Some Definition of Hartley-Hilbert and Fourier-Hilbert Transforms in a Quotient Space of Boehmians

We investigate the Hartley-Hilbert and Fourier-Hilbert transforms on a quotient space of Boehmians. The investigated transforms are well-defined and linear mappings in the space of Boehmians. Further properties are also obtained. 1. Introduction The Hilbert transform of a function via the Hartley transform is defined in [1, 2] as where and are, respectively, the even and odd components of the Hartley transform given as [3] where . Let be a casual function; that is, , for , and then and are related by a Hilbert transform pair as [3] The Hilbert transform of via the Fourier transform is defined by where and are, respectively, the real and imaginary components of the Fourier transform given as The Hartley transform is extended to Boehmians in [4] and to strong Boehmians in [5]. The Hartley-Hilbert and Fourier-Hilbert transforms were discussed in various spaces of distributions and spaces of Boehmians in [1, 6]. In this paper, aim at investigating the Hartley-Hilbert transform on the context of Boehmians. Investigating the later transform is analogous. 2. Spaces of Quotients (Spaces of Boehmians) One of the most youngest generalizations of functions and more particularly of distributions is the theory of Boehmians. The idea of the construction of Boehmians was initiated by the concept of regular operators [7]. Regular operators form a subalgebra of the field of Mikusinski operators and they include only such functions whose support is bounded from the left. In a concrete case, the space of Boehmians contains all regular operators, all distributions, and some objects which are neither operators nor distributions. The construction of Boehmians is similar to the construction of the field of quotients and, in some cases, it gives just the field of quotients. On the other hand, the construction is possible where there are zero divisors, such as space (the space of continuous functions) with the operations of pointwise additions and convolution. A number of integral transforms have been extended to Boehmian spaces in the recent past by many authors such as Roopkumar in [8, 9]; Mikusinski and Zayed in [10]; Al-Omari and Kilicman in [1, 6, 11]; Karunakaran and Vembu [12]; Karunakaran and Roopkumar [13]; Al-Omari et al. in [14]; and many others. For abstract construction of Boehmian spaces we refer to [14–16]. By denote the Mellin-type convolution product of first kind defined by [8] Properties of are presented as follows [8]:(i);(ii);(iii); is complex number;(iv). By , we denote the convolution product defined by By denote the space of test functions of bounded