On generalizations of Boehmian space and Hartley transform

TL;DR

This paper introduces a generalized Boehmian space called G-Boehmian space where the denominator set need not be a commutative semigroup, and extends the Hartley transform to this new setting, comparing it with existing approaches.

Contribution

It proposes a new G-Boehmian space framework relaxing the commutativity requirement and extends the Hartley transform within this context.

Findings

Defined G-Boehmian space without commutative semigroup requirement.

Provided an example of G-Boehmian space.

Extended and compared the Hartley transform in this new setting.

Abstract

Boehmians are quotients of sequences which are constructed by using a set of axioms. In particular, one of these axioms states that the set $S$ from which the {\it denominator} sequences are formed should be a commutative semigroup with respect to a binary operation. In this paper, we introduce a generalization of abstract Boehmian space, called $G$-Boehmian space, in which $S$ is not necessarily a commutative semigroup. Next, we provide an example of a $G$-Boehmian space and we discuss an extension of the Hartley transform on it. Finally, we compare the Hartley transform in this paper with the existing works on Hartley transform of Boehmians.