Journal of Mathematics
Volume 2014 (2014), Article ID 790161, 5 pages
http://dx.doi.org/10.1155/2014/790161

Research Article

Some Definition of Hartley-Hilbert and Fourier-Hilbert Transforms in a Quotient Space of Boehmians

S. K. Q. Al-Omari

Department of Applied Sciences, Faculty of Engineering Technology, Al-Balqa’ Applied University, Amman 11134, Jordan

Received 20 May 2014; Accepted 16 August 2014; Published 6 November 2014

Academic Editor: Tepper L Gill

Copyright © 2014 S. K. Q. Al-Omari. 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.

Abstract.

We investigate the Hartley-Hilbert and Fourier-Hilbert transforms on a quotient space of Boehmians. The investigated transforms are well-defined and linear mappings in the space of Boehmians. Further properties are also obtained.

1. Introduction

The Hilbert transform of a function via the Hartley transform is defined in [1, 2] as

where and are, respectively, the even and odd components of the Hartley transform given as [3]

where .

Let be a casual function; that is, , for , and then and are related by a Hilbert transform pair as [3]

The Hilbert transform of via the Fourier transform is defined by

where and are, respectively, the real and imaginary components of the Fourier transform given as

The Hartley transform is extended to Boehmians in [4] and to strong Boehmians in [5]. The Hartley-Hilbert and Fourier-Hilbert transforms were discussed in various spaces of distributions and spaces of Boehmians in [1, 6].

In this paper, aim at investigating the Hartley-Hilbert transform on the context of Boehmians. Investigating the later transform is analogous.

2. Spaces of Quotients (Spaces of Boehmians)

One of the most youngest generalizations of functions and more particularly of distributions is the theory of Boehmians. The idea of the construction of Boehmians was initiated by the concept of regular operators [7]. Regular operators form a subalgebra of the field of Mikusinski operators and they include only such functions whose support is bounded from the left. In a concrete case, the space of Boehmians contains all regular operators, all distributions, and some objects which are neither operators nor distributions.

The construction of Boehmians is similar to the construction of the field of quotients and, in some cases, it gives just the field of quotients. On the other hand, the construction is possible where there are zero divisors, such as space (the space of continuous functions) with the operations of pointwise additions and convolution.

A number of integral transforms have been extended to Boehmian spaces in the recent past by many authors such as Roopkumar in [8, 9]; Mikusinski and Zayed in [10]; Al-Omari and Kilicman in [1, 6, 11]; Karunakaran and Vembu [12]; Karunakaran and Roopkumar [13]; Al-Omari et al. in [14]; and many others.

For abstract construction of Boehmian spaces we refer to [14–16].

By denote the Mellin-type convolution product of first kind defined by [8]

Properties of are presented as follows [8]:

(i);(ii);(iii); is complex number;(iv).

By , we denote the convolution product defined by

By denote the space of test functions of bounded supports defined on that vanish more rapidly than every power of as . Denote by the subset of defined by

Let be the set of delta sequences satisfying the following properties:

:;:, ;:, ;:, and as , where

Now we establish the following theorem.

Theorem 1. Let ; then one has .

Proof. Let and be in ; then we have

By the change of variables , we get that

Once again, by the change of variables , we get

This completes the proof of the theorem.

Now, we establish the following theorem.

Theorem 2. Let , , and be test functions in ; then .

Proof. Let be given. Then, from (6) and (7), we have

Hence, the theorem is completely proved.

Theorem 3. Let ; then .

Proof. Proof of this theorem follows from technique similar to that of Theorem 2. Therefore, we omit the details.

Theorem 4. Let and ; then the following hold:

(1);(2)if as , then as ;(3);(4).

Proof. Proof of (1) and (2) follows from elementary integral calculus. Proof of (3) is already obtained in Theorem 2. Proof of (4) is obvious by the properties of the product (7). Further details are, thus, avoided.
This completes the proof of the theorem.

Theorem 5. Let and ; then as .

Proof. Let be a compact set containing the support of ; then, by Property 2 of delta sequences, we have

Hence, (14) goes to zero as .
Hence the theorem is completely proved.

The Boehmian space is constructed.

The sum of two Boehmians and multiplication by a scalar can be defined in a natural way:

where , the field of complex numbers.

The operation and the differentiation are defined by

Similarly, the space can be proved.

The sum of two Boehmians and multiplication by a scalar can be defined in a natural way:

The operation and the differentiation are defined by

As next, let ; then .

For some details, we by definitions have that

But since

for every power of as , where is a compact set containing the support of , similarly we see that

for every power of as . Therefore, we get that .

3. Hartley-Hilbert Transform of Quotients

Let ; then we define the extended Hartley-Hilbert transform of as

in the quotient space .

Theorem 6. The operator is well defined and linear from into .

Proof. We show that is well defined.
Let in ; then, by the concept of quotients of the space we have

Hence, by Theorem 1, we get from (23) that

Hence, from (24), it follows that

in .
Therefore,

in .
That is,

To show that is linear, let ; then

Linearity of the classical Hartley-Hilbert transform implies

Theorem 1 gives

Thus, by addition of Boehmians

Hence,

The proof of the theorem is completed.

Theorem 7. The necessary and sufficient condition for to be in the range of is that belongs to range of the classical , for every .

Proof. Let be in the range of ; then of course belongs to the range of the classical , .
To establish the converse, let be in the range of , . Then there is such that , .
Since , we get , .
Therefore, Theorem 1 yields

where and , .
Thus, , .
Hence,

The theorem is, therefore, completely proved.

Theorem 8. The transform is consistent with .

Proof. For every , let be its representative; then we have , , . For all it is clear that is independent from the representative.
By Theorem 1 we have

which is the representative of in the space .
Hence the proof of this theorem is completed.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The author expresses his gratitude to the referees for valuable suggestions.

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