# The Fractional Hilbert Transform on the Real Line

Authors: Naheed Abdullah; Saleem Iqbal; Asma Khalid; Amnah S. Al Johani; Ilyas Khan; Abdul Rehman; Mulugeta Andualem
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

This paper examines some special properties and important results of the fractional Hilbert transform (FHT) on the real lineℝ. In this rigorous study, we modify certain theorems of classical Hilbert transform for FHT and develop new theorems. Moreover, FHT of some common functions is given and eigenfunctions are also studied. We prove that FHT, denoted by Hα, is an isomorphism on L2ℝ. We also show that in L2ℝ, FHT is an isometry. Furthermore, we investigate Riesz inequality on Lpℝ for p>1 to establish Hilbert formulae for FHT.


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## Body

## 1. Introduction

Integral transforms are applied for solving a variety of problems in mathematics, physics, engineering science, optics, and signal processing. Mathematicians and engineers have been using integral transforms for more than two centuries. Many integral transforms—Laplace, Fourier, z-transform, Hilbert, Mellin, Laplace–Carson, Hankel, Sumudu, natural, and Shehu to name a few—exist in the literature. The best-known integral transform is perhaps the Fourier transform (FT), but specialists in many fields use the Hilbert transform (HT) [1–4]. Due to limitations of most integral transforms, many authors introduced alternative approaches to solve advanced problems. Their fractional counterparts provide new level of freedom, i.e., the non-integer order. As an example, the fractional Hilbert transform has been successfully used for a wide range of problems from optics to signal processing. The higher-dimensional HT which is known as Riesz–Hilbert transform was used to construct the monogenic signal [5].The Hilbert transform, defined by David Hilbert and named after him in 1924, was first discussed by G. H. Hardy mathematically. This transform arose while Hilbert was working on a problem called Riemann–Hilbert problem in 1905. Hilbert’s earlier work on discrete Hilbert transform was published in Hermann Weyl’s dissertation. For a complete historical discussion, the reader is referred to King [6].Mathematically, Hilbert integral transform onℝ is defined as(1)Hft=1π∫−∞∞fxt−xdx,where the integral is taken as Cauchy principal value at t=x. From the definition of Hilbert transform, it is clear that Hilbert transform of a function f has a non-local dependence on f.The introduction of fractional Fourier transform in 1929 as powers of FT opened the door not only to many problem-solving techniques in mathematics, signal processing, and optics to name a few but also to fractionalization of many other transforms [2, 7–10]. FHT has been studied extensively based on its application in signal processing and optics [11–16]. Many authors have investigated the properties of fractional Hilbert transform. Bernstein [5] constructed fractional monogenic signals on the basis of fractional Riesz–Hilbert transform. Lohmann et al. [17] introduced a generalization of the classical Hilbert transform called fractional Hilbert transform. Zayed [18] fractionalized the standard Hilbert transform by a different approach, i.e., negative frequency of signal is suppressed in the fractional Fourier domain; for this, FHT is defined in terms of chirp signal. We need a detailed mathematical theory of FHT to explore its applications in applied mathematics as well as other sciences. So, the goal of this paper is to mathematically analyze Zayed’s definition of FHT in order to make the theory of FHT well understood. The reader is referred to Zayed [18] for a detailed discussion on the definition of FHT.The organization of the paper is as follows. In Section2, we recall some key concepts, properties of HT, and some related theorems. In Section 3, we define properties of FHT. In Section 4, we discuss FHT of Gaussian function and some other functions. In Section 5, we study the boundedness and continuity of FHT on L2 and L∞. Finally, in Section 6, we consider the Riesz inequality and Hilbert formulae.


## 2. Preliminary

We begin this section by defining the FHT as in [18]. The fractional Hilbert transform of a function f, denoted by Hαft, is defined as(2)Hαft=1π∫−∞∞e−it2−x2/2cotαt−xfxdxforα≠0,π2,π.From the above definition, it follows that [19](3)Hαft=e−it2/2cotαHeit2/2cotαft.Letℱα denote the fractional Fourier transform; then, for 0<α<π [6],(4)ℱαHαft=−isgntℱαft.From equation (3), we arrive at the conclusion that if f∈Lp, for p>1, then Hαf∈Lp. Also, recall that ℱα is an isometry on L2ℝ [20], i.e.,(5)ℱαf2=f2.It is clear thatHα reduces to classical HT for α=2n+1π/2 where n∈ℤ, i.e.,(6)Hα=Hforα=2n+1π2.The FHT does not have the semigroup property, i.e.,HαHβ≠Hα+β [18]. Now we recall some results that will be of use in later applications.Theorem 1 (see [10, Chapter 4]).
Letf,g∈Lp,1<p<∞; then, the Hilbert transform satisfies the following properties:(1)
Linearity:Hf+g=Hf+Hg.(2)
Derivative:Hf′=Hf.(3)
Iteration:H2f=−f,a.e.(4)
Fourier transform ofH: ℱHft=−isgntℱft.Theorem 2 (see [10, p. 203]).
Iff∈Lpℝ for p>1, then there exists a constant ℛp such that(7)∫−∞∞Hftp≤ℛpp∫−∞∞ftp.
Throughout this paper, we deduce results for FHT by using the properties of HT andℱα.


## 3. Properties of Fractional Hilbert Transform

In this section, we state and prove some important properties of FHT which will be used in later analysis. Throughout this paper, “∗” and “†” will be used to denote the complex conjugate operation.


### 3.1. Complex Conjugation Property

Unlike Hilbert transform, FHT is not complex conjugate invariant. Letf be a real-valued function; then,(a)
Letf be a real-valued function; then,(8)Hαf∗=1π∫−∞∞e−it2−x2/2cotαt−xfxdx∗=1π∫−∞∞eit2−x2/2cotαt−xf∗xdx.Asfis real−valued,sof∗=f,thereforeHαf∗=1π∫−∞∞e−it2−x2/2cot−αt−xfxdx=H−αftHαf∗=H−αf.(b)
Suppose thatf is a complex-valued function; then, Hαf∗=H−αf∗.


### 3.2. Parity Property

Parity property of FHT follows from that of HT.Letf be an even function denoted by fe; then,(9)Hαft=Hαfet=e−it2/2cotαHeit2/2cotαfe,using parity property of Hilbert transform,=godd.Thus, iff is an even function, then Hαf is odd. Similarly, if f is odd, then Hα is even.


### 3.3. Iteration Property

In order to prove this special property, we use the relation between FHT and fractional Fourier transform (4).(10)FαHαHαft=−isgntFαHαft=−isgnt−isgntFαft=i2sgnt2Fαft=−FαftFαHαHαft=Fα−ft.This implies that(11)Hα2ft=−ft.Thus,(12)Hα2=−I.From the iteration property, we can conclude that(13)Hα=−Hα−1orHα−1=−Hα.


### 3.4. Eigenvalues

Let(14)Hαf=λf,HαHαf=λHαf,Hα2f=λλf,−f=λ2f.So,(15)λ2=−1.This implies that(16)λ=±i.Thus,±i are eigenvalues of f.


### 3.5. Eigenfunctions

Eigenfunctions corresponding to eigenvaluesi and −i are f=g−iHαg and f=g+iHαg, respectively.Proof.
An eigenfunction corresponding to the eigenvaluei is(17)Hαf=Hαg−iHαg=Hαg−iHα2g=Hαg+ig=−i2Hαg+ig=ig−iHαg,Hαf=if.
An eigenfunction corresponding to the eigenvalue−i is(18)Hαf=Hαg+iHαg=Hαg+iHα2g=Hαg−ig=−i2Hαg−ig=−ig−iHαg,Hαf=−if.
This completes the proof.


### 3.6. Fractional Hilbert Transform of the Producttnft

The FHT oftft is given by(19)Hαtft=1π∫−∞∞e−it2−x2/2cotαt−xxfxdx.Takingx=x−t+t=t−t−x,=1π∫−∞∞e−it2−x2/2cotαt−xtfxdx−1π∫−∞∞e−it2−x2/2cotαt−xt−xfxdx=tHαft−1π∫−∞∞e−it2−x2/2cotαfxdx.By using mathematical induction, we have(20)Hαtnft=tnHαft−1π∑k=0n−1tk∫−∞∞tn−1−ke−it2−x2/2cotαfxdx.As an example, considerHα−t=Hα−1t:(21)=e−it2/2cotαtsgntScotαπt1+i−Ccotαπt1−i+e−it2/2cotα1πiScotαπt+Ccotαπt.


## 3.1. Complex Conjugation Property

Unlike Hilbert transform, FHT is not complex conjugate invariant. Letf be a real-valued function; then,(a)
Letf be a real-valued function; then,(8)Hαf∗=1π∫−∞∞e−it2−x2/2cotαt−xfxdx∗=1π∫−∞∞eit2−x2/2cotαt−xf∗xdx.Asfis real−valued,sof∗=f,thereforeHαf∗=1π∫−∞∞e−it2−x2/2cot−αt−xfxdx=H−αftHαf∗=H−αf.(b)
Suppose thatf is a complex-valued function; then, Hαf∗=H−αf∗.


## 3.2. Parity Property

Parity property of FHT follows from that of HT.Letf be an even function denoted by fe; then,(9)Hαft=Hαfet=e−it2/2cotαHeit2/2cotαfe,using parity property of Hilbert transform,=godd.Thus, iff is an even function, then Hαf is odd. Similarly, if f is odd, then Hα is even.


## 3.3. Iteration Property

In order to prove this special property, we use the relation between FHT and fractional Fourier transform (4).(10)FαHαHαft=−isgntFαHαft=−isgnt−isgntFαft=i2sgnt2Fαft=−FαftFαHαHαft=Fα−ft.This implies that(11)Hα2ft=−ft.Thus,(12)Hα2=−I.From the iteration property, we can conclude that(13)Hα=−Hα−1orHα−1=−Hα.


## 3.4. Eigenvalues

Let(14)Hαf=λf,HαHαf=λHαf,Hα2f=λλf,−f=λ2f.So,(15)λ2=−1.This implies that(16)λ=±i.Thus,±i are eigenvalues of f.


## 3.5. Eigenfunctions

Eigenfunctions corresponding to eigenvaluesi and −i are f=g−iHαg and f=g+iHαg, respectively.Proof.
An eigenfunction corresponding to the eigenvaluei is(17)Hαf=Hαg−iHαg=Hαg−iHα2g=Hαg+ig=−i2Hαg+ig=ig−iHαg,Hαf=if.
An eigenfunction corresponding to the eigenvalue−i is(18)Hαf=Hαg+iHαg=Hαg+iHα2g=Hαg−ig=−i2Hαg−ig=−ig−iHαg,Hαf=−if.
This completes the proof.


## 3.6. Fractional Hilbert Transform of the Producttnft

The FHT oftft is given by(19)Hαtft=1π∫−∞∞e−it2−x2/2cotαt−xxfxdx.Takingx=x−t+t=t−t−x,=1π∫−∞∞e−it2−x2/2cotαt−xtfxdx−1π∫−∞∞e−it2−x2/2cotαt−xt−xfxdx=tHαft−1π∫−∞∞e−it2−x2/2cotαfxdx.By using mathematical induction, we have(20)Hαtnft=tnHαft−1π∑k=0n−1tk∫−∞∞tn−1−ke−it2−x2/2cotαfxdx.As an example, considerHα−t=Hα−1t:(21)=e−it2/2cotαtsgntScotαπt1+i−Ccotαπt1−i+e−it2/2cotα1πiScotαπt+Ccotαπt.


## 4. Fractional Hilbert Transform of Some Common Functions

### 4.1. Constant Function

For anya∈ℝ,(22)Hαa=sgntae−it2/2cotαCcotαπt1−i−Scotαπt1+i.Graphs offx=a, its HT, and FHT are displayed in Figure 1 for α=π/4.Figure 1
Forα=π/4. (a) Plot of ft=1. (b) Hilbert transform of ft. (c) Real part of Hαft. (d) Imaginary part of Hαft.
(a)(b)(c)(d)


### 4.2. Gaussian Function

The fractional Hilbert transform of the Gaussian function forb>0, defined as ft=e−bt2, is(23)Hαft=1π∫−∞∞eix2−t2/2cotαt−xe−bt2dx=eit2/2cotαπ∫−∞∞eicotα/2−bx2t−xdx=eicotα/2−bt2ebt2π∫−∞∞eicotα/2−bx2t−xdx=e−bt2Hβ1,wherecotβ=cotα−2b.Hαe−bt2=e−bt2sgnte−it2/2cotα−2bCcotα−2bπt1−i−e−bt2sgnte−it2/2cotα−2bScotα−2bπt1+i.Graphs offt=e−bt2, its Hilbert transform, and its FHT are shown in Figure 2 for α=π/4.Figure 2
Forα=π/4. (a) Plot of the Gaussian function. (b) Real part of its Hilbert transform. (c) Imaginary part of its HT. (d) Real part of its FHT. (e) Imaginary part of its FHT.
(a)(b)(c)(d)(e)To find the FHT of second derivative of the Gaussian function, we need FHT of the following function.(24)Hαte−t2/2t−x=e−i−t2/2cotαπ∫−∞∞xe−x2/2ei−x2/2cotαt−x2dx=e−it2/2cotαπ∫−∞∞e−t2/2eit2/2cotαt−x2−e−t2/2eit2/2cotαtt−xdx=πe−t2/2δt−x.Example 1.
The second derivative of the Gaussian function is(25)ft=−d2dt2e−t2/2=1−t2e−t2/2.
The FHT offt is(26)Hαft=−Hαd2dt2e−t2/2,=−Hαg″t=icotαHαtg′t−Hαg′tt−x,=icotαHα−t2e−t2/2+Hαte−t2/2t−x,=icotβ+1e−t2/2Hβ−t+πe−t2/2δt−x,wherecotβ=cotα−1,=icotαe−t2/2e−i−t2/2cotα−1tsgntScotα−1πt1+i−Ccotα−1πt1−i+e−i−t2/2cotα1πiScotα−1πt+Ccotα−1πt.
Graphs offt=1−t2e−t2/2, its Hilbert transform, and its FHT are plotted in Figure 3 for α=π/4.Figure 3
Forα=π/4. (a) Plot of the second derivative of Gaussian function. (b) Real part of its Hilbert transform. (c) Imaginary part of its HT. (d) Real part of its FHT. (e) Imaginary part of its FHT.
(a)(b)(c)(d)(e)


## 4.1. Constant Function

For anya∈ℝ,(22)Hαa=sgntae−it2/2cotαCcotαπt1−i−Scotαπt1+i.Graphs offx=a, its HT, and FHT are displayed in Figure 1 for α=π/4.Figure 1
Forα=π/4. (a) Plot of ft=1. (b) Hilbert transform of ft. (c) Real part of Hαft. (d) Imaginary part of Hαft.
(a)(b)(c)(d)


## 4.2. Gaussian Function

The fractional Hilbert transform of the Gaussian function forb>0, defined as ft=e−bt2, is(23)Hαft=1π∫−∞∞eix2−t2/2cotαt−xe−bt2dx=eit2/2cotαπ∫−∞∞eicotα/2−bx2t−xdx=eicotα/2−bt2ebt2π∫−∞∞eicotα/2−bx2t−xdx=e−bt2Hβ1,wherecotβ=cotα−2b.Hαe−bt2=e−bt2sgnte−it2/2cotα−2bCcotα−2bπt1−i−e−bt2sgnte−it2/2cotα−2bScotα−2bπt1+i.Graphs offt=e−bt2, its Hilbert transform, and its FHT are shown in Figure 2 for α=π/4.Figure 2
Forα=π/4. (a) Plot of the Gaussian function. (b) Real part of its Hilbert transform. (c) Imaginary part of its HT. (d) Real part of its FHT. (e) Imaginary part of its FHT.
(a)(b)(c)(d)(e)To find the FHT of second derivative of the Gaussian function, we need FHT of the following function.(24)Hαte−t2/2t−x=e−i−t2/2cotαπ∫−∞∞xe−x2/2ei−x2/2cotαt−x2dx=e−it2/2cotαπ∫−∞∞e−t2/2eit2/2cotαt−x2−e−t2/2eit2/2cotαtt−xdx=πe−t2/2δt−x.Example 1.
The second derivative of the Gaussian function is(25)ft=−d2dt2e−t2/2=1−t2e−t2/2.
The FHT offt is(26)Hαft=−Hαd2dt2e−t2/2,=−Hαg″t=icotαHαtg′t−Hαg′tt−x,=icotαHα−t2e−t2/2+Hαte−t2/2t−x,=icotβ+1e−t2/2Hβ−t+πe−t2/2δt−x,wherecotβ=cotα−1,=icotαe−t2/2e−i−t2/2cotα−1tsgntScotα−1πt1+i−Ccotα−1πt1−i+e−i−t2/2cotα1πiScotα−1πt+Ccotα−1πt.
Graphs offt=1−t2e−t2/2, its Hilbert transform, and its FHT are plotted in Figure 3 for α=π/4.Figure 3
Forα=π/4. (a) Plot of the second derivative of Gaussian function. (b) Real part of its Hilbert transform. (c) Imaginary part of its HT. (d) Real part of its FHT. (e) Imaginary part of its FHT.
(a)(b)(c)(d)(e)


## 5. Fractional Hilbert Transform onL2 and L∞

Now we prove some useful results forHα on L2 which clearly show that Hα is bounded on L2. We then apply this theory to prove some generalized relations. Here we make use of the formula in [10, p. 58].Iff∈Lpandg∈Lqsuch thatp−1+q−1=1,then(27)∫−∞∞fxdx∫−∞∞gyy−xdx=∫−∞∞gydy∫−∞∞fxy−xdy,to justify the change of order of integration.Theorem 3.
Multiplication formula: for allf,g∈L2ℝandα∈ℝ, we have(28)∫−∞∞Hαftgtdt=−∫−∞∞ftH−αgtdt.Proof.
(29)∫−∞∞Hαftgtdt=∫−∞∞1π∫−∞∞e−it2−x2/2cotαt−xfxdxgtdt=−∫−∞∞fx1π∫−∞∞e−ix2−t2/2cot−αx−tgtdtdx=∫−∞∞fxH−αgxdx=∫−∞∞ftH−αgtdt.
The multiplication formula will serve as the main technical theorem of this paper.Theorem 4.
For allα∈ℝ and f,g∈L2ℝ,(1)
Hα is an isomorphism on L2ℝ.(2)
Hαf2=f2.(3)
Hα is a unitary operator on L2ℝ.Proof.
Letf∈L2ℝ; then, Hαf∈L2ℝ.(1)
Linearity ofHα is trivial.One-to-one: consider(30)Hαft=Hαgt.By the linearity ofHα, this implies that(31)Hαft−gt=0Hα2ft−gt=Hα0,therefore, by the iteration property,−Ift−gt=0ft−gt=0,ft=gt.So,Hα is one-to-one.Onto:For everyf∈L2ℝ, there exists −Hαft∈L2ℝ such that(32)Hα−Hαft=f.Thus,Hα is an isomorphism on L2ℝ.(33)2Hαf2=F−α−isgntFαf2=−isgntFαf2=−isgntFαf2=Fαf2=f2.(3)
From multiplication property of FHT, we can write(34)Hαf,g=f,H−αg=f,−Hα†g.So,(35)−Hαf,g=f,Hα†g.Therefore,(36)Hα†=−Hα=Hα−1.Also,(37)Hαf2=f2.Thus,Hα is a unitary operator on L2.We can deduce the following.Corollary 1.
Hα:L2ℝ⟶L2ℝ is a homeomorphism.Proof.
SinceHα:L2ℝ⟶L2ℝ is an isomorphism and the continuity of Hα follows from the relation,(38)Hαft=e−it2/2cotαHeit2/2cotαfℝ.
We know thatHα on L2ℝ is bounded, and we now discuss the conditions on FHT which give boundedness and continuity of Hα on L∞ℝ.Proposition 1.
Letf∈L∞ℝandH−α be bounded on L1; then, Hαf∈L∞ℝ.Proof.
(39)HαfL∞=supg∈L1g1=1∫−∞∞Hαfxgxdx=supg∈L1g1=1∫−∞∞fxH−αgxdx≤supg∈L1‖g‖1=1fL∞H−αgL1,hence,HαfL∞≤CfL∞.Proposition 2.
Letf∈L∞ and H−α be bounded on L1; then, Hαf is continuous on ℝ.Proof.
For anyh>0, we have(40)supt∈ℝ|Hαft+h−Hαft|=supt∈ℝe−it+h2−x2/2cotαπt+h−xfx+h−e−it2−x2/2cotαπt−xfxdx=supt∈ℝe−it2+2ht+h2−x2/2cotαπt+h−xfx+h−e−it2−x2/2cotαπt−xfxdx≤supt∈ℝ∫−∞∞1πe−it2−x2/2cotαe−iht+h/2cotαt+h−x−1t−xfxdxe−iht+h/2cotαt+h−x−1t−xfx⟶0ash⟶0.
That is,(41)limh⟶0supt∈ℝHαft+h−Hαft=0.
Thus,Hαf is continuous on ℝ.


## 6. The Riesz Inequality

In this section, we modify few results of Hilbert transform [21] for FHT. First, we study Riesz inequality which gives a bounded linear map of Hα on Lp.Theorem 5.
Forf∈Lps for p>1, the fractional Hilbert transform satisfies(42)Hαfp≤ℛpfp,where(43)ℛp=tanπ2p,1<p≤2,cotπ2p,2≤p<∞.Proof.
Letf∈Lpℝ for p>1; then,(44)∫−∞∞Hαftpdt=∫−∞∞e−it2/2cotαHe−it2/2cotαftpdt=∫−∞∞e−it2/2cotαpHe−it2/2cotαftpdt≤∫−∞∞He−it2/2cotαftpdt.
Letgt=e−it2/2cotαft; then, from Theorem 2, we arrive at the following inequality:(45)∫−∞∞Hαftpdt≤ℛpp∫−∞∞gtpdt=ℛpp∫−∞∞ftpdt,Thus,Hαfp≤ℛpfp,with the constant ℛp independent of f. Finally, with the help of Riesz inequality and multiplication formula, we can give generalized relations known as Hilbert formulae as follows.Proposition 3.
Letf∈Lp for p>1, and g∈Lp′ such that 1/p+1/p′=1. Then,(46)1∫−∞∞Hαftgtdt=−∫−∞∞ftH−αgtdt.(47)2∫−∞∞ftg∗tdt=−∫−∞∞HαftHα∗gtdt.Proof.
(1)
HereHαf∈Lp and Hαg∈Lp′ for α∈ℝ and S00 is the set of all simple functions on ℝ. Let us consider two sequences fn,gn⊂S00 such that limn⟶∞f−fnp=0 and limn⟶∞g−gnp′=0.By the Riesz inequality, this implieslimn⟶∞Hαf−Hαfnp=0 and limn⟶∞Hαg−Hαgnp′=0.Sincefn,gn∈L2, by the multiplication formula,(48)∫−∞∞Hαfntgmtdt=−∫−∞∞ftH−αgmtdt∀m,n∈ℕ.Now by Holder’s inequality,limn⟶∞Hαf−Hαfnp=0 implies limn⟶∞Hαf−Hαfn1=0, and this implies(49)limn⟶∞Hαfn=Hαf,Similarly,limn⟶∞fn=f,limm⟶∞Hαgm=Hαg,andlimm⟶∞gm=g.Thus equation (48) becomes∫−∞∞Hαftgtdt=−∫−∞∞ftH−αgtdt(2)
Since(50)f+g2=Hαf+Hαg2,and(51)f+ig2=Hαf+iHαg2,equation (50) can be written in the form of inner product as(52)f+g,f+g=Hαf+Hαg,Hαf+Hαgf22+f,g+g,f+g22=Hαf22+Hαf,Hαg+Hαg,Hαf+Hαg22,Asf2=Hαf2and|g‖2=Hαg2.Therefore,(53)f,g+g,f=Hαf,Hαg+Hαg,Hαf.Also, from equation (53), we can write(54)g,f−f,g=Hαg,Hαf−Hαf,Hαg.Subtracting (54) from (53), we arrive at the following equation:(55)f,g=Hαf,Hαg.Thus,(56)∫−∞∞ftgtdt=∫−∞∞HαftHαgtdtforp=2.Again, consider the sequencesfn and gm as in (1). Then ∀m,n∈ℕ,(57)∫−∞∞fntgm∗tdt=∫−∞∞HαfntHα∗gmtdt.Hence, by Holder’s inequality,(58)∫−∞∞ftg∗tdt=∫−∞∞HαftHα∗gtdt.
We next prove FHT inversion onLp for p>1. Recall that if f∈Lp, for p>1, then Hαf∈Lp; also, from the linear and iteration properties of FHT, we can deduce that Hα is 1−1.Theorem 6.
Inverse ofHαft is given by Hα−1=−Hα.Proof.
(59)HαH−αft=1π∫−∞∞e−it2−x2/2cotαt−xHα−1fxdx=1π∫−∞∞e−it2−x2/2cotαt−x1π∫−∞∞−e−ix2−y2/2cotαx−yfydydx=e−it2/2cotα1π∫−∞∞eix2/2cotαt−xdx1π∫−∞∞e−ix2/2cotαeiy2/2cotαx−yfydy=e−it2/2cotα1π∫−∞∞dxt−x1π∫−∞∞eiy2/2cotαx−yfydy=e−it2/2cotαHH−1eit2/2cotαft=e−it2/2cotαeit2/2cotαft=ft,Thus,Hα−1=−Hα.


## 7. Conclusion and Future Work

In the present work, we have investigated important properties of FHT by generalizing those of HT. In fact, we have recognized sufficient conditions to study the boundedness and inversion problem. Also, some appropriate examples are given to confirm the crucial results. Ultimately, this paper contains plots of relevant examples to confirm the authenticity of the obtained results.In future work, extension of FHT on space of generalized functions can be established by extending the current results to includeℒ1, space of distributions with compact support, and the space of ultra-distributions. Further, it will be of interest to introduce and study quaternion FHT on space of generalized functions.



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