On the generalized Hartley and Hartley-Hilbert transformations
© Al-Omari and Kılıçman; licensee Springer 2013
Received: 12 March 2013
Accepted: 25 June 2013
Published: 23 July 2013
In this paper, we extend Hartley and Hartley-Hilbert transformations (HT and HHT, respectively) to a certain space of tempered distributions. We then establish a certain convolution theorem for the HHT. The convolution theorem, obtained in this way, has been shown to possess a factorization property of Fourier convolution type. Proving the new convolution theorem for the HHT, by the usual convolution product, the transform is investigated on a certain space of Boehmians. Its properties of linearity and convergence are also discussed in the context of Boehmian spaces.
MSC:54C40, 14E20, 46E25, 20C20.
1 Test function spaces and distributions
The idea of specifying a function not by its values but by its behavior as a functional on some space of testing functions is a concept that is quite familiar to scientists and engineers through their experience with the classical Fourier and Laplace transformations. Test functions, on which distributions operate, cannot in general be written down in an explicit form. The advantage of distributions over classical functions is that the distribution concept provides a better mechanism for analyzing certain physical phenomena than the function concept does because, for one reason, various entities such as the delta function δ can be correctly described as a distribution but not as a function. Furthermore, physical quantities that can be adequately represented as a function can also be characterized as a distribution. In addition, distributions attain an infinite number of derivatives and those derivatives always exist, which is not applied to functions.
The space of testing functions, denoted by , consists of all complex-valued functions φ that are infinitely smooth and zero outside some finite interval. The set of continuous linear forms (conjugates or dual space) on constitutes a space of distributions, denoted by .
The space of complex-valued smooth functions is denoted by ℰ and its dual space is denoted by .
By we denote the space of all complex-valued functions φ that are infinitely smooth and are such that as , they and their partial derivatives decay to zero faster than all powers of . Elements of are called testing functions of rapid descents. is indeed a linear space.
If , then its partial derivatives are in . In fact, is dense in and is dense in ℰ. The dual space of is called the space of tempered distributions and denoted by with a property that , being the (conjugate of ℰ) space of distributions of compact support. For the convergence on , ℰ and and their topologies, we refer to [1, 2].
2 HT and HHT of tempered distributions
2.1 Introduction to HT and HH transforms
- (iv)Convolution: The convolution theorem of HT is given as(3)
are the odd and even components of the HT.
The HHT, which permits some attractive applications in geophysics and signal processing, has been extended to a specific space of generalized functions (Boehmian spaces) in .
2.2 First convolution theorem of HHT
Convolutions of integral transforms which possess the factorization property of Fourier convolution type have become of interest to many authors and have been applied to solving systems of integral equations. In , we studied the convolution theorem for HHT in some detail. In this paper we make the idea more precise. We define some generalized convolution of HHT that permits a factorization property of Fourier convolution type.
Let U be a linear space and V be a commutative algebra on the field K. Let be a linear operator from U to V. A bilinear map is called the convolution for T if for any . The image is denoted by .
Theorem 2.1 (First convolution theorem of HHT)
Similarly, we proceed for to get , is the integral equation given in (7).
Hence the theorem. □
- (iii)Let , then using the definition of HHT and that of ⋆, we get
Hence, the properties of HT odd and even parts, , , and that of the integral operator ∫ imply that .
Proof of (iv) is analogous to that given for part (ii).
This completes the proof. □
Next is a straightforward corollary of Theorem 2.2.
2.3 HT and HHT of distributions
In this subsection we discuss HT and HHT on a tempered distribution space.
This is because the right-hand side of (9) converges uniformly for each x.
for every pair of non-negative m and k. This completes the proof. □
for each .
This can be stated in similar words as: HT and HHT of tempered distributions are tempered distributions. □
Corollary 2.7 If , then .
Theorem 2.8 Let , then H f and HH f are both linear mappings from into .
Similarly, we proceed for HH f, . Hence the theorem. □
3 Generalized distributions
One of the youngest generalizations of functions, and more particularly of distributions, is the theory of Boehmians. The name Boehmian space is given to all objects defined by an abstract construction similar to that of field of quotients. The construction applied to function spaces yields various spaces of generalized functions.
If and , , then .
The elements of Δ are called delta sequences.
for each .
The relation ∼ is an equivalent relation on A and hence splits A into equivalence classes. The equivalence class containing is denoted by . These equivalence classes are called Boehmians and the space of all Boehmians is denoted by , see .
- (1)If as in Y and, is any fixed element, then
If as in Y and , then in Y as .
The operation ∗ is extended to by the following definition.
Definition 3.1 If and , then .
In , two types of convergence, δ and Δ convergence, are defined as follows.
The following lemma is equivalent to the statement of δ-convergence.
4 The spaces and
Theorem 4.1 (Second convolution theorem of HHT)
where ∗ is the usual convolution product of f and g, see .
with and .
Hence, invoking (16) and (17) in (13), our theorem follows. □
Let us consider another space of Boehmians:
Denote by the space of HHT s of distributions from . Indeed, is also a subspace of by (12). A member is said to converge in to a value ξ if there are such that reaches τ for large values of n.
for , .
where , . This proves the theorem. □
Theorem 4.3 Let , then .
where , .
Hence the theorem. □
The proof of (ii) and (iii) follows from simple computations. The proof is therefore completed. □
Theorem 4.5 Let , then .
The proof is completed. □
Proof Follows from similar computations to those above.
Hence our theorem is completely proved. □
Theorem 4.7 Let and and , then .
Hence . This completes the proof. □
as since .
Thus the theorem. □
The Boehmian space is completely established.
5 HHT of Boehmians
It is clear that EHHT is well defined.
Theorem 5.1 EHHT is linear.
Theorem 5.2 EHHT is one-to-one.
Theorem 5.3 EHHT is continuous with respect to δ convergence.
Proof Let in as . We show that in as .
and and as , .
The proof is completed. □
Theorem 5.4 EHHT s are continuous with respect to Δ convergence.
Hence the theorem. □
in the space .
It is of interest to see that the inverse transform preserves all the above properties that Λ does such as linearity, one-to-one and continuity of with respect to convergence in . Proofs are avoided.
The authors are very grateful to the referees for their valuable suggestions and comments that helped to improve the paper.
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