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.
Keywords:distributions; test function; HH transform; Boehmians
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 .
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:
(iv) Convolution: The convolution theorem of HT is given as
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.
Theorem 2.1 (First convolution theorem of HHT)
LetHHf, HHgbe theHHTs offandg, respectively, then
Proof Under the hypothesis of the theorem, we write
which can be written as
Then, with a simple modification, we get
Hence the theorem. □
Proof of (ii) is analogous to that of the first part.
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.
Theorem 2.4Iffis in, thenHfis also in.
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
for every pair of non-negative m and k. This completes the proof. □
Corollary 2.6Iffis in, thenHHfis also in.
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
This can be stated in similar words as: HT and HHT of tempered distributions are tempered distributions. □
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.
The elements of Δ are called delta sequences.
Consider the class A of pairs of sequences defined by
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:
The following lemma is equivalent to the statement of δ-convergence.
Theorem 4.1 (Second convolution theorem of HHT)
where ∗ is the usual convolution product offandg, see.
Proof Using the definition of HHT implies
Fubini’s theorem implies
in (14) then multiplying and canceling similar quantities yield
Hence, invoking (16) and (17) in (13), our theorem follows. □
Let us consider another space of Boehmians:
Proof Using (17) we get
By (18) and Theorem 4.2, we get
Hence the theorem. □
Proof (i) The linearity of HHTs and (17) implies
The proof of (ii) and (iii) follows from simple computations. The proof is therefore completed. □
The proof is completed. □
Proof Follows from similar computations to those above.
Hence our theorem is completely proved. □
Thus the theorem. □
The concept of quotients of sequences is justified by
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
It is clear that EHHT is well defined.
Theorem 5.1EHHTis linear.
Theorem 5.2EHHTis one-to-one.
Theorem 5.3EHHTis continuous with respect toδconvergence.
For each we, by , can find such that
The continuity of HHTs implies
The proof is completed. □
Theorem 5.4EHHTs are continuous with respect to Δ convergence.
Hence the theorem. □
The authors declare that they have no competing interests.
Both authors contributed equally to the manuscript and read and approved the final draft.
The authors are very grateful to the referees for their valuable suggestions and comments that helped to improve the paper.
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