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On the generalized Hartley-Hilbert and Fourier-Hilbert transforms

Abstract

In this paper, we discuss Hartley-Hilbert and Fourier-Hilbert transforms on a certain class of generalized functions. The extended transforms considered in this article are shown to be well-defined, one-to-one, linear and continuous mappings with respect to δ and Δ convergence. Certain theorems are also established.

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

1 Introduction

The Hartley transform of a function f(t) of one variable is defined in [1]

H(ξ):=(Hf)(ξ)= f(t)cas(ξt)dt,

where cas(ξt)=cos(ξt)+sin(ξt) is the transform kernel. The Hartley transform is a spectral transform closely related to the Fourier transform. It contains the same information as the Fourier transform does, and no advantage accrues in its use for complex signals. For real signals, the Hartley transform is real and this can offer computational advantages in the applications of signal processing that traditionally make use of Fourier transforms.

The Hilbert transform of a function f(t) via the Hartley transform (Hartley-Hilbert transform) is described in [2] as

hh(t):=(hhf)(t):= 1 π 0 ( H o ( ξ ) cos ( ξ t ) + H e ( ξ ) sin ( ξ t ) ) dξ,
(1)

where

H(ξ):=(Hf)(ξ)= f(t)cas(ξt)dt:= H e (ξ)+ H o (ξ),
(2)

H e (ξ) and H o (ξ) being the even and odd components of the Hartley transform [3].

If a casual function f(t) (f(t)=0, t<0) is considered, then even and odd components H e (ξ) (respectively, H o (ξ)) of the Hartley transform of f(t) constitute a Hilbert transform pair:

H o (ξ)= 1 π H e ( u ) ξ u du
(3)

and

H e (ξ)= 1 π H o ( u ) ξ u du;
(4)

see [2, 4].

On the other hand, the Hilbert transform of f(t) via the Fourier transform (Fourier-Hilbert transform) is defined by

fh(t):=(fhf)(t)= 1 π 0 ( FI ( ξ ) cos ( ξ t ) FR ( ξ ) sin ( ξ t ) ) dξ,
(5)

where

F(ξ)= f(t) e i ξ t dt:=FR(ξ)iFI(ξ),
(6)

FR(ξ) and FI(ξ) being the real and imaginary components of the Fourier transform of f(t).

In [5], Fourier and Hartley transforms are coined as mathematical twins. This suggests that the corresponding properties of Fourier-Hilbert and Hartley-Hilbert transforms can simultaneously be developed and the inherited applications may nicely be replaced. It is interesting to know that Hartley-Hilbert and Fourier-Hilbert transforms enjoy wide applications in signal processing, network theory and some other geophysical applications. In [6], it has been shown that Fourier-Hilbert and Hartley-Hilbert transforms, while possessing the same magnitude, differ in phase by 270. On the other hand, the inverse Hartley-Hilbert transform returns the original function unlike the Fourier-Hilbert transform, which results in the negative of the original function.

In the present article, we further discuss this pair of transforms on certain spaces of generalized functions. We spread results over three sections. Fourier-Hilbert and Hartley-Hilbert transforms are reviewed in Section 1. The distributional definition of the cited transforms is presented in Section 2. The Hartley-Hilbert transform is extended to the context of Boehmians in Section 3, where some properties are also described.

2 The extended Hartley-Hilbert and Fourier-Hilbert transforms

Let E(R), in short , be the space of all infinitely smooth complex-valued functions φ(t) over . Each φ(t)E should satisfy the set of inequalities

γ k (ϕ)= sup t K | D t k ϕ(t)|<,
(7)

where kN (k=0,1,2,) and K run through compact subsets of , D t d d t . The topology of E(R) is generated by the sequence ( γ k ( ϕ ) ) 0 of multinorms which makes E(R) a locally convex Hausdörff topological vector space. Also,

D(R)E(R),
(8)

D(R) is the Schwartz space of test functions. Denoting by E (R), or E , the space of continuous linear forms on E(R), the topology of D(R) is stronger than that induced on D(R) by E (R), and the restriction of any element of E(R) to D(R) is in D (R), the space of Schwartz distributions; see [7] and [8]. Elements of E (R) are the so-called distributions of compact supports; see, for example, [4, 911] and [6].

Theorem 1 Let (H φ j )0 as j in E(R), then H e φ j 0 and H o φ j 0 in E(R) as j.

Proof The proof of this remark is straightforward from the fact [[9], (78), (79)]

( H e φ j ) (t)= 1 2 ( H φ j ( t ) + H φ j ( t ) )

and

( H o φ j ) (t)= 1 2 ( H φ j ( t ) H φ j ( t ) ) ,

thus detailed proof is avoided. □

We note that the space of distributions of compact support is closed under differentiation and multiplication by smooth functions, thus we have the following theorem.

Theorem 2 Let φE(R), then H e (ξ)E(R).

Proof For every φE(R), we have

| D ξ k H e (ξ)|=| K φ(t) D ξ k cos(ξt)dt| K | t k φ(t)|dt.

Then by considering supremum over compact subsets K of , we get

sup ξ K | D ξ k H e (ξ)|<,

for all nonnegative integers k. This completes the proof of the theorem. □

Theorem 3 Let φE(R), then H o (ξ)E(R).

Proof Proof is similar to that of Theorem 2 and therefore we prefer to omit details. □

Theorem 4 Let φE(R), then hhφE(R).

Proof Let φE(R) be arbitrary, then simple computation yields

| D t k hhφ(t)| 1 π 0 | ξ k || H o (ξ)+ H e (ξ)|dξ,
(9)

where H o (ξ) and H e (ξ) are the odd and even components of Hφ. Theorem 2 and Theorem 3 imply H o (ξ) and H e (ξ) are in E(R). The linearity of E(R) implies H o (ξ)+ H e (ξ)E(R). Hence,

| ξ k || H o (ξ)+ H e (ξ)|E(R).

This implies

0 | ξ k || H o (ξ)+ H e (ξ)|dξE(R).

Hence, from (9), we get

sup t K | D t k hhφ(t)|<

for every kN, where K varies over compact subsets of . Therefore, the theorem is proved. □

Now, by considering proofs which are alike to those used for Theorem 1, Theorem 2 and Theorem 3 and further employing the fact that [2]

FR(ξ)= H e (ξ)andFI(ξ)= H o (ξ),

we state the following theorem.

Theorem 5 Let φE(R), then

(i) FR(ξ)E(R),

(ii) FI(ξ)E(R),

(iii) fhφE(R).

As a consequence of Theorem 4 and Theorem 5, we have

Definition 6 Let f E (R), then the extended Hartley-Hilbert and Fourier-Hilbert transforms of a distribution f of compact support are respectively defined as

hhf,φ=f,hhφ
(10)

and

fhf,φ=f,fhφ
(11)

for every φE(R).

From (10) and (11), we establish that hhf,fhf E (R) for each f E (R).

Theorem 7 Let f E (R), then the mappings

hhf,φ=f,hhφ,φE(R)
(12)

and

fhf,φ=f,fhφ,φE(R)
(13)

are continuous mappings from E (R) onto E (R).

Proof We prove (12) since the proof of (13) is quite similar. Let φE(R), then Theorem 3 implies that

hhφE(R).

Hence, f,hhφ is well defined.

Let f 1 , f 2 E (R), then f 1 + f 2 E (R) and

hh ( f 1 + f 2 ) , φ = f 1 + f 2 ,hhφ,

i.e.,

hh ( f 1 + f 2 ) , φ = f 1 ,hhφ+ f 2 ,hhφ.

Therefore, hhf,φ=f,hhφ defines a linear functional on E(R). If φ j 0 in E(R) as j, then from Theorem 3, hh φ j 0 in E(R) as j.

Thus, f,hh φ j 0 as j, and hence

hhf, φ j 0as j.

Therefore, (12) defines a continuous linear functional on E(R) and thus distribution in E (R).

Moreover, f j f in E (R) as j implies f j ,φf,φ as j, where φE(R) is arbitrary. Thus,

hh f j , φ = f j , hh φ f , hh φ = hh f , φ ,

i.e.,

hh f j hhf

as j in E (R).

This proves (12). Similarly, (13) can be proved. The proof of this theorem is completed. □

Denote by L 1 the Lebesgue space of integrable functions. Between two L 1 functions, φ 1 and φ 2 , the convolution product is defined by

( φ 1 φ 2 )(ξ)= φ 1 (t) φ 2 (ξt)dt.
(14)

The Fourier transform of the convolution product φ 1 φ 2 is defined by [4]

F( φ 1 φ 2 )(ξ)=F φ 1 (ξ)F φ 2 (ξ).
(15)

The most complicated convolution theorem of the Hartley transform, compared to the Fourier transform, is that [1]

H( φ 1 φ 2 )(ξ)= 1 2 G(H φ 1 ×H φ 2 ),
(16)

where

G ( φ 1 × φ 2 ) ( ξ ) = φ 1 ( ξ ) φ 2 ( ξ ) + φ 1 ( ξ ) φ 2 ( ξ ) + φ 1 ( ξ ) φ 2 ( ξ ) φ 1 ( ξ ) φ 2 ( ξ ) .
(17)

Following the manipulation theorem of Hartley transform for the Hartley-Hilbert transform is essential for the following investigation.

Theorem 8 Let φ 1 and φ 2 be L 1 functions, then

( hh ( φ 1 φ 2 ) ) ( t ) = 0 ( H 1 e ( ξ ) H 2 o ( ξ ) + H 1 o ( ξ ) H 2 e ( ξ ) ) cos ( ξ t ) d ξ + 0 ( H 1 e ( ξ ) H 2 e ( ξ ) H 1 o ( ξ ) H 2 o ( ξ ) ) sin ( ξ t ) d ξ ,

where the pair ( H 1 e , H 1 o ) (( H 2 e , H 2 o )) is the even and odd components of Hartley transforms of φ 1 ( φ 2 ), respectively.

Proof For every φ 1 and φ 2 , hh( φ 1 φ 2 ) is given by

( hh ( φ 1 φ 2 ) ) ( t ) = 0 ( H o ( φ 1 φ 2 ) ( ξ ) cos ( ξ t ) + H e ( φ 1 φ 2 ) ( ξ ) sin ( ξ t ) ) d ξ = I 1 + I 2 ,

where

I 1 (t)= 0 H o ( φ 1 φ 2 )(ξ)cos(ξt)dξ
(18)

and

I 2 (t)= 0 H e ( φ 1 φ 2 )(ξ)sin(ξt)dξ.
(19)

It suffices to prove that

I 1 (t)= 0 ( H 1 e ( ξ ) H 2 o ( ξ ) + H 1 o ( ξ ) H 2 e ( ξ ) ) cos(ξt)dξ
(20)

and

I 2 (t)= 0 ( H 1 e ( ξ ) H 2 e ( ξ ) H 1 o ( ξ ) H 2 o ( ξ ) ) sin(ξt)dξ.
(21)

To establish (20), we invoke (14) in (18) to get

I 1 (t)= 0 ( φ 1 ( y ) φ 2 ( x y ) d y ) sin(ξx)cos(ξt)dxdξ.
(22)

Fubini’s theorem and the substitution xy=z imply that

I 1 (t)= 0 cos(ξt) φ 1 (y) φ 2 (z)sin(ξz+ξy)dzdydξ.
(23)

Applying the identity

sin(ξz+ξy)=sin(ξz)cos(ξy)+sin(ξy)cos(ξz)

to (23) and using simple computation jointly with Fubini’s theorem, we establish (20).

Similarly, by using the Fubini’s theorem and the identity

cos(ξz+ξy)=cos(ξz)cos(ξy)sin(ξz)sin(ξy)

together with technique which is similar to that of (20), we derive (21). Hence, the theorem is completely proved. □

3 The Hartley-Hilbert transform of Boehmians

In this section, we investigate Hartley-Hilbert transform since proofs involving Fourier-Hilbert transform are analogous. It is assumed the reader is acquainted with the general construction of general Boehmian spaces. For more details, we refer to [1, 5, 722] and the references cited therein.

Let G be the subspace of Hartley-Hilbert transforms of members of . Then we say gG if there is φE such that hhφ=g. Denote by HD the space of Hartley transforms of D functions. Convergence in G is defined as follows: g i =hh φ i G, for some φ i E, is said to converge to gG if there is φE, g=hhφ and φ i φ as i.

In view of the convolution theorem, we need the following definition.

Definition 9 Let D be the Schwartz space of test functions of bounded support. To G and D, we assign the product × defined by

(g×ψ)(ξ)= 0 ( p ( φ , ψ ) ( ξ ) cos ( ξ t ) + q ( φ , ψ ) ( ξ ) sin ( ξ t ) ) dt,
(24)

where

p(φ,ψ)(ξ)= H e (ξ) ψ o (ξ)+ H o (ξ) ψ e (ξ)

and

q(φ,ψ)(ξ)= H e (ξ) ψ e (ξ) H o (ξ) ψ o (ξ).

H e ( H o ) are the even and odd parts of the Hartley transform of φ, g=hhφ, and ψ e ( ψ o ) are the even and odd parts of ψ, respectively.

Thus, we have the following remark.

Remark 10 Let g=hhφE and HψHD for some ψD, then

g×Hψ=hh(φψ).

By Δ, we mean the family of all delta sequences; see [11]. Denote by HΔ the set of Hartley transforms of all members of Δ. That is, if ( δ i )Δ, then (H δ i )HΔ. Let ( H e δ i ) and ( H o δ i ) be the even and odd parts of (H δ i ), then it is easy to see that

( H e δ i ) (ξ)= δ i (x)cos(xξ)dx1as i,
(25)

and, similarly,

( H o δ i ) (ξ)= δ i (x)sin(xξ)dx0as i
(26)

on compact subsets of .

Theorem 11 Let g=hhφ and ( δ i )Δ, then g×H δ i g as i.

Proof This theorem can be automatically established by invoking (25) and (26) in (24). For some detail, from (24) we have

(g×H δ i )(ξ)= 0 ( p ( φ , δ i ) ( ξ ) cos ( ξ t ) + q ( φ , δ i ) ( ξ ) sin ( ξ t ) ) dt,

where

p(φ, δ i )(ξ)= H e (ξ) H o δ i (ξ)+ H o (ξ) H e δ i (ξ)

and

q(φ, δ i )(ξ)= H e (ξ) H e δ i (ξ) H o (ξ) H o δ i (ξ).

Hence, employing (25) and (26) in the above equation establishes the theorem. □

Theorem 12 Let g i =hh φ i , g=hhφ, g i g as i, and HψHD, then

g i ×Hψg×Hψas i.

Proof of this theorem simply follows from (24).

Now, we state the following theorem without proof and, in fact, the proof of this theorem follows from simple computation.

Theorem 13 Let g 1 =hh φ 1 and g 2 =hh φ 2 , then for every HψHD and kR, we have(i)

( g 1 + g 2 )×Hψ= g 1 ×Hψ+ g 2 ×Hψ;

(ii)

k( g 1 ×Hψ)=k g 1 ×Hψ= g 1 ×kHψ.

The Boehmian space G 2 (G,HD,HΔ,×), or G 2 , is described. Convergence in G 2 (G,HD,HΔ,×) is defined as follows.

δ-convergence A sequence of Boehmians ( β j ) in G 2 (G,HD,HΔ,×) is said to be δ-convergent to a Boehmian β in G 2 (G,HD,HΔ,×), denoted by β j δ β, if there exists a delta sequence ( ϵ i ) such that

( β j × ϵ i ),(β× ϵ i )G,j,iN,

and ( β j × ϵ i )(β× ϵ i ) as j, in G, for every iN.

Δ-convergent A sequence of Boehmians ( β j ) in G 2 (G,HD,HΔ,×) is said to be Δ-convergent to a Boehmian β in G 2 (G,HD,HΔ,×) and denoted by β j Δ β, if there exists a ( ϵ j )Δ such that ( β j β) ϵ j G, jN, and ( β j β) ϵ j 0 as j in G.

Now, let G 1 (E,D,Δ,), G 1 , be the usual Boehmian space obtained from the group and D as a subgroup of .

Definition 14 Let β=[ φ i δ i ] G 1 (E,D,Δ,), then we define the Hartley-Hilbert transform of β=[ φ i δ i ] by

hh E β= [ hh φ i H δ i ]
(27)

in the space G 2 (G,HD,HΔ,×).

Theorem 15 The extended Hartley-Hilbert transform hh E is well defined.

Proof Assume for some φ i , φ i E and δ i , δ i Δ, i=1,2,3, , we have

β:= [ φ i δ i ] = [ φ i δ i ] := β G 1 .

This yields φ i δ i φ i δ i ; i.e., φ i δ i = φ i δ i . An application of the Hartley-Hilbert transform on both sides and Theorem 8 together with (24) yields

hh φ i ×H δ i =hh δ i ×H δ i

for iN. This means hh φ i H δ i hh φ i H δ i , i.e.,

[ hh φ i H δ i ] = [ hh φ i H δ i ] .

Hence,

hh E [ φ i δ i ] = hh E [ φ i δ i ] .

This proves the theorem. □

Theorem 16 The operator hh E : G 1 G 2 is linear.

Proof of this theorem is obvious.

Theorem 17 hh E : G 1 G 2 is one-one.

Proof Assume [ hh φ i H δ i ]=[ hh φ i H δ i ] in G 2 , then certainly,

hh φ i ×H δ i =hh δ i ×H δ i .

By the aid of (24), we get hh( φ i δ i )=hh( φ i δ i ). This yields φ i δ i = φ i δ i , i N. Thus, we have reached the identity

[ φ i δ i ] = [ φ i δ i ] .

This completes the proof of the theorem. □

Theorem 18 The mapping hh E : G 1 G 2 is continuous with respect to Δ convergence.

Proof If β i Δ β in G 1 as i, then we find ( φ i )E and ( δ i )Δ such that

( β i β) δ i = [ φ i δ i δ i ] , φ i 0 as i.

Upon employing hh E to both sides, we get

hh E ( β i β)×H δ i = [ hh φ i × H δ i H δ i ] hh φ i 0
(28)

as i, since φ i 0 as i. Hence, from (28) and Theorem 11, hh E β i hh E β as i. Thus, the proof of the theorem is completed. □

Theorem 19 hh E is continuous with respect to δ-convergence.

Proof If β i δ β as i, then by [9], there are φ j and φ i , j in such that β i =[ φ i , j δ i ], β=[ φ j δ j ], where φ i , j φ j as i. Continuity of hh implies hh φ i , j hh φ j as i. Hence,

[ hh φ i , j δ i ] [ hh φ j δ i ] as j.

This theorem is completely proved. □

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Acknowledgements

The authors express their sincere thanks to the referees for the careful and detailed reading of the manuscript and very helpful suggestions. The second author also acknowledges that the present work was partially supported by University Putra Malaysia under the grand Science Fund No: 5450536.

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Al-Omari, S., Kılıçman, A. On the generalized Hartley-Hilbert and Fourier-Hilbert transforms. Adv Differ Equ 2012, 232 (2012). https://doi.org/10.1186/1687-1847-2012-232

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Keywords

  • Hilbert transform
  • Hartley transform
  • Fourier transform
  • Hartley-Hilbert transform
  • Boehmian space