On the generalized Hartley-Hilbert and Fourier-Hilbert transforms
© Al-Omari and Kılıçman; licensee Springer 2012
Received: 2 October 2012
Accepted: 30 November 2012
Published: 28 December 2012
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.
where 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.
and being the even and odd components of the Hartley transform .
and being the real and imaginary components of the Fourier transform of .
In , 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 , 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
is the Schwartz space of test functions. Denoting by , or , the space of continuous linear forms on , the topology of is stronger than that induced on by , and the restriction of any element of to is in , the space of Schwartz distributions; see  and . Elements of are the so-called distributions of compact supports; see, for example, [4, 9–11] and .
Theorem 1 Let as in , then and in as .
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 , then .
for all nonnegative integers k. This completes the proof of the theorem. □
Theorem 3 Let , then .
Proof Proof is similar to that of Theorem 2 and therefore we prefer to omit details. □
Theorem 4 Let , then .
for every , where varies over compact subsets of ℛ. Therefore, the theorem is proved. □
we state the following theorem.
Theorem 5 Let , then
As a consequence of Theorem 4 and Theorem 5, we have
for every .
From (10) and (11), we establish that for each .
are continuous mappings from onto .
Hence, is well defined.
Therefore, defines a linear functional on . If in as , then from Theorem 3, in as .
Therefore, (12) defines a continuous linear functional on and thus distribution in .
as in .
This proves (12). Similarly, (13) can be proved. The proof of this theorem is completed. □
Following the manipulation theorem of Hartley transform for the Hartley-Hilbert transform is essential for the following investigation.
where the pair () is the even and odd components of Hartley transforms of (), respectively.
to (23) and using simple computation jointly with Fubini’s theorem, we establish (20).
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, 7–22] and the references cited therein.
Let be the subspace of Hartley-Hilbert transforms of members of ℰ. Then we say if there is such that . Denote by the space of Hartley transforms of functions. Convergence in is defined as follows: , for some , is said to converge to if there is , and as .
In view of the convolution theorem, we need the following definition.
() are the even and odd parts of the Hartley transform of φ, , and () are the even and odd parts of ψ, respectively.
Thus, we have the following remark.
on compact subsets of ℛ.
Theorem 11 Let and , then as .
Hence, employing (25) and (26) in the above equation establishes the theorem. □
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.
The Boehmian space , or , is described. Convergence in is defined as follows.
and as , in , for every .
Δ-convergent A sequence of Boehmians in is said to be Δ-convergent to a Boehmian β in and denoted by , if there exists a such that , , and as in .
Now, let , , be the usual Boehmian space obtained from the group ℰ and as a subgroup of ℰ.
in the space .
Theorem 15 The extended Hartley-Hilbert transform is well defined.
This proves the theorem. □
Theorem 16 The operator is linear.
Proof of this theorem is obvious.
Theorem 17 is one-one.
This completes the proof of the theorem. □
Theorem 18 The mapping is continuous with respect to Δ convergence.
as , since as . Hence, from (28) and Theorem 11, as . Thus, the proof of the theorem is completed. □
Theorem 19 is continuous with respect to δ-convergence.
This theorem is completely proved. □
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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