/
ϵ
n
,
g
n
/
τ
n
∈
B
ϑ
,
ϑ
1
,
⊛
,
⊛
,
Δ
,
κ
,
η
∈
R; then
(28)
κ
f
n
ϵ
n
+
η
g
n
τ
n
h
h
=
κ
f
n
ϵ
n
+
η
g
n
τ
n
h
h
,
i
.
e
.
=
κ
f
n
⊛
τ
n
+
η
g
n
⊛
ϵ
n
ϵ
n
⊛
τ
n
h
h
,
i
.
e
.
=
κ
f
n
⊛
τ
n
+
η
g
n
⊛
ϵ
n
h
h
ϵ
n
⊛
τ
n
.
Linearity of the classical Hartley-Hilbert transform implies(29)
κ
f
n
ϵ
n
+
η
g
n
τ
n
h
h
=
κ
f
n
⊛
τ
n
h
h
+
η
g
n
⊛
ϵ
n
h
h
ϵ
n
⊛
τ
n
.
Theorem1 gives
(30)
κ
f
n
ϵ
n
+
η
g
n
τ
n
h
h
=
κ
f
n
h
h
*
τ
n
+
η
g
n
h
h
*
ϵ
n
ϵ
n
⊛
τ
n
.
Thus, by addition of Boehmians(31)
κ
f
n
ϵ
n
+
η
g
n
τ
n
h
h
=
κ
f
n
h
h
ϵ
n
+
η
g
n
h
h
τ
n
.
Hence,(32)
κ
f
n
ϵ
n
+
η
g
n
τ
n
h
h
=
κ
f
n
h
h
ϵ
n
+
η
g
n
h
h
τ
n
.
The proof of the theorem is completed.Theorem 7.
The necessary and sufficient condition forg
n
/
ψ
n
∈
B
ϑ
,
ϑ
1
,
⊛
,
*
,
Δ to be in the range of g
n
/
ψ
n
h
h is that g
n belongs to range of the classical h
h, for every n
∈
N.Proof.
Letg
n
/
ψ
n be in the range of g
n
/
ψ
n
h
h; then of course g
n belongs to the range of the classical h
h, ∀
n
∈
N.
To establish the converse, letg
n be in the range of h
h, ∀
n
∈
N. Then there is ϕ
n
∈
ϑ
(
R
+
) such that h
h
ϕ
n
=
g
n, n
∈
N.
Sinceg
n
/
ψ
n
∈
B
ϑ
,
ϑ
1
,
⊛
,
*
,
Δ, we get g
n
*
ψ
m
=
g
m
*
ψ
n, ∀
m
,
n
∈
N.
Therefore, Theorem1 yields
(33)
h
h
ϕ
n
⊛
ψ
m
=
h
h
ϕ
m
⊛
ψ
n
,
∀
m
,
n
∈
N
,
where ϕ
n
∈
ϑ
(
R
+
) and ψ
n
∈
Δ, ∀
n
∈
N.
Thus,ϕ
n
⊛
ψ
m
=
ϕ
m
⊛
ψ
n, m
,
n
∈
N.
Hence,(34)
ϕ
n
ψ
n
∈
B
ϑ
,
ϑ
1
,
⊛
,
⊛
,
Δ
,
ϕ
n
ψ
n
h
h
=
g
n
ψ
n
.
The theorem is, therefore, completely proved.Theorem 8.
The transformϕ
n
/
ψ
n
h
h is consistent with h
h
:
ϑ
(
R
+
)
→
ϑ
(
R
+
).Proof.
For everyϕ
∈
ϑ
(
R
+
), let β
∈
B
ϑ
,
ϑ
1
,
⊛
,
⊛
,
Δ be its representative; then we have β
=
ϕ
⊛
ψ
n
/
ψ
n, ∀
n
∈
N, ψ
n
∈
Δ. For all n
∈
N it is clear that ψ
n is independent from the representative.
By Theorem1 we have
(35)
β
h
h
=
ϕ
⊛
ψ
n
ψ
n
h
h
=
h
h
ϕ
⊛
ψ
n
ψ
n
=
h
h
ϕ
*
ψ
n
ψ
n
,
which is the representative ofh
h
ϕ in the space ϑ
(
R
+
).
Hence the proof of this theorem is completed.



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*Source: 790161-2014-11-06.xml*