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 Correspondence should be addressed to S. K. Q. Al-Omari; s.k.q.alomari@fet.edu.jo 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.


Introduction
The Hilbert transform of a function () via the Hartley transform is defined in [1,2] as where () e h and () o h are, respectively, the even and odd components of the Hartley transform () h given as [3] () h () = ∫ where cas() = cos() + sin().
Let () be a casual function; that is, () = 0, for  > 0, and then () o h () and () e h () are related by a Hilbert transform pair as [3] () ( The Hilbert transform of () via the Fourier transform is defined by where () r f () and () i f () 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.

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 2 Journal of Mathematics 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 C (the space of continuous functions) with the operations of pointwise additions and convolution.
By ⊛ denote the Mellin-type convolution product of first kind defined by [8] Properties of ⊛ are presented as follows [8]: By * , we denote the convolution product defined by By (R + ) denote the space of test functions of bounded supports defined on R + that vanish more rapidly than every power of  as  → ∞.Denote by  1 the subset of (R + ) defined by Let Δ be the set of delta sequences satisfying the following properties: (1): Now we establish the following theorem.
Once again, by the change of variables  = , we get This completes the proof of the theorem.Now, we establish the following theorem.Proof.Let  > 0 be given.Then, from ( 6) and ( 7), we have Hence, the theorem is completely proved.
Proof.Proof of this theorem follows from technique similar to that of Theorem 2. Therefore, we omit the details.
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 B (, ( 1 , ⊛) , * , Δ) is constructed.The sum of two Boehmians and multiplication by a scalar can be defined in a natural way: where  ∈ C, the field of complex numbers.
The operation * and the differentiation are defined by Similarly, the space B (, ( 1 , ⊛) , ⊛, Δ) 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  ∈ (R + ); then () hh () ∈ (R + ).
For some details, we by definitions have that for every power of  as  → ∞.Therefore, we get that () hh () ∈ (R + ).