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On the generalized Hartley and Hartley-Hilbert transformations
Advances in Difference Equations volume 2013, Article number: 222 (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 .
A distribution f in can be represented, corresponding to , through the convergent integral
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
The Hartley transform (HT) was introduced originally by Hartley 1942 as an integral transform with a number of properties similar to those of the Fourier transform (FT). The HT of a function over R is a real function defined by [3, 4]
Some properties of HT are:
Convolution: The convolution theorem of HT is given as(3)
The Hilbert transform via the Hartley transform, the Hartley-Hilbert transform (HHT), is defined by 
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)
Let HH f, HH g be the HHT s of f and g, respectively, then
Proof Under the hypothesis of the theorem, we write
which can be written as
Then, with a simple modification, we get
Similarly, we proceed for to get , is the integral equation given in (7).
Hence the theorem. □
Theorem 2.2 Let f, g and h be functions, then:
Proof (i) Let . By the aid of Theorem 2.1, we get
Proof of (ii) is analogous to that of the first part.
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.
Corollary 2.3 Let f, g and h be functions, then:
2.3 HT and HHT of distributions
In this subsection we discuss HT and HHT on a tempered distribution space.
Theorem 2.4 If f is in , then H f is also in .
Proof If , then HT certainly exists. Differentiating, in the ordinary sense, the right-hand side of the integral equation
with respect to x yields
This is because the right-hand side of (9) converges uniformly for each x.
Indeed, integrating by parts m times and by the fact that
as for each , we get
Since , the integral on the right-hand side of (10) is bounded by constants, say . Hence
for every pair of non-negative m and k. This completes the proof. □
Corollary 2.5 If f is in , then and are in .
Corollary 2.6 If f is in , then HH f is also in .
Proof By Corollary 2.5, . Thus, the linearity of and the fact that for every imply .
Let , then, by the aid of Corollary 2.5 and Corollary 2.6, we are led to the following definitions:
The right-hand sides of (11) and (12) are well defined and, therefore, from the left-hand sides of (11) and (12), we get that
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 .
Proof Let and , be arbitrary, then
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.
For a linear space Y and a subspace X of Y, assume that to all pairs , of elements, , , the products , are assigned such that the following conditions are satisfied:
Let Δ be a family of sequences from X such that for then:
If and , , then .
The elements of Δ are called delta sequences.
Consider the class A of pairs of sequences defined by
for each .
The pair is said to be quotient of sequences, denoted by , if
Two quotients of sequences and are said to be equivalent, , if
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 .
The sum and multiplication by a scalar of two Boehmians can be defined in a natural way
The operation ∗ and differentiation are defined by and . Many times, Y is equipped with the notion of convergence. The intrinsic relationship between the notion of convergence and the product ∗ are given by:
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.
Definition 3.2 A sequence of Boehmians in is said to be δ-convergent to a Boehmian β in , denoted by , if there exists a delta sequence such that , , and
The following lemma is equivalent to the statement of δ-convergence.
Lemma 3.3 () in if and only if there are and such that , and for each ,
4 The spaces and
Theorem 4.1 (Second convolution theorem of HHT)
Let f and g be functions, then
where ∗ is the usual convolution product of f and g, see .
Proof Using the definition of HHT implies
with and .
Fubini’s theorem implies
The substitution and the fact
in (14) then multiplying and canceling similar quantities yield
Similarly, we proceed for to get
Hence, invoking (16) and (17) in (13), our theorem follows. □
Denote by the usual Boehmian space with the convolution product ∗ as an operation, as a group, as a subgroup of ( dense in ) and, Δ as the collection of delta sequences from such that
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.
Also, denote by the set of HHT s of test functions from , then is a subspace of by Corollary 2.6. In similar notations .
Next, let us consider an operation defined by
for , .
Theorem 4.2 Let and , then for and ,
Proof For every , , we get
where , . This proves the theorem. □
Theorem 4.3 Let , then .
Proof Using (17) we get
where , .
By (18) and Theorem 4.2, we get
Hence the theorem. □
Theorem 4.4 Let and , then:
Proof (i) The linearity of HHT s and (17) implies
The proof of (ii) and (iii) follows from simple computations. The proof is therefore completed. □
Theorem 4.5 Let , then .
Proof For ,
Since , we get
The proof is completed. □
Theorem 4.6 Let , , then
Proof Follows from similar computations to those above.
In detail, for , and , we see that
Hence our theorem is completely proved. □
Theorem 4.7 Let and and , then .
Proof Assume that , then
Hence, . Allowing implies
Hence . This completes the proof. □
Theorem 4.8 Let and , then
Proof Since , , there are , such that and . Hence
as since .
Thus the theorem. □
The Boehmian space is completely established.
A typical element in is given as
The concept of quotients of sequences is justified by
Two quotients and are said to be equivalent in the sense of if
Sum and multiplication by a scalar of two Boehmians can be defined in a natural way
The operation • and differentiation are defined by
5 HHT of Boehmians
Let us define the EHHT of a Boehmian by
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 .
For each we, by , can find such that
and and as , .
The continuity of HHT s implies
The proof is completed. □
Theorem 5.4 EHHT s are continuous with respect to Δ convergence.
Proof Let in as , then there are and such that
and as . Hence, by Theorem 4.1,
Hence the theorem. □
Remark 5.5 Let , then it is so natural to define the inverse HHT of β as
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.
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The authors are very grateful to the referees for their valuable suggestions and comments that helped to improve the paper.
The authors declare that they have no competing interests.
Both authors contributed equally to the manuscript and read and approved the final draft.