A Lebesgue Integrable Space of Boehmians for a Class of Dk Transformations

Abstract

Boehmians are objects obtained by an abstract algebraic construction similar to that of field of quotients and it in some cases just gives the field of quotients. As Boehmian spaces are represented by convolution quotients, integral transforms have a natural extension onto appropriately defined spaces of Boehmians. In this paper, we have defined convolution products and a class of delta sequences and have examined the axioms necessary for generating the Dk spaces of Boehmians. The extended Dk transformation has therefore been defined as a one-to-one onto mapping continuous with respect to Δ and δ convergences. Over and above, it has been asserted that the necessary and sufficient conditions for an integrable sequence to be in the range of the Dk transformation is that the class of quotients belongs to the range of the representative. Further results related to the inverse problem are also discussed. © 2018 EUDOXUS PRESS, LLC.