# On the generalized Hartley and Hartley-Hilbert transformations

- Shrideh Khalaf Qasem Al-Omari
^{1}and - Adem Kılıçman
^{2}Email author

**2013**:222

https://doi.org/10.1186/1687-1847-2013-222

© Al-Omari and Kılıçman; licensee Springer 2013

**Received: **12 March 2013

**Accepted: **25 June 2013

**Published: **23 July 2013

## Abstract

In this paper, we extend Hartley and Hartley-Hilbert transformations (HT and HHT, respectively) to a certain space of tempered distributions. We then establish a certain convolution theorem for the HHT. The convolution theorem, obtained in this way, has been shown to possess a factorization property of Fourier convolution type. Proving the new convolution theorem for the HHT, by the usual convolution product, the transform is investigated on a certain space of Boehmians. Its properties of linearity and convergence are also discussed in the context of Boehmian spaces.

**MSC:**54C40, 14E20, 46E25, 20C20.

## Keywords

## 1 Test function spaces and distributions

The idea of specifying a function not by its values but by its behavior as a functional on some space of testing functions is a concept that is quite familiar to scientists and engineers through their experience with the classical Fourier and Laplace transformations. Test functions, on which distributions operate, cannot in general be written down in an explicit form. The advantage of distributions over classical functions is that the distribution concept provides a better mechanism for analyzing certain physical phenomena than the function concept does because, for one reason, various entities such as the delta function *δ* can be correctly described as a distribution but not as a function. Furthermore, physical quantities that can be adequately represented as a function can also be characterized as a distribution. In addition, distributions attain an infinite number of derivatives and those derivatives always exist, which is not applied to functions.

The space of testing functions, denoted by
, consists of all complex-valued functions *φ* that are infinitely smooth and zero outside some finite interval. The set of continuous linear forms (conjugates or dual space) on
constitutes a space of distributions, denoted by ${\mathcal{D}}^{\mathrm{\prime}}$.

*f*in ${\mathcal{D}}^{\mathrm{\prime}}$ can be represented, corresponding to $f(t)$, through the convergent integral

The space of complex-valued smooth functions is denoted by ℰ and its dual space is denoted by ${\mathcal{E}}^{\mathrm{\prime}}$.

By
we denote the space of all complex-valued functions *φ* that are infinitely smooth and are such that as $|t|\to \mathrm{\infty}$, they and their partial derivatives decay to zero faster than all powers of $|t{|}^{-1}$. Elements of
are called testing functions of rapid descents.
is indeed a linear space.

If $\varphi \in \mathcal{S}$, then its partial derivatives are in . In fact, is dense in and is dense in ℰ. The dual space of is called the space of tempered distributions and denoted by ${\mathcal{S}}^{\mathrm{\prime}}$ with a property that ${\mathcal{E}}^{\mathrm{\prime}}\subset {\mathcal{S}}^{\mathrm{\prime}}\subset {\mathcal{D}}^{\mathrm{\prime}}$, ${\mathcal{E}}^{\mathrm{\prime}}$ being the (conjugate of ℰ) space of distributions of compact support. For the convergence on , ℰ and and their topologies, we refer to [1, 2].

## 2 HT and HHT of tempered distributions

### 2.1 Introduction to HT and HH transforms

- (i)
Shift: $(\mathtt{H}f(y-{y}_{o}))(x)=cos(2\pi xy{y}_{o})(\mathtt{H}f)(x)+sin(2\pi x{y}_{o})(\mathtt{H}f)(-x)$.

- (ii)
Modulation: $\mathtt{H}(cos(2\pi {x}_{o}y)f(y))(x)=\frac{1}{2}\mathtt{H}f(x-{x}_{o})+\frac{1}{2}\mathtt{H}f(x-{x}_{o})$.

- (iii)
Derivative: $\mathtt{H}(\frac{d}{dy}f(y))(x)=-2\pi x(\mathtt{H}f)(-x)$.

- (iv)Convolution: The convolution theorem of HT is given as$\begin{array}{rl}(\mathtt{H}(f\ast g)(t))(x)=& \frac{1}{2}((\mathtt{H}f)(x)(\mathtt{H}g)(x)+(\mathtt{H}f)(-x)(\mathtt{H}g)(x))\\ +\frac{1}{2}((\mathtt{H}f)(x)(\mathtt{H}g)(-x)-(\mathtt{H}f)(-x)(\mathtt{H}g)(x)).\end{array}$(3)

are the odd and even components of the HT.

The HHT, which permits some attractive applications in geophysics and signal processing, has been extended to a specific space of generalized functions (Boehmian spaces) in [6].

### 2.2 First convolution theorem of HHT

Convolutions of integral transforms which possess the factorization property of Fourier convolution type have become of interest to many authors and have been applied to solving systems of integral equations. In [7], we studied the convolution theorem for HHT in some detail. In this paper we make the idea more precise. We define some generalized convolution of HHT that permits a factorization property of Fourier convolution type.

Let U be a linear space and V be a commutative algebra on the field K. Let $\mathtt{T}\in \mathtt{L}(\mathtt{U},\cdot )$ be a linear operator from U to V. A bilinear map $\star :\mathtt{U}\times \mathtt{U}:\to \mathtt{U}$ is called the convolution for T if $\mathtt{T}(\star (f,g))=\mathtt{T}(f)\mathtt{T}(g)$ for any $f,g\in \mathtt{U}$. The image $\star (f,g)$ is denoted by $f\star g$.

**Theorem 2.1** (First convolution theorem of HHT)

*Let*HH

*f*, HH

*g*

*be the*HHT

*s of*

*f*

*and*

*g*,

*respectively*,

*then*

*where*

*Proof*Under the hypothesis of the theorem, we write

Similarly, we proceed for $\mathtt{B}(\xi )$ to get $\mathtt{B}(\xi )={\mathtt{H}}^{e}(f\star g)(\xi )$, $f\star g$ is the integral equation given in (7).

Hence the theorem. □

**Theorem 2.2**

*Let*

*f*,

*g*

*and*

*h*

*be*${L}^{1}$

*functions*,

*then*:

- (i)
$\mathtt{HH}(f\star g)=\mathtt{HH}(g\star f)$.

- (ii)
$\mathtt{HH}((f\star g)\star h)=\mathtt{HH}(f\star (g\star h))=\mathtt{HH}(g\star (f\star h))=\mathtt{HH}(h\star (f\star g))$.

- (iii)
$\mathtt{HH}(f\star (g+h))=\mathtt{HH}(f\star g)+\mathtt{HH}(f\star h)$.

- (iv)
$\mathtt{HH}(f+(g\star h))=\mathtt{HH}((f+g)\star (f\star h))$.

*Proof*(i) Let $f,g\in {L}^{1}$. By the aid of Theorem 2.1, we get

- (iii)Let $f,g\in {L}^{1}$, then using the definition of HHT and that of ⋆, we get$\begin{array}{r}\mathtt{HH}(f\star (g+h))(x)\\ \phantom{\rule{1em}{0ex}}=\frac{1}{\pi}{\int}_{0}^{\mathrm{\infty}}({\mathtt{H}}^{o}(f\star (g+h))(y)cos(xy)+{\mathtt{H}}^{e}(f\star (g+h))(y)sin(xy))\phantom{\rule{0.2em}{0ex}}dy\\ \phantom{\rule{1em}{0ex}}=\frac{1}{\pi}{\int}_{0}^{\mathrm{\infty}}({\mathtt{H}}^{o}(f\star g+f\star h)(y)cos(xy)+{\mathtt{H}}^{e}(f\star g+f\star h)(y)sin(xy))\phantom{\rule{0.2em}{0ex}}dy.\end{array}$

Hence, the properties of HT odd and even parts, ${\mathtt{H}}^{o}$, ${\mathtt{H}}^{e}$, and that of the integral operator ∫ imply that $\mathtt{HH}(f\star (g+h))(x)=\mathtt{HH}(f\star g+f\star h)(x)$.

Proof of (iv) is analogous to that given for part (ii).

This completes the proof. □

Next is a straightforward corollary of Theorem 2.2.

**Corollary 2.3**

*Let*

*f*,

*g*

*and*

*h*

*be*${L}^{1}$

*functions*,

*then*:

- (i)
$f\star g=g\star f$.

- (ii)
$(f\star g)\star h=f\star (g\star h)$.

- (iii)
$f\star (g+h)=f\star g+f\star h$.

- (iv)
$f+(g\star h)=(f+g)\star (f\star h)$.

### 2.3 HT and HHT of distributions

In this subsection we discuss HT and HHT on a tempered distribution space.

**Theorem 2.4** *If* *f* *is in*
, *then* H *f* *is also in*
.

*Proof*If $f\in \mathcal{S}$, then HT certainly exists. Differentiating, in the ordinary sense, the right-hand side of the integral equation

*x*yields

This is because the right-hand side of (9) converges uniformly for each *x*.

*m*times and by the fact that

for every pair of non-negative *m* and *k*. This completes the proof. □

**Corollary 2.5** *If* *f* *is in*
, *then* ${\mathtt{H}}^{o}f$ *and* ${\mathtt{H}}^{e}f$ *are in*
.

**Corollary 2.6** *If* *f* *is in*
, *then* HH *f* *is also in*
.

*Proof* By Corollary 2.5, ${\mathtt{H}}^{o}f,{\mathtt{H}}^{e}f\in \mathcal{S}$. Thus, the linearity of
and the fact that $\int \varphi (\xi )\phantom{\rule{0.2em}{0ex}}d\xi \in \mathcal{S}$ for every $\varphi \in \mathcal{S}$ imply $\mathtt{HH}f\in \mathcal{S}$.

for each $f\in {\mathcal{S}}^{\mathrm{\prime}}$.

This can be stated in similar words as: HT and HHT of tempered distributions are tempered distributions. □

**Corollary 2.7** *If* $\varphi \in \mathcal{S}$, *then* $(\mathtt{H}\varphi )(\mathtt{HH}\varphi )\in \mathcal{S}$.

**Theorem 2.8** *Let* $f\in {\mathcal{S}}^{\mathrm{\prime}}$, *then* H *f* *and* HH *f* *are both linear mappings from* ${\mathcal{S}}^{\mathrm{\prime}}$ *into* ${\mathcal{S}}^{\mathrm{\prime}}$.

*Proof*Let $f,g\in {\mathcal{S}}^{\mathrm{\prime}}$and $\phi \in \mathcal{S}$, $\alpha \in \mathtt{R}$ be arbitrary, then

Similarly, we proceed for HH *f*, $\mathrm{\forall}f\in {\mathcal{S}}^{\mathrm{\prime}}$. Hence the theorem. □

## 3 Generalized distributions

One of the youngest generalizations of functions, and more particularly of distributions, is the theory of Boehmians. The name Boehmian space is given to all objects defined by an abstract construction similar to that of field of quotients. The construction applied to function spaces yields various spaces of generalized functions.

- (1)
$\varphi \ast \psi \in \mathtt{X}$ and $\varphi \ast \psi =\psi \ast \varphi $.

- (2)
$(f\ast \varphi )\ast \psi =f\ast (\varphi \ast \psi )$.

- (3)
$(f+g)\ast \varphi =f\ast \varphi +g\ast \varphi $

- (4)
$k(f\ast \varphi )=(kf)\ast \varphi =f\ast (k\varphi )$, $k\in \mathtt{R}$.

- (5)
If $({\u03f5}_{n})\in \mathrm{\Delta}$ and $f\ast {\u03f5}_{n}=g\ast {\u03f5}_{n}$, $n=1,2,\dots $ , then $f=g$.

- (6)
$({\u03f5}_{n}),({\tau}_{n})\in \mathrm{\Delta}\Rightarrow ({\u03f5}_{n}\ast {\tau}_{n})\in \mathrm{\Delta}$.

The elements of Δ are called *delta* sequences.

for each $n\in \mathtt{N}$.

The relation ∼ is an equivalent relation on A and hence splits A into equivalence classes. The equivalence class containing $\frac{{f}_{n}}{{\u03f5}_{n}}$ is denoted by $[\frac{{f}_{n}}{{\u03f5}_{n}}]$. These equivalence classes are called *Boehmians* and the space of all Boehmians is denoted by $\mathtt{B}(\mathtt{Y},\mathtt{X},\mathrm{\Delta},\ast )$, see [8].

- (1)If ${f}_{n}\to f$ as $n\to \mathrm{\infty}$ in Y and, $\varphi \in \mathtt{X}$ is any fixed element, then${f}_{n}\ast \varphi \to f\ast \varphi \phantom{\rule{1em}{0ex}}\text{in}\mathtt{Y}\text{as}n\to \mathrm{\infty}.$
- (2)
If ${f}_{n}\to f$ as $n\to \mathrm{\infty}$ in Y and $({\u03f5}_{n})\in \mathrm{\Delta}$, then ${f}_{n}\ast {\u03f5}_{n}\to f$ in Y as $n\to \mathrm{\infty}$.

The operation ∗ is extended to $\mathtt{B}(\mathtt{Y},\mathtt{X},\mathrm{\Delta},\ast )\times \mathtt{X}$ by the following definition.

**Definition 3.1** If $[\frac{{f}_{n}}{{\u03f5}_{n}}]\in \mathtt{B}(\mathtt{Y},\mathtt{X},\mathrm{\Delta},\ast )$ and $\varphi \in \mathtt{X}$, then $[\frac{{f}_{n}}{{\u03f5}_{n}}]\ast \varphi =[\frac{{f}_{n}\ast \varphi}{{\u03f5}_{n}}]$.

In $\mathtt{B}(\mathtt{Y},\mathtt{X},\mathrm{\Delta},\ast )$, two types of convergence, *δ* and Δ convergence, are defined as follows.

**Definition 3.2**A sequence of Boehmians $({\beta}_{n})$ in $\mathtt{B}(\mathtt{Y},\mathtt{X},\mathrm{\Delta},\ast )$ is said to be

*δ*-convergent to a Boehmian

*β*in $\mathtt{B}(\mathtt{Y},\mathtt{X},\mathrm{\Delta},\ast )$, denoted by ${\beta}_{n}\stackrel{\delta}{\to}\beta $, if there exists a delta sequence $({\u03f5}_{n})$ such that $({\beta}_{n}\ast {\u03f5}_{n}),(\beta \ast {\u03f5}_{n})\in \mathtt{Y}$, $\mathrm{\forall}k,n\in \mathtt{N}$, and

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

**Lemma 3.3**${\beta}_{n}\stackrel{\delta}{\to}\beta $ ($n\to \mathrm{\infty}$)

*in*$\mathtt{B}(\mathtt{Y},\mathtt{X},\mathrm{\Delta},\ast )$

*if and only if there are*${f}_{n,k},{f}_{k}\in \mathtt{Y}$

*and*${\u03f5}_{k}\in \mathrm{\Delta}$

*such that*${\beta}_{n}=[\frac{{f}_{n,k}}{{\u03f5}_{k}}]$, $\beta =[\frac{{f}_{k}}{{\u03f5}_{k}}]$

*and for each*$k\in \mathtt{N}$,

**Definition 3.4** A sequence of Boehmians $({\beta}_{n})$ in $\mathtt{B}(\mathtt{Y},\mathtt{X},\mathrm{\Delta},\ast )$ is said to be Δ-convergent to a Boehmian *β* in $\mathtt{B}(\mathtt{Y},\mathtt{X},\mathrm{\Delta},\ast )$, denoted by ${\beta}_{n}\stackrel{\mathrm{\Delta}}{\to}\beta $, if there exists an $({\u03f5}_{n})\in \mathrm{\Delta}$ such that $({\beta}_{n}-\beta )\ast {\u03f5}_{n}\in \mathtt{Y}$, $\mathrm{\forall}n\in \mathtt{N}$, and $({\beta}_{n}-\beta )\ast {\u03f5}_{n}\to 0$ as $n\to \mathrm{\infty}$ in Y. See, for example, [7, 9–12] and [2].

## 4 The spaces ${\mathtt{B}}_{1}({\mathcal{S}}^{\mathrm{\prime}},\mathcal{S},\mathrm{\Delta},\ast )$ and ${\mathtt{B}}_{2}({\mathrm{\Psi}}_{1},{\mathrm{\Psi}}_{2},{\mathrm{\Psi}}_{3},\u2022)$

**Theorem 4.1** (Second convolution theorem of HHT)

*Let*

*f*

*and*

*g*

*be*${\mathtt{L}}^{1}$

*functions*,

*then*

*where* ∗ *is the usual convolution product of* *f* *and* *g*, *see* [1].

*Proof*Using the definition of HHT implies

with $\mathtt{A}(\xi )={\mathtt{H}}^{o}(f\ast g)(\xi )$ and $\mathtt{B}(\xi )={\mathtt{H}}^{e}(f\ast g)(\xi )$.

Hence, invoking (16) and (17) in (13), our theorem follows. □

Let us consider another space of Boehmians:

Denote by ${\mathrm{\Psi}}_{1}$ the space of HHT s of distributions from ${\mathcal{S}}^{\mathrm{\prime}}$. Indeed, ${\mathrm{\Psi}}_{1}$ is also a subspace of ${\mathcal{S}}^{\mathrm{\prime}}$ by (12). A member ${\xi}_{n}\in {\mathrm{\Psi}}_{1}$ is said to converge in ${\mathrm{\Psi}}_{1}$ to a value *ξ* if there are ${\tau}_{n},\tau \in {\mathcal{S}}^{\mathrm{\prime}}$ such that ${\tau}_{n}$ reaches *τ* for large values of *n*.

Also, denote by ${\mathrm{\Psi}}_{2}$ the set of HHT s of test functions from , then ${\mathrm{\Psi}}_{2}$ is a subspace of ${\mathrm{\Psi}}_{1}$ by Corollary 2.6. In similar notations ${\mathrm{\Psi}}_{3}=\mathtt{HH}\mathrm{\Delta}$.

for $\xi =\mathtt{HH}{\xi}^{\ast}$, $\varphi =\mathtt{HH}{\varphi}^{\ast}$.

**Theorem 4.2**

*Let*$\xi \in {\mathrm{\Psi}}_{1}$

*and*$\varphi \in {\mathrm{\Psi}}_{2}$,

*then for*$\xi =\mathtt{HH}{\xi}^{\ast}$

*and*$\varphi =\mathtt{HH}{\varphi}^{\ast}$,

*Proof*For every $\xi \in {\mathrm{\Psi}}_{1}$, $\varphi \in {\mathrm{\Psi}}_{2}$, we get

where $\xi =\mathtt{HH}{\xi}^{\ast}$, $\varphi =\mathtt{HH}{\varphi}^{\ast}$. This proves the theorem. □

**Theorem 4.3** *Let* ${\varphi}_{1},{\varphi}_{2}\in {\mathrm{\Psi}}_{2}$, *then* $\u2022({\varphi}_{1},{\varphi}_{2})=\u2022({\varphi}_{1},{\varphi}_{2})$.

*Proof*Using (17) we get

where ${\varphi}_{1}=\mathtt{HH}{\varphi}_{1}^{\ast}$, ${\varphi}_{2}=\mathtt{HH}{\varphi}_{2}^{\ast}$.

Hence the theorem. □

**Theorem 4.4**

*Let*${\xi}_{1},{\xi}_{2},{\xi}_{n},\xi \in {\mathrm{\Psi}}_{1}$

*and*$\varphi \in {\mathrm{\Psi}}_{2}$,

*then*:

- (i)
$\u2022(k{\xi}_{1},\varphi )(x)=\u2022({\xi}_{1},k\varphi )(x)=k(\u2022({\xi}_{1},\varphi )(x))$, $k\in \mathtt{R}$.

- (ii)
$\u2022({\xi}_{1}+{\xi}_{2},\varphi )(x)=\u2022({\xi}_{1},\varphi )(x)+\u2022({\xi}_{2},\varphi )(x)$.

- (iii)
$\u2022({\xi}_{n},\varphi )(x)\to \u2022(\xi ,\varphi )(x)$

*as*$n\to \mathrm{\infty}$.

*Proof*(i) The linearity of HHT s and (17) implies

The proof of (ii) and (iii) follows from simple computations. The proof is therefore completed. □

**Theorem 4.5** *Let* $({\alpha}_{n}),({\epsilon}_{n})\in {\mathrm{\Psi}}_{3}$, *then* $\u2022({\alpha}_{n},{\epsilon}_{n})\in {\mathrm{\Psi}}_{3}$.

*Proof*For $({\alpha}_{n}),({\epsilon}_{n})\in {\mathrm{\Psi}}_{3}$,

The proof is completed. □

**Theorem 4.6**

*Let*$\xi \in {\mathrm{\Psi}}_{1}$, ${\varphi}_{1},{\varphi}_{2}\in {\mathrm{\Psi}}_{2}$,

*then*

*Proof* Follows from similar computations to those above.

Hence our theorem is completely proved. □

**Theorem 4.7** *Let* ${\xi}_{1},{\xi}_{2}\in {\mathrm{\Psi}}_{1}$ *and* $({\delta}_{n})\in {\mathrm{\Psi}}_{3}$ *and* $\u2022({\xi}_{1},{\delta}_{n})(x)=\u2022({\xi}_{2},{\delta}_{n})(x)$, *then* ${\xi}_{1}={\xi}_{2}$.

*Proof*Assume that $\u2022({\xi}_{1},{\delta}_{n})(x)=\u2022({\xi}_{2},{\delta}_{n})(x)$, then

Hence ${\xi}_{1}={\xi}_{2}$. This completes the proof. □

**Theorem 4.8**

*Let*$({\delta}_{n})\in {\mathrm{\Psi}}_{3}$

*and*$\xi \in {\mathrm{\Psi}}_{1}$,

*then*

*Proof*Since $\xi \in {\mathrm{\Psi}}_{1}$, $({\delta}_{n})\in {\mathrm{\Psi}}_{3}$, there are ${\xi}^{\ast}\in \mathcal{S}$, ${\delta}_{n}^{\ast}\in \mathrm{\Delta}$ such that $\mathtt{HH}{\xi}^{\ast}=\xi $ and ${\delta}_{n}=\mathtt{HH}{\delta}_{n}^{\ast}$. Hence

as $n\to \mathrm{\infty}$ since ${\delta}_{n}^{\ast}\in \mathrm{\Delta}$.

Thus the theorem. □

The Boehmian space ${\mathtt{B}}_{2}({\mathrm{\Psi}}_{1},{\mathrm{\Psi}}_{2},{\mathrm{\Psi}}_{3},\u2022)$ is completely established.

Hence, $\u2022(\mathtt{HH}{f}_{n},\mathtt{HH}{\varphi}_{m})=\u2022(\mathtt{HH}{f}_{m},\mathtt{HH}{\varphi}_{n})$.

## 5 HHT of Boehmians

It is clear that EHHT is well defined.

**Theorem 5.1** EHHT *is linear*.

**Theorem 5.2** EHHT *is one*-*to*-*one*.

**Theorem 5.3** EHHT *is continuous with respect to* *δ* *convergence*.

*Proof* Let ${\beta}_{n}\stackrel{\delta}{\to}\beta $ in ${\mathtt{B}}_{1}({\mathcal{S}}^{\mathrm{\prime}},\mathcal{S},\mathrm{\Delta},\ast )$ as $n\to \mathrm{\infty}$. We show that $\mathrm{\Lambda}{\beta}_{n}\to \mathrm{\Lambda}\beta $ in ${\mathtt{B}}_{2}({\mathrm{\Psi}}_{1},{\mathrm{\Psi}}_{2},{\mathrm{\Psi}}_{3},\u2022)$ as $n\to \mathrm{\infty}$.

and $\beta =[\frac{{f}_{k}}{{\varphi}_{k}}]$ and ${f}_{n,k}\to {f}_{k}$ as $n\to \mathrm{\infty}$, $\mathrm{\forall}k\in \mathtt{N}$.

The proof is completed. □

**Theorem 5.4** EHHT *s are continuous with respect to* Δ *convergence*.

*Proof*Let ${\beta}_{n}\stackrel{\mathrm{\Delta}}{\to}\beta $ in ${\mathtt{B}}_{1}({\mathcal{S}}^{\mathrm{\prime}},\mathcal{S},\mathrm{\Delta},\ast )$ as $n\to \mathrm{\infty}$, then there are ${f}_{n}\in {\mathcal{S}}^{\mathrm{\prime}}$ and ${\varphi}_{n}\in \mathrm{\Delta}$ such that

Hence the theorem. □

**Remark 5.5**Let $\beta =[\frac{\mathtt{HH}{f}_{n}}{\mathtt{HH}{\delta}_{n}}]\in {\mathtt{B}}_{2}({\mathrm{\Psi}}_{1},{\mathrm{\Psi}}_{2},{\mathrm{\Psi}}_{3},\u2022)$, then it is so natural to define the inverse HHT of

*β*as

in the space ${\mathtt{B}}_{1}({\mathcal{S}}^{\mathrm{\prime}},\mathcal{S},\mathrm{\Delta},\ast )$.

It is of interest to see that the inverse transform ${\mathrm{\Lambda}}^{-1}$ preserves all the above properties that Λ does such as linearity, one-to-one and continuity of ${\mathrm{\Lambda}}^{-1}$ with respect to convergence in ${\mathtt{B}}_{1}({\mathcal{S}}^{\mathrm{\prime}},\mathcal{S},\mathrm{\Delta},\ast )$. Proofs are avoided.

## Declarations

### Acknowledgements

The authors are very grateful to the referees for their valuable suggestions and comments that helped to improve the paper.

## Authors’ Affiliations

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