Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 348701, 6 pages
http://dx.doi.org/10.1155/2013/348701
Research Article
Some Remarks on the Extended Hartley-Hilbert and
Fourier-Hilbert Transforms of Boehmians
S. K. Q. Al-Omari1 and A. KJlJçman2
1
2
Department of Applied Sciences, Faculty of Engineering Technology, Al-Balqa’ Applied University, Amman 11134, Jordan
Department of Mathematics and Institute of Mathematical Research, Universiti Putra Malaysia (UPM),
43400 Serdang, Selangor, Malaysia
Correspondence should be addressed to A. Kılıçman; akilicman@putra.upm.edu.my
Received 19 January 2013; Accepted 15 March 2013
Academic Editor: Mustafa Bayram
Copyright © 2013 S. K. Q. Al-Omari and A. Kılıçman. This is an open access article distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited.
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.
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 realvalued frequency function via the kernel
𝑘 (𝜐; 𝑥) = cas (𝜐𝑥) .
(1)
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 function 𝑓(𝑥) can be
expressed as either [5]
A (𝜐) =:
∞
1
∫ 𝑓 (𝑥) cas (𝜐𝑥) 𝑑𝑥
√2𝜋 −∞
(2)
or
∞
A (𝑓) =: ∫
−∞
𝑓 (𝑥) cas (2𝜋𝑓𝑥) 𝑑𝑥,
(3)
where the angular or radian frequency variable 𝜐 is related to
the frequency variable 𝑓 by 𝜐 = 2𝜋𝑓 and
A (𝑓) = √2𝜋A (2𝜋𝑓) = √2𝜋A (𝜐) .
(4)
The integral kernel, known as cosine-sine function, is
defined as
cas (𝜐𝑥) = cos 𝜐𝑥 + sin (𝜐𝑥) .
(5)
Inverse Hartley transform may be defined as either
𝑓 (𝑥) =
∞
1
∫ A (𝜐) cas (𝜐𝑥) 𝑑𝜐
√2𝜋 −∞
(6)
2
Abstract and Applied Analysis
Theorem 1 (Convolution Theorem). Let 𝑓 and 𝑔 ∈ C then
or
𝑓 (𝑥) = ∫
∞
−∞
A (𝑓) cas (2𝜋𝑓𝑥) 𝑑𝑓.
(7)
∞
BA (𝑓 ∗ 𝑔) (𝑦) = ∫ (𝑘1 (𝑥) cos (𝑦𝑥) + 𝑘2 (𝑥) sin (𝑦𝑥)) 𝑑𝑥,
0
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]
(𝑓 ∗ 𝑔) (𝑦) = ∫
∞
−∞
𝑓 (𝑥) 𝑔 (𝑥 − 𝑦) 𝑑𝑦
(8)
has a relationship with the Fourier transform with the factorization property
F (𝑓 ∗ 𝑔) (𝑦) = F (𝑓) (𝑦) F (𝑔) (𝑦) .
where
𝑘1 (𝑥) = A𝑒 𝑓 (𝑥) A𝑜 𝑔 (𝑥) + A𝑜 𝑓 (𝑥) A𝑒 𝑔 (𝑥) ,
𝑘2 (𝑥) = A𝑒 𝑓 (𝑥) A𝑒 𝑔 (𝑥) − A𝑜 𝑓 (𝑥) A𝑜 𝑔 (𝑥) .
1
A (𝑓 ∗ 𝑔) (𝑦) = G (A𝑓 × A𝑔) (𝑦) ,
2
(10)
(11)
=∫
𝑎, 𝑏 ∈ R.
(12)
(ii) Scaling: if 𝑓 is a real function then
∫
−∞
𝑓 (𝛼𝜍) cas (2𝜋𝑦𝜍) 𝑑𝜍 =
𝑦
1
(A𝑓) ( ) .
𝑎
𝑎
𝑘2 (𝑥) = A𝑒 (𝑓 ∗ 𝑔) (𝑥) .
(19)
(∫
∞
−∞
𝑓 (𝛾) 𝑔 (𝑦 − 𝛾) 𝑑𝑟) cas (𝑥𝑦) 𝑑𝑦
𝑓 (𝛾) ∫
∞
−∞
(20)
𝑔 (𝑦 − 𝛾) cas (𝑥𝑦) 𝑑𝑦𝑑𝑟.
The substitution 𝑦 − 𝛾 = 𝑧 and using of (1) together with
Fubini theorem imply
A𝑜 (𝑓 ∗ 𝑔) (𝑥)
=∫
∞
−∞
∞
∞
−∞
(i) Linearity: if 𝑓 and 𝑔 are real functions then
A (𝑎𝑓 + 𝑏𝑔) (𝑦) = 𝑎A (𝑓) (𝑦) + 𝑏A (𝑔) (𝑦) ,
∞
−∞
Some properties of Hartley transforms can be listed as
follows.
(18)
A𝑜 (𝑓 ∗ 𝑔) (𝑥)
=∫
+ 𝑓 (𝑦) 𝑔 (𝑦) − 𝑓 (−𝑦) 𝑔 (−𝑦) .
𝑘1 (𝑥) = A𝑜 (𝑓 ∗ 𝑔) (𝑥) ,
Therefore, we have
where
G (𝑓 ∗ 𝑔) (𝑦) = 𝑓 (𝑦) 𝑔 (𝑦) + 𝑓 (𝑦) 𝑔 (−𝑦)
(17)
Proof. To prove this theorem it is sufficient to establish that
(9)
The more complicated convolution theorem of Hartley transforms, compared to that of Fourier transforms, is that
(16)
𝑓 (𝛾) ∫
∞
−∞
𝑔 (𝑧) (cos (𝑥 (𝑧 + 𝛾)) + sin (𝑥 (𝑧 + 𝛾)))
× 𝑑𝑧𝑑𝑟.
(21)
(13)
By invoking the formulae
2. Distributional Hartley-Hilbert Transform
of Compact Support
cos (𝑥 (𝑧 + 𝛾)) = cos (𝑥𝑧) cos (𝑥𝛾) − sin (𝑥𝑧) sin (𝑥𝛾) ,
The Hilbert transform via the Hartley transform is defined by
[6, 7]
sin (𝑥 (𝑧 + 𝛾)) = sin (𝑥𝑧) cos (𝑥𝛾) + cos (𝑥𝑧) sin (𝑥𝛾) ,
(22)
A
then (18) follows from simple computation. Proof of (19) has
a similar technique. Hence, the theorem is completely proved.
where
It is of interest to know that cos(𝑥𝑦) and sin(𝑥𝑦) are
members of C and, therefore, A𝑜 𝑓, A𝑒 𝑓 ∈ C󸀠 . This leads to
the following statement.
1 ∞
B (𝑦) = − ∫ (A𝑜 (𝑥) cos (𝑥𝑦) + A𝑒 (𝑥) sin (𝑥𝑦)) 𝑑𝑥,
𝜋 0
(14)
A𝑜 (𝑥) =
A (𝑥) − A (−𝑥)
,
2
A (𝑥) + A (−𝑥)
A𝑒 (𝑥) =
2
(15)
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 of BA .
Definition 2. Let 𝑓 ∈ C󸀠 then we define the distributional
Hartley-Hilbert transform of 𝑓 as
̂
BA 𝑓 (𝑦) = ⟨A𝑜 𝑓 (𝑥) , cos (𝑥𝑦)⟩ + ⟨A𝑒 𝑓 (𝑥) , sin (𝑥𝑦)⟩ .
(23)
̂
A
The extended transform B
𝑓 is clearly well defined for
󸀠
each 𝑓 ∈ C .
Abstract and Applied Analysis
3
Theorem 3. The distributional Hartley-Hilbert transform
̂
BA 𝑓 is linear.
Proof. Let 𝑓, 𝑔 ∈ C󸀠 then their components A𝑒 𝑓, A𝑜 𝑓, A𝑒 𝑔,
A𝑜 𝑔 ∈ C󸀠 . Hence,
̂
BA (𝑓 + 𝑔) (𝑦) = ⟨A𝑜 (𝑓 + 𝑔) (𝑥) , cos (𝑥𝑦)⟩
+ ⟨A𝑒 (𝑓 + 𝑔) (𝑥) , sin (𝑥𝑦)⟩ .
(24)
By factoring and rearranging components we get that
̂
̂
̂
BA (𝑓 + 𝑔) (𝑦) = BA 𝑓 (𝑦) + BA 𝑔 (𝑦) .
(25)
+ ⟨𝑘A𝑒 𝑓 (𝑥) , sin (𝑥𝑦)⟩ .
(26)
Hence,
̂
̂
BA (𝑘𝑓) (𝑦) = 𝑘BA 𝑓 (𝑦) .
(27)
This completes the proof of the theorem.
Theorem 4. Let 𝑓 ∈ C󸀠 then ̂
BA 𝑓 is a continuous mapping
󸀠
on C .
Proof. Let 𝑓𝑛 , 𝑓 ∈ C󸀠 , 𝑛 ∈ N and 𝑓𝑛 → 𝑓 as 𝑛 → ∞. Then,
̂
BA 𝑓𝑛 (𝑦) = ⟨A𝑜 𝑓𝑛 (𝑥) , cos (𝑥𝑦)⟩ + ⟨A𝑒 𝑓𝑛 (𝑥) , sin (𝑥𝑦)⟩
󳨀→ ⟨A𝑜 𝑓 (𝑥) , cos (𝑥𝑦)⟩ + ⟨A𝑒 𝑓 (𝑥) , sin (𝑥𝑦)⟩
̂
= BA 𝑓 (𝑦)
̂
D𝑘𝑦 BA 𝑓 (𝑦) = ⟨A𝑜 𝑓 (𝑥) , D𝑘𝑦 cos (𝑥𝑦)⟩
+ ⟨A𝑒 𝑓 (𝑥) , D𝑘𝑦 sin (𝑥𝑦)⟩ .
as 𝑛 󳨀→ ∞.
(28)
Proof of this theorem is analogous to that of the previous
theorem and is thus avoided.
Denote by 𝛿 the dirac delta function then it is easy to see
that
A𝑒 𝛿 (𝑦) = 1,
∫
∞
Proof. Let 𝑓, 𝑔 ∈ C󸀠 and that ̂
BA 𝑓 = ̂
BA 𝑔 then, using (23)
we get
󵄩󵄩 󵄩󵄩
󵄩󵄩𝛿𝑛 󵄩󵄩 = ∫
−∞
lim ∫
𝑛 → ∞ |𝑥|>𝜀
𝑒
= ⟨A𝑜 𝑔 (𝑥) , cos 𝑥𝑦⟩ + ⟨A𝑒 𝑔 (𝑥) , sin 𝑥𝑦⟩ .
+ ⟨A𝑒 𝑓 (𝑥) − A𝑒 𝑔 (𝑥) , sin (𝑥𝑦)⟩ = 0.
A 𝑓 (𝑥) = A 𝑔 (𝑥) ,
𝑒
𝑒
A 𝑓 (𝑥) = A 𝑔 (𝑥) .
∞
−∞
(31)
∞
A𝑜 𝛿𝑛 (𝑦) = ∫
−∞
= A𝑜 𝑔 (𝑥) + A𝑒 𝑔 (𝑥) = A𝑔 (𝑥)
for all 𝑥. This completes the proof of the theorem.
0 < M ∈ R,
(35)
󵄨
󵄨󵄨
󵄨󵄨𝛿𝑛 (𝑥)󵄨󵄨󵄨 𝑑𝑥 = 0 for each 𝜀 > 0,
Proposition 7. Let (𝛿𝑛 ) ∈ Δ then
Therefore,
A𝑓 (𝑥) = A𝑜 𝑓 (𝑥) + A𝑒 𝑓 (𝑥)
󵄨
󵄨󵄨
󵄨󵄨𝛿𝑛 (𝑥)󵄨󵄨󵄨 𝑑𝑥 < M,
(30)
A𝑒 𝛿𝑛 (𝑦) = ∫
𝑜
𝛿𝑛 (𝑥) 𝑑𝑥 = 1,
where 𝜀(𝛿𝑛 )(𝑥) = {𝑥 ∈ R : 𝛿𝑛 (𝑥) ≠ 0}.
The set of all such delta sequences is usually denoted as Δ.
Each element in Δ corresponds to the dirac delta function 𝛿,
for large values of 𝑛.
Hence,
𝑜
(34)
(29)
Basic properties of inner product implies
⟨A𝑜 𝑓 (𝑥) − A𝑜 𝑔 (𝑥) , cos (𝑥𝑦)⟩
∞
−∞
Theorem 5. The mapping ̂
BA 𝑓 is one-to-one.
⟨A 𝑓 (𝑥) , cos 𝑥𝑦⟩ + ⟨A 𝑓 (𝑥) , sin 𝑥𝑦⟩
A𝑒 𝛿 (𝑦) = 0.
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].
Let D be the space of test functions of bounded support.
By delta sequence, we mean a subset of D of sequences (𝛿𝑛 )
such that
Hence we have the following theorem.
𝑜
(33)
3. Lebesgue Space of Boehmians for
Hartley-Hilbert Transforms
Furthermore,
̂
BA (𝑘𝑓) (𝑦) = ⟨𝑘A𝑜 𝑓 (𝑥) , cos (𝑥𝑦)⟩
Theorem 6. Let 𝑓 ∈ C󸀠 then 𝑓 is analytic and
(32)
𝛿𝑛 (𝑥) cos (𝑥𝑦) 𝑑𝑥 󳨀→ 1 as 𝑛 󳨀→ ∞,
𝛿𝑛 (𝑥) sin (𝑥𝑦) 𝑑𝑥 󳨀→ 0
as 𝑛 󳨀→ ∞.
(36)
Let L1 (R), L1 (R) = L1 , be the space of complex
valued Lebesgue integrable functions. From Proposition 7 we
establish the following theorem.
4
Abstract and Applied Analysis
Theorem 8. Let 𝑓 ∈ L1 then BA (𝑓 ∗ 𝛿𝑛 )(𝑦) → BA 𝑓(𝑦)
as 𝑛 → ∞.
Proof. For 𝑓 ∈ L1 , (𝛿𝑛 ) ∈ Δ, then using of (14) implies
The following has importance in the sense of analysis.
BA (𝑓 ∗ 𝛿𝑛 ) (𝑦)
=∫
∞
−∞
Example 9. Every infinitely smooth function 𝑓(𝑥) ∈ L1 such
that D𝑘 𝑓(𝑥) ∉ L1 is integrable as Boehmian but not integrable as function.
𝑜
𝑒
(A (𝑓 ∗ 𝛿𝑛 ) (𝑥) cos 𝑥𝑦 + A (𝑓 ∗ 𝛿𝑛 ) (𝑥) sin 𝑥𝑦) 𝑑𝑥.
(37)
Since
Theorem 10. Let [𝑓𝑛 /𝛿𝑛 ] ∈ 𝜌L1 , then the sequence
BA (𝑓𝑛 ) (𝑦)
∞
=∫
−∞
(𝑓 ∗ 𝛿𝑛 ) (𝜁) = ∫
∞
𝑓 (𝑡) 𝛿𝑛 (𝜁 − 𝑡) 𝑑𝑡 󳨀→ 𝑓 (𝜁)
−∞
(38)
∞
−∞
=∫
(44)
converges uniformly on each compact subset K of R.
Proof. By aid of the Theorem 8 and the concept of quotient of
sequences, we have,
as 𝑛 → ∞, we see that
A𝑜 (𝑓 ∗ 𝛿𝑛 ) (𝑥) = ∫
(A𝑜 𝑓𝑛 (𝑥) cos (𝑥𝑦) + A𝑒 𝑓𝑛 (𝑥) sin (𝑥𝑦)) 𝑑𝑥
∞
∞
−∞
󳨀→ ∫
(𝑓 ∗ 𝛿𝑛 ) (𝜁) sin 𝑥𝜁𝑑𝜁
𝑓 (𝑡) ∫
∞
−∞
−∞
BA (𝑓𝑛 ) (𝑦) = BA (𝑓𝑛 ∗
𝛿𝑛 (𝜁 − 𝑡) sin (𝑥𝜁) 𝑑𝜁𝑑𝑡
= BA (
𝑓 (𝑡) sin (𝑥𝑡) 𝑑𝑡
𝛿𝑘
) (𝑦)
𝛿𝑘
𝑓𝑛 ∗ 𝛿𝑘
) (𝑦)
𝛿𝑘
𝑓
= B ( 𝑘 ∗ 𝛿𝑛 ) (𝑦)
𝛿𝑘
(45)
A
𝑜
= A (𝑓) (𝑥) .
󳨀→ BA
(39)
Similarly,
𝑓𝑘
(𝑦)
𝛿𝑘
as 𝑛 󳨀→ ∞,
(40)
where convergence ranges over compact subsets of R. The
theorem is completely proved.
Therefore, invoking the above equations in (37), we get
BA (𝑓 ∗ 𝛿𝑛 )(𝑦) → BA 𝑓(𝑦) as 𝑛 → ∞. Hence the
theorem.
Let [𝑓𝑛 /𝛿𝑛 ] ∈ 𝜌L1 , then by virtue of Theorem 10 we define
the Hartley-Hilbert transform of the Lebesgue Boehmian
[𝑓𝑛 /𝛿𝑛 ] as
Denote by 𝜌L1 the space of integrable Boehmians, then
𝜌L1 is a convolution algebra when multiplication by scalar,
addition, and convolution is defined as [9]
̃ 𝑓
BA [ 𝑛 ] = lim BA 𝑓𝑛 .
𝑛→∞
𝛿𝑛
A𝑒 (𝑓 ∗ 𝛿𝑛 ) (𝑥) 󳨀→ A𝑒 (𝑓) (𝑥)
as 𝑛 󳨀→ ∞.
𝑓
𝑘𝑓
𝑘[ 𝑛] = [ 𝑛],
𝛿𝑛
𝛿𝑛
[
𝑔
𝑓 ∗ 𝛾𝑛 + 𝑔𝑛 ∗ 𝛿𝑛
𝑓𝑛
] + [ 𝑛] = [ 𝑛
],
𝛿𝑛
𝛾𝑛
𝛿𝑛 ∗ 𝛾𝑛
[
(41)
𝑓𝑛
𝑔
𝑓 ∗ 𝑔𝑛
] ∗ [ 𝑛] = [ 𝑛
].
𝛿𝑛
𝛾𝑛
𝛿𝑛 ∗ 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
D𝑘 𝜌 = 𝜌 ∗ D𝑘 𝛿.
(42)
The integral of a Boehmian 𝜌 = [𝑓𝑛 /𝛿𝑛 ] ∈ 𝜌L1 is defined as
[11]
∫
∞
−∞
𝜌 (𝑥) 𝑑𝑥 = ∫
∞
−∞
𝑓1 (𝑥) 𝑑𝑥.
It is of great interest to observe the following example.
(43)
(46)
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
𝑓𝑛 ∗ 𝛾𝑚 = 𝑔𝑚 ∗ 𝛿𝑛 ,
for every 𝑚, 𝑛 ∈ N.
(47)
Hence, applying the Hartley-Hilbert transform to both sides
of the above equation and using the concept of quotients of
sequences imply
BA (𝑓𝑛 ∗ 𝛾𝑚 ) = BA (𝑔𝑚 ∗ 𝛿𝑛 ) = BA (𝑔𝑛 ∗ 𝛿𝑚 ) .
(48)
Thus, in particular, for 𝑛 = 𝑚, and considering Theorem 3,
we get
lim BA 𝑓𝑛 = lim BA 𝑔𝑛 .
(49)
𝑓
̃
̃ 𝑔
BA [ 𝑛 ] = BA [ 𝑛 ] .
𝛿𝑛
𝛾𝑛
(50)
𝑛→∞
𝑛→∞
Hence,
Definition (46) is therefore well defined.
Theorem 11. The generalized transform ̃
BA is linear.
Abstract and Applied Analysis
5
Proof. Let 𝜌1 = [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
̃
BA (𝜌1 + 𝜌2 ) = lim (BA (𝑓𝑛 ∗ 𝛾𝑛 ) + BA (𝑔𝑛 ∗ 𝛿𝑛 )) .
𝑛→∞
Δ
→ 𝜌 as 𝑛 → ∞ in 𝜌L1 , then there is 𝑓𝑛 ∈ L1
Proof. Let 𝜌𝑛 󳨀
and 𝛿𝑛 ∈ Δ such that
(𝜌𝑛 − 𝜌) ∗ 𝛿𝑛 = [
𝑓𝑛 ∗ 𝛿𝑛
],
𝛿𝑘
𝑓𝑛 󳨀→ 0 as 𝑛 󳨀→ ∞.
(57)
(51)
Thus by the aid of Theorem 3 and the hypothesis of the
theorem we have
By Theorem 8, we get
̃
BA (𝜌1 + 𝜌2 ) = lim BA 𝑓𝑛 + lim BA 𝑔𝑛 .
(52)
̃
̃
̃
BA (𝜌1 + 𝜌2 ) = BA 𝜌1 + BA 𝜌2 .
(53)
𝑛→∞
𝑛→∞
Hence,
̃
̃ 𝑓 ∗ 𝛿𝑛
BA ((𝜌𝑛 − 𝜌) ∗ 𝛿𝑛 ) = BA [ 𝑛
]
𝛿𝑘
󳨀→ BA (𝑓𝑛 ∗ 𝛿𝑛 )
󳨀→ BA 𝑓𝑛
Also, for each complex number 𝛼, we have
𝑛→∞
(58)
(54)
̃
= 𝛼BA 𝜌1 .
Theorem 12. Let 𝜌 ∈ 𝜌L1 and (𝜖𝑛 ) ∈ Δ, then
𝑛→∞
̃ 𝑓
̃ 𝑓
BA ([ 𝑛 ] ∗ 𝛿) = BA [ 𝑛 ] .
𝛿𝑛
𝛿𝑛
(55)
BA (𝜌 ∗ 𝜖𝑛 ) = ̃
BA [𝑓𝑛 ∗
Proof. Let 𝜌 = [𝑓𝑛 /𝛿𝑛 ] ∈ 𝜌L1 , then ̃
A
𝜖𝑛 /𝛿𝑛 ] = lim𝑛 → ∞ B (𝑓𝑛 ∗ 𝜖𝑛 ).
BA 𝑓 = ̃
BA 𝜌.
Hence, ̃
BA (𝜌 ∗ 𝜖 ) = lim
𝑛
Δ
Therefore, ̃
BA (𝜌𝑛 − 𝜌) → 0 as 𝑛 → ∞. Thus, ̃
BA 𝜌𝑛 󳨀
→
̃
A
𝜌 as 𝑛 → ∞.
B
This completes the proof.
Lemma 16. Let [𝑓𝑛 /𝛿𝑛 ] ∈ 𝜌L1 , and 𝛿 has the usual meaning
of (34) then
Hence we have the following theorem.
̃
̃
̃
BA (𝜌 ∗ 𝜖𝑛 ) = BA 𝜌 = BA (𝜖𝑛 ∗ 𝜌) .
as 𝑛 󳨀→ ∞
󳨀→ 0 as 𝑛 󳨀→ ∞.
̃
̃ 𝛼𝑓
BA (𝛼𝜌1 ) = BA [ 𝑛 ]
𝛿𝑛
= 𝛼 lim BA 𝑓𝑛
as 𝑛 󳨀→ ∞
(59)
Proof. Let 𝜌 = [𝑓𝑛 /𝛿𝑛 ] ∈ 𝜌L1 , then
̃ 𝑓
̃ 𝑓 ∗𝛿
BA ([ 𝑛 ] ∗ 𝛿) = BA [ 𝑛
]
𝛿𝑛
𝛿𝑛
𝑛
̃
A
𝜌=̃
BA (𝜖𝑛 ∗ 𝜌).
Similarly, we proceed for B
This completes the theorem. The following theorem is
obvious.
= lim BA (𝑓𝑛 ∗ 𝛿)
𝑛→∞
(60)
= lim BA 𝑓𝑛 .
𝑛→∞
Theorem 13. If ̃
BA 𝜌1 = 0, then 𝜌1 = 0.
Theorem 14. The Hartley-Hilbert transform ̃
BA is continuous
with respect to the 𝛿-convergence.
Hence,
̃ 𝑓
̃ 𝑓
BA ([ 𝑛 ] ∗ 𝛿) = BA [ 𝑛 ] .
𝛿𝑛
𝛿𝑛
(61)
𝛿
󳨀 𝜌 in 𝜌L1 as 𝑛 → ∞, then we show that
Proof. Let 𝜌𝑛 →
𝛿 ̃
̃
A
A
B 𝜌𝑛 →
󳨀 B 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
implies BA 𝑓𝑛,𝑘 → BA 𝑓𝑘 in the space of continuous
functions. Therefore, considering limits we get
̃ 𝑓
̃ 𝑓𝑛,𝑘
] 󳨀→ BA [ 𝑘 ] .
BA [
𝛿𝑘
𝛿𝑘
̃
A
Theorem 17. The Hartley-Hilbert transform B
is one-toone.
̃
A
[𝑓𝑛 /𝛿𝑛 ] = ̃
BA [𝑔𝑛 /𝛾𝑛 ], then by the aid of (46),
Proof. Let B
we get
lim BA 𝑓𝑛 = lim BA 𝑔𝑛 .
(62)
BA ( lim 𝑓𝑛 ) = BA ( lim 𝑔𝑛 ) ,
(63)
𝑛→∞
(56)
This completes the proof of the theorem.
Theorem 15. The Hartley-Hilbert transform ̃
BA is continuous
with respect to the Δ-convergence.
𝑛→∞
Hence,
𝑛→∞
𝑛→∞
that is, BA 𝑓 = BA 𝑔. The fact that BA is one-to-one implies
𝑓 = 𝑔. Hence we have the following theorem.
6
Abstract and Applied Analysis
4. A Comparative Study: Fourier-Hilbert
Transform
̃
F
is continuTheorem 23. The Fourier-Hilbert transform B
ous with respect to the 𝛿-convergence.
In [6, 7], the Hilbert transform via the Fourier transform of
𝑓(𝑥) is defined as
̃
F
Theorem 24. The Fourier-Hilbert transform B
is continuous with respect to the Δ-convergence.
BF (𝑓) (𝑦)
=
∞
1
∫ (F𝑖 (𝑓) (𝑥) cos (𝑥𝑦) − F𝑟 (𝑓) (𝑥) sin (𝑥𝑦)) 𝑑𝑥,
𝜋 0
(64)
where
∞
F𝑟 (𝑓) (𝑥) = ∫ 𝑓 (𝑡) cos (𝑥𝑡) 𝑑𝑡,
0
∞
(65)
F𝑖 (𝑓) (𝑥) = ∫ 𝑓 (𝑡) sin (𝑥𝑡) 𝑑𝑡
0
are, respectively, the real and imaginary components of the
Fourier transform of 𝑓, which are related by
F (𝑓) (𝑥) = F𝑟 (𝑓) (𝑥) − 𝑖F𝑖 (𝑓) (𝑥) .
(66)
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 FourierHilbert transforms of (𝑓𝑛 ) satisfies
BF (𝑓𝑛 ) (𝑦)
=
1 ∞
∫ (F𝑖 (𝑓𝑛 ) (𝑥) cos (𝑥𝑦) − F𝑟 (𝑓𝑛 ) (𝑥) sin (𝑥𝑦)) 𝑑𝑥
𝜋 0
(67)
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
̃ 𝑓
BF [ 𝑛 ] = lim BF 𝑓𝑛
𝑛→∞
𝛿𝑛
(68)
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.
̃
F
Theorem 19. The generalized Fourier-Hilbert transform B
is linear.
̃
̃
̃
F
F
F
Theorem 20. B
(𝜌 ∗ 𝜖𝑛 ) = B
𝜌=B
(𝜖 ∗ 𝜌), (𝜖𝑛 ) ∈ Δ.
̃
F
Theorem 21. If B
𝜌1 = 0, then 𝜌1 = 0.
Δ
Δ
̃
F
→ 𝜌 as 𝑛 → ∞ in 𝜌L1 , then B
𝜌𝑛 󳨀
→
Theorem 22. If 𝜌𝑛 󳨀
̃
F
B
𝜌 as 𝑛 → ∞ in 𝜌L1 on compact subsets.
Proofs of the above theorems are similar to that given for
the corresponding ones of Hartley-Hilbert transform.
References
[1] A. H. Zemanian, Generalized Integral Transformations, Dover
Publications, New York, NY, USA, 2nd edition, 1987.
[2] R. S. Pathak, Integral Transforms of Generalized Functions and
Their Applications, Gordon and Breach Science Publishers,
North Vancouver, Canada, 1997.
[3] S. K. Q. Al-Omari, D. Loonker, P. K. Banerji, and S. L. Kalla,
“Fourier sine (cosine) transform for ultradistributions and their
extensions to tempered and ultraBoehmian spaces,” Integral
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[4] S. Al-Omari and A. Kılıçman, “On the generalized HartleyHilbert and fourier-Hilbert transforms,” Advances in Difference
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[5] R. P. Millane, “Analytic properties of the Hartley transform and
their applications,” Proceedings of the IEEE, vol. 82, no. 3, pp.
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[6] N. Sundararajan and Y. Srinivas, “Fourier-Hilbert versus
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2010.
[7] N. Sundarajan, “Fourier and hartley transforms: a mathematical
twin,” Indian Journal of Pure and Applied Mathematics, vol. 8,
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[8] T. K. Boehme, “The support of Mikusiński operators,” Transactions of the American Mathematical Society, vol. 176, pp. 319–334,
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[9] P. Mikusiński, “Fourier transform for integrable Boehmians,”
The Rocky Mountain Journal of Mathematics, vol. 17, no. 3, pp.
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[10] S. K. Q. Al-Omari and A. Kılıçman, “On diffraction Fresnel
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Proceedings of the American Mathematical Society, vol. 123, no.
3, pp. 813–817, 1995.
[12] S. K. Q. Al-Omari and A. Kılıçman, “Note on Boehmians for
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