Some Remarks on the Extended Hartley-Hilbert and Fourier-Hilbert Transforms of Boehmians

and Applied Analysis 3 Theorem 3. The distributional Hartley-Hilbert transform ̂ Bf is linear. Proof. Let f, g ∈ C󸀠 then their components Af,Af,Ag, Aog ∈ C. Hence, ̂ B A (f + g) (y) = ⟨A o (f + g) (x) , cos (xy)⟩ + ⟨A e (f + g) (x) , sin (xy)⟩ . (24) By factoring and rearranging components we get that ̂ B A (f + g) (y) = ̂ B A f (y) + ̂ B A g (y) . (25) Furthermore, ̂ B A (kf) (y) = ⟨kA o f (x) , cos (xy)⟩ + ⟨kA e f (x) , sin (xy)⟩ . (26)


Introduction
The classical theory of integral transforms and their applications have been studied for a long time, and they are applied in many fields of mathematics.Later, after [1], the extension of classical integral transformations to generalized functions has comprised an active area of research.Several integral transforms are extended to various spaces of generalized functions, distributions [2], ultradistributions, Boehmians [3,4], and many more.
In recent years, many papers are devoted to those integral transforms which permit a factorization identity (of Fourier convolution type) such as Fourier transform, Mellin transform, Laplace transform, and few others that have a lot of attraction, the reason that the theory of integral transforms, generally speaking, became an object of study of integral transforms of Boehmian spaces.
The Hartley transform is an integral transformation that maps a real-valued temporal or spacial function into a realvalued frequency function via the kernel  (; ) = cas () . ( This novel symmetrical formulation of the traditional Fourier transform, attributed to Hartley 1942, leads to a parallelism that exists between a function of the original variable and that of its transform.In any case, signal and systems analysis and design in the frequency domain using the Hartley transform may be deserving an increased awareness due to the necessity of the existence of a fast algorithm that can substantially lessen the computational burden when compared to the classical complex-valued fast Fourier transform.The Hartley transform of a function () can be expressed as either [5] or where the angular or radian frequency variable  is related to the frequency variable  by  = 2 and The integral kernel, known as cosine-sine function, is defined as cas () = cos  + sin () . ( Inverse Hartley transform may be defined as either or The theory of convolutions of integral transforms has been developed for a long time and is applied in many fields of mathematics.Historically, the convolution product [2] ( * ) has a relationship with the Fourier transform with the factorization property The more complicated convolution theorem of Hartley transforms, compared to that of Fourier transforms, is that where Some properties of Hartley transforms can be listed as follows.
(i) Linearity: if  and  are real functions then

Distributional Hartley-Hilbert Transform of Compact Support
The Hilbert transform via the Hartley transform is defined by [6, 7] where are the respective odd and even components of (2).We denote, C(R), C(R) = C, the space of smooth functions and Following is the convolution theorem of B A .
Theorem 1 (Convolution Theorem).Let  and  ∈ C then where Proof.To prove this theorem it is sufficient to establish that Therefore, we have The substitution  −  =  and using of (1) together with Fubini theorem imply then (18) follows from simple computation.Proof of (19) has a similar technique.Hence, the theorem is completely proved.
It is of interest to know that cos() and sin() are members of C and, therefore, A  , A   ∈ C  .This leads to the following statement. Hence, Therefore, for all .This completes the proof of the theorem.(33) Proof of this theorem is analogous to that of the previous theorem and is thus avoided.
Denote by  the dirac delta function then it is easy to see that

Lebesgue Space of Boehmians for Hartley-Hilbert Transforms
The original construction of Boehmians produce a concrete space of generalized functions.Since the space of Boehmians was introduced, many spaces of Boehmians were defined.In references, we list selected papers introducing different spaces of Boehmians.One of the main motivations for introducing different spaces of Boehmians was the generalization of integral transforms.The idea requires a proper choice of a space of functions for which a given integral transform is well defined, a choice of a class of delta sequences that is transformed by that integral transform to a well-behaved class of approximate identities, and finally a convolution product that behaves well under the transform.If these conditions are met, the transform has usually an extension to the constructed space of Boehmians and the extension has desirable properties.For general construction of Boehmians, see [8][9][10][11][12].
Let D be the space of test functions of bounded support.By delta sequence, we mean a subset of D of sequences where (  )() = { ∈ R :   () ̸ = 0}.The set of all such delta sequences is usually denoted as Δ.Each element in Δ corresponds to the dirac delta function , for large values of .
, be the space of complex valued Lebesgue integrable functions.From Proposition 7 we establish the following theorem.(51) By Theorem 8, we get Hence, Also, for each complex number , we have Hence we have the following theorem.Similarly, we proceed for BA  = BA (  * ).This completes the theorem.The following theorem is obvious.
Theorem 13.If BA  1 = 0, then  1 = 0. Hence, that is, B A  = B A .The fact that B A is one-to-one implies  = .Hence we have the following theorem.