# Some Remarks on the Extended Hartley-Hilbert and Fourier-Hilbert Transforms of Boehmians

Authors: S. K. Q. Al-Omari; A. Kılıçman
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We obtain generalizations of Hartley-Hilbert and Fourier-Hilbert transforms on classes of distributions having compact support. Furthermore, we also study extension to certain space of Lebesgue integrable Boehmians. New characterizing theorems are also established in an adequate performance.


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## Body

## 1. Introduction

The classical theory of integral transforms and their applications have been studied for a long time, and they are applied in many fields of mathematics. Later, after [1], the extension of classical integral transformations to generalized functions has comprised an active area of research. Several integral transforms are extended to various spaces of generalized functions, distributions [2], ultradistributions, Boehmians [3, 4], and many more.In recent years, many papers are devoted to those integral transforms which permit a factorization identity (of Fourier convolution type) such as Fourier transform, Mellin transform, Laplace transform, and few others that have a lot of attraction, the reason that the theory of integral transforms, generally speaking, became an object of study of integral transforms of Boehmian spaces.The Hartley transform is an integral transformation that maps a real-valued temporal or spacial function into a real-valued frequency function via the kernel(1)k(υ;x)=cas(υx).
This novel symmetrical formulation of the traditional Fourier transform, attributed to Hartley 1942, leads to a parallelism that exists between a function of the original variable and that of its transform. In any case, signal and systems analysis and design in the frequency domain using the Hartley transform may be deserving an increased awareness due to the necessity of the existence of a fast algorithm that can substantially lessen the computational burden when compared to the classical complex-valued fast Fourier transform.The Hartley transform of a functionf(x) can be expressed as either [5]
(2)𝒜(υ)=:12π∫-∞∞‍f(x)cas(υx)dx
or
(3)𝒜(f)=:∫-∞∞‍f(x)cas(2πfx)dx,
where the angular or radian frequency variable υ is related to the frequency variable f by υ=2πf and
(4)𝒜(f)=2π𝒜(2πf)=2π𝒜(υ).The integral kernel, known as cosine-sine function, is defined as(5)cas(υx)=cosυx+sin(υx).
Inverse Hartley transform may be defined as either
(6)f(x)=12π∫-∞∞‍𝒜(υ)cas(υx)dυ
or
(7)f(x)=∫-∞∞‍𝒜(f)cas(2πfx)df.
The theory of convolutions of integral transforms has been developed for a long time and is applied in many fields of mathematics. Historically, the convolution product [2]
(8)(f*g)(y)=∫-∞∞‍f(x)g(x-y)dy
has a relationship with the Fourier transform with the factorization property
(9)ℱ(f*g)(y)=ℱ(f)(y)ℱ(g)(y).
The more complicated convolution theorem of Hartley transforms, compared to that of Fourier transforms, is that
(10)𝒜(f*g)(y)=12𝒢(𝒜f×𝒜g)(y),
where
(11)𝒢(f*g)(y)=f(y)g(y)+f(y)g(-y)+f(y)g(y)-f(-y)g(-y).
Some properties of Hartley transforms can be listed as follows.(i)
Linearity: iff and g are real functions then
(12)𝒜(af+bg)(y)=a𝒜(f)(y)+b𝒜(g)(y),a,b∈ℛ.(ii)
Scaling: iff is a real function then
(13)∫-∞∞‍f(ας)cas(2πyς)dς=1a(𝒜f)(ya).


## 2. Distributional Hartley-Hilbert Transform of Compact Support

The Hilbert transform via the Hartley transform is defined by [6, 7]
(14)ℬ𝒜(y)=-1π∫0∞‍(𝒜o(x)cos(xy)+𝒜e(x)sin(xy))dx,
where
(15)𝒜o(x)=𝒜(x)-𝒜(-x)2,𝒜e(x)=𝒜(x)+𝒜(-x)2
are the respective odd and even components of (2).We denote,𝒞(ℛ),  𝒞(ℛ)=𝒞, the space of smooth functions and 𝒞′(ℛ),  𝒞′(ℛ)=𝒞′, the strong dual of 𝒞 of distributions of compact support over ℛ.Following is the convolution theorem ofℬ𝒜.Theorem 1 (convolution Theorem).
Letf and g∈𝒞 then
(16)ℬ𝒜(f*g)(y)=∫0∞‍(k1(x)cos(yx)+k2(x)sin(yx))dx,
where
(17)k1(x)=𝒜ef(x)𝒜og(x)+𝒜of(x)𝒜eg(x),k2(x)=𝒜ef(x)𝒜eg(x)-𝒜of(x)𝒜og(x).Proof.
To prove this theorem it is sufficient to establish that(18)k1(x)=𝒜o(f*g)(x),(19)k2(x)=𝒜e(f*g)(x).
Therefore, we have
(20)𝒜o(f*g)(x)=∫-∞∞‍(∫-∞∞‍f(γ)g(y-γ)dr)cas(xy)dy=∫-∞∞‍f(γ)∫-∞∞‍g(y-γ)cas(xy)dydr.
The substitution y-γ=z and using of (1) together with Fubini theorem imply
(21)𝒜o(f*g)(x)=∫-∞∞‍f(γ)∫-∞∞‍g(z)(cos(x(z+γ))+sin(x(z+γ)))×dzdr.
By invoking the formulae
(22)cos(x(z+γ))=cos(xz)cos(xγ)-sin(xz)sin(xγ),sin(x(z+γ))=sin(xz)cos(xγ)+cos(xz)sin(xγ),
then (18) follows from simple computation. Proof of (19) has a similar technique. Hence, the theorem is completely proved.It is of interest to know thatcos(xy) and sin(xy) are members of 𝒞 and, therefore, 𝒜of,𝒜ef∈𝒞′. This leads to the following statement.Definition 2.
Letf∈𝒞′ then we define the distributional Hartley-Hilbert transform of f as
(23)ℬ𝒜^f(y)=〈𝒜of(x),cos(xy)〉+〈𝒜ef(x),sin(xy)〉.The extended transformℬ𝒜^f is clearly well defined for each f∈𝒞′.Theorem 3.
The distributional Hartley-Hilbert transformℬ𝒜^f is linear.Proof.
Letf,g∈𝒞′ then their components 𝒜ef,𝒜of,𝒜eg,𝒜og∈𝒞′. Hence,
(24)ℬ𝒜^(f+g)(y)=〈𝒜o(f+g)(x),cos(xy)〉+〈𝒜e(f+g)(x),sin(xy)〉.
By factoring and rearranging components we get that
(25)ℬ𝒜^(f+g)(y)=ℬ𝒜^f(y)+ℬ𝒜^g(y).
Furthermore,
(26)ℬ𝒜^(kf)(y)=〈k𝒜of(x),cos(xy)〉+〈k𝒜ef(x),sin(xy)〉.
Hence,
(27)ℬ𝒜^(kf)(y)=kℬ𝒜^f(y).
This completes the proof of the theorem.Theorem 4.
Letf∈𝒞′ then ℬ𝒜^f is a continuous mapping on 𝒞′.Proof.
Letfn,f∈𝒞′,  n∈𝒩 and fn→f as n→∞. Then,
(28)ℬ𝒜^fn(y)=〈𝒜ofn(x),cos(xy)〉+〈𝒜efn(x),sin(xy)〉⟶〈𝒜of(x),cos(xy)〉+〈𝒜ef(x),sin(xy)〉=ℬ𝒜^f(y)asn⟶∞.
Hence we have the following theorem.Theorem 5.
The mappingℬ𝒜^f is one-to-one.Proof.
Letf,g∈𝒞′ and that ℬ𝒜^f=ℬ𝒜^g then, using (23) we get
(29)〈𝒜of(x),cosxy〉+〈𝒜ef(x),sinxy〉=〈𝒜og(x),cosxy〉+〈𝒜eg(x),sinxy〉.
Basic properties of inner product implies
(30)〈𝒜of(x)-𝒜og(x),cos(xy)〉+〈𝒜ef(x)-𝒜eg(x),sin(xy)〉=0.
Hence,
(31)𝒜of(x)=𝒜og(x),𝒜ef(x)=𝒜eg(x).
Therefore,(32)𝒜f(x)=𝒜of(x)+𝒜ef(x)=𝒜og(x)+𝒜eg(x)=𝒜g(x)
for all x. This completes the proof of the theorem.Theorem 6.
Letf∈𝒞′ then f is analytic and
(33)𝒟ykℬ𝒜^f(y)=〈𝒜of(x),𝒟ykcos(xy)〉+〈𝒜ef(x),𝒟yksin(xy)〉.Proof of this theorem is analogous to that of the previous theorem and is thus avoided.Denote byδ the dirac delta function then it is easy to see that
(34)𝒜eδ(y)=1,𝒜eδ(y)=0.


## 3. Lebesgue Space of Boehmians for Hartley-Hilbert Transforms

The original construction of Boehmians produce a concrete space of generalized functions. Since the space of Boehmians was introduced, many spaces of Boehmians were defined. In references, we list selected papers introducing different spaces of Boehmians. One of the main motivations for introducing different spaces of Boehmians was the generalization of integral transforms. The idea requires a proper choice of a space of functions for which a given integral transform is well defined, a choice of a class of delta sequences that is transformed by that integral transform to a well-behaved class of approximate identities, and finally a convolution product that behaves well under the transform. If these conditions are met, the transform has usually an extension to the constructed space of Boehmians and the extension has desirable properties. For general construction of Boehmians, see [8–12].Let𝒟 be the space of test functions of bounded support. By delta sequence, we mean a subset of 𝒟 of sequences (δn) such that
(35)∫-∞∞‍δn(x)dx=1,∥δn∥=∫-∞∞‍|δn(x)|dx<ℳ,0<ℳ∈ℛ,limn→∞∫|x|>ε‍|δn(x)|dx=0foreachε>0,
where ε(δn)(x)={x∈ℛ:δn(x)≠0}.The set of all such delta sequences is usually denoted asΔ. Each element in Δ corresponds to the dirac delta function δ, for large values of n.Proposition 7.
Let(δn)∈Δ then
(36)𝒜eδn(y)=∫-∞∞‍δn(x)cos(xy)dx⟶1asn⟶∞,𝒜oδn(y)=∫-∞∞‍δn(x)sin(xy)dx⟶0asn⟶∞.Letℒ1(ℛ), ℒ1(ℛ)=ℒ1, be the space of complex valued Lebesgue integrable functions. From Proposition 7 we establish the following theorem.Theorem 8.
Letf∈ℒ1  then  ℬ𝒜(f*δn)(y)→ℬ𝒜f(y)  as  n→∞.Proof.
Forf∈ℒ1,  (δn)∈Δ, then using of (14) implies
(37)ℬ𝒜(f*δn)(y)=∫-∞∞‍(𝒜o(f*δn)(x)cosxy+𝒜e(f*δn)(x)sinxy)dx.
Since
(38)(f*δn)(ζ)=∫-∞∞‍f(t)δn(ζ-t)dt⟶f(ζ)
as n→∞, we see that
(39)𝒜o(f*δn)(x)=∫-∞∞‍(f*δn)(ζ)sinxζdζ=∫-∞∞‍f(t)∫-∞∞‍δn(ζ-t)sin(xζ)dζdt⟶∫-∞∞‍f(t)sin(xt)dt=𝒜o(f)(x).
Similarly,
(40)𝒜e(f*δn)(x)⟶𝒜e(f)(x)asn⟶∞.
Therefore, invoking the above equations in (37), we get ℬ𝒜(f*δn)(y)→ℬ𝒜f(y) as n→∞. Hence the theorem.Denote byρℒ1 the space of integrable Boehmians, then ρℒ1 is a convolution algebra when multiplication by scalar, addition, and convolution is defined as [9]
(41)k[fnδn]=[kfnδn],[fnδn]+[gnγn]=[fn*γn+gn*δnδn*γn],[fnδn]*[gnγn]=[fn*gnδn*γn].
Each function f∈ℒ1 is identified with the Boehmian [f*δn/δn]. Also, [fn/δn]*δn=fn∈ℒ1, for every n∈𝒩. Since [δn/δn] corresponds to Dirac delta distribution δ, the kth-derivative of each ρ∈ρℒ1 is defined as
(42)𝒟kρ=ρ*𝒟kδ.
The integral of a Boehmian ρ=[fn/δn]∈ρℒ1 is defined as [11]
(43)∫-∞∞‍ρ(x)dx=∫-∞∞‍f1(x)dx.
It is of great interest to observe the following example.Example 9.
Every infinitely smooth functionf(x)∈ℒ1 such that 𝒟kf(x)∉ℒ1 is integrable as Boehmian but not integrable as function.The following has importance in the sense of analysis.Theorem 10.
Let[fn/δn]∈ρℒ1, then the sequence
(44)ℬ𝒜(fn)(y)=∫-∞∞‍(𝒜ofn(x)cos(xy)+𝒜efn(x)sin(xy))dx
converges uniformly on each compact subset 𝒦 of ℛ.Proof.
By aid of the Theorem8 and the concept of quotient of sequences, we have,
(45)ℬ𝒜(fn)(y)=ℬ𝒜(fn*δkδk)(y)=ℬ𝒜(fn*δkδk)(y)=ℬ𝒜(fkδk*δn)(y)⟶ℬ𝒜fkδk(y)asn⟶∞,
where convergence ranges over compact subsets of ℛ. The theorem is completely proved.Let[fn/δn]∈ρℒ1, then by virtue of Theorem 10 we define the Hartley-Hilbert transform of the Lebesgue Boehmian [fn/δn] as
(46)ℬ𝒜~[fnδn]=limn→∞ℬ𝒜fn.
on compact subsets of ℛ.The next objective is to establish that our definition is well defined. Let[fn/δn]=[gn/γn] in ρℒ1, then
(47)fn*γm=gm*δn,foreverym,n∈𝒩.
Hence, applying the Hartley-Hilbert transform to both sides of the above equation and using the concept of quotients of sequences imply
(48)ℬ𝒜(fn*γm)=ℬ𝒜(gm*δn)=ℬ𝒜(gn*δm).
Thus, in particular, for n=m, and considering Theorem 3, we get
(49)limn→∞ℬ𝒜fn=limn→∞ℬ𝒜gn.
Hence,
(50)ℬ𝒜~[fnδn]=ℬ𝒜~[gnγn].
Definition (46) is therefore well defined.Theorem 11.
The generalized transformℬ𝒜~ is linear.Proof.
Letρ1=[fn/δn] and ρ2=[gn/γn] be arbitrary in ρℒ1 and α∈ℂ then ρ1+ρ2=[(fn*γn+gn*δn)/δn*γn]. Hence, employing (46) yields
(51)ℬ𝒜~(ρ1+ρ2)=limn→∞(ℬ𝒜(fn*γn)+ℬ𝒜(gn*δn)).
By Theorem 8, we get
(52)ℬ𝒜~(ρ1+ρ2)=limn→∞ℬ𝒜fn+limn→∞ℬ𝒜gn.
Hence,
(53)ℬ𝒜~(ρ1+ρ2)=ℬ𝒜~ρ1+ℬ𝒜~ρ2.
Also, for each complex number α, we have
(54)ℬ𝒜~(αρ1)=ℬ𝒜~[αfnδn]=αlimn→∞ℬ𝒜fn=αℬ𝒜~ρ1.
Hence we have the following theorem.Theorem 12.
Letρ∈ρℒ1 and (ϵn)∈Δ, then
(55)ℬ𝒜~(ρ*ϵn)=ℬ𝒜~ρ=ℬ𝒜~(ϵn*ρ).Proof.
Letρ=[fn/δn]∈ρℒ1, then ℬ𝒜~(ρ*ϵn)=ℬ𝒜~[fn*ϵn/δn]=limn→∞ℬ𝒜(fn*ϵn).
Hence,ℬ𝒜~(ρ*ϵn)=limn→∞ℬ𝒜fn=ℬ𝒜~ρ.
Similarly, we proceed forℬ𝒜~ρ=ℬ𝒜~(ϵn*ρ).
This completes the theorem. The following theorem is obvious.Theorem 13.
Ifℬ𝒜~ρ1=0, then ρ1=0.Theorem 14.
The Hartley-Hilbert transformℬ𝒜~ is continuous with respect to the δ-convergence.Proof.
Letρn→δρ in ρℒ1 as n→∞, then we show that ℬ𝒜~ρn→δℬ𝒜~ρ as n→∞. Using ([11, Theorem 2.6]), we find fn,k,fk∈ℒ1,(δk)∈Δ such that [fn,k/δk]=ρn,[fk/δk]=ρ and fn,k→fk as n→∞,k∈𝒩.
Applying the Hartley-Hilbert transform for both sides impliesℬ𝒜fn,k→ℬ𝒜fk in the space of continuous functions. Therefore, considering limits we get
(56)ℬ𝒜~[fn,kδk]⟶ℬ𝒜~[fkδk].
This completes the proof of the theorem.Theorem 15.
The Hartley-Hilbert transformℬ𝒜~ is continuous with respect to the Δ-convergence.Proof.
Letρn→Δρ as n→∞ in ρℒ1, then there is fn∈ℒ1 and δn∈Δ such that
(57)(ρn-ρ)*δn=[fn*δnδk],fn⟶0asn⟶∞.
Thus by the aid of Theorem 3 and the hypothesis of the theorem we have
(58)ℬ𝒜~((ρn-ρ)*δn)=ℬ𝒜~[fn*δnδk]⟶ℬ𝒜(fn*δn)asn⟶∞⟶ℬ𝒜fnasn⟶∞⟶0asn⟶∞.
Therefore, ℬ𝒜~(ρn-ρ)→0 as n→∞. Thus, ℬ𝒜~ρn→Δℬ𝒜~ρ as n→∞.
This completes the proof.Lemma 16.
Let[fn/δn]∈ρℒ1, and δ has the usual meaning of (34) then
(59)ℬ𝒜~([fnδn]*δ)=ℬ𝒜~[fnδn].Proof.
Letρ=[fn/δn]∈ρℒ1, then
(60)ℬ𝒜~([fnδn]*δ)=ℬ𝒜~[fn*δδn]=limn→∞ℬ𝒜(fn*δ)=limn→∞ℬ𝒜fn.
Hence,
(61)ℬ𝒜~([fnδn]*δ)=ℬ𝒜~[fnδn].Theorem 17.
The Hartley-Hilbert transformℬ𝒜~ is one-to-one.Proof.
Letℬ𝒜~[fn/δn]=ℬ𝒜~[gn/γn], then by the aid of (46), we get
(62)limn→∞ℬ𝒜fn=limn→∞ℬ𝒜gn.
Hence,
(63)ℬ𝒜(limn→∞fn)=ℬ𝒜(limn→∞gn),
that is, ℬ𝒜f=ℬ𝒜g. The fact that ℬ𝒜 is one-to-one implies f=g. Hence we have the following theorem.


## 4. A Comparative Study: Fourier-Hilbert Transform

In [6, 7], the Hilbert transform via the Fourier transform of f(x) is defined as
(64)ℬℱ(f)(y)=1π∫0∞‍(ℱi(f)(x)cos(xy)-ℱr(f)(x)sin(xy))dx,
where
(65)ℱr(f)(x)=∫0∞‍f(t)cos(xt)dt,ℱi(f)(x)=∫0∞‍f(t)sin(xt)dt
are, respectively, the real and imaginary components of the Fourier transform of f, which are related by
(66)ℱ(f)(x)=ℱr(f)(x)-iℱi(f)(x).
It is interesting to know that a concrete relationship between ℱr,𝒜e and ℱi,𝒜o is described as ℱr(x)=𝒜e(x) and ℱi(x)=𝒜o(x) [2]. Those equations, above, justify the following statements of the next theorems.Theorem 18.
Let[fn/δn]∈ρℒ1, then the sequence of Fourier-Hilbert transforms of (fn) satisfies
(67)ℬℱ(fn)(y)=1π∫0∞‍(ℱi(fn)(x)cos(xy)-ℱr(fn)(x)sin(xy))dx
and converges uniformly on each compact subset 𝒦 of ℛ.Thus, for[fn/δn]∈ρℒ1, the Fourier-Hilbert transform of [fn/δn] is similarly defined by
(68)ℬℱ~[fnδn]=limn→∞ℬℱfn
on compact subsets of ℛ.The following theorems are stated and their proofs are justified for similar reasons. We prefer to we omit the details.Theorem 19.
The generalized Fourier-Hilbert transformℬℱ~ is linear.Theorem 20.
ℬ
ℱ
~
(
ρ
*
ϵ
n
)
=
ℬ
ℱ
~
ρ
=
ℬ
ℱ
~
(
ϵ
*
ρ
),  (ϵn)∈Δ.Theorem 21.
Ifℬℱ~ρ1=0, then ρ1=0.Theorem 22.
Ifρn→Δρ as n→∞ in ρℒ1, then ℬℱ~ρn→Δℬℱ~ρ as n→∞ in ρℒ1 on compact subsets.Theorem 23.
The Fourier-Hilbert transformℬℱ~ is continuous with respect to the δ-convergence.Theorem 24.
The Fourier-Hilbert transformℬℱ~ is continuous with respect to the Δ-convergence.Proofs of the above theorems are similar to that given for the corresponding ones of Hartley-Hilbert transform.



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