Certain Spaces of Functions over the Field of Non-Newtonian Complex Numbers

Abstract

This paper is devoted to investigate some characteristic features of complex numbers and functions in terms of non-Newtonian calculus. Following Grossman and Katz, (Non-Newtonian Calculus, Lee Press, Piegon Cove, Massachusetts, 1972), we construct the field ${ℂ}^{*}$ of $*$-complex numbers and the concept of $*$-metric. Also, we give the definitions and the basic important properties of $*$-boundedness and $*$-continuity. Later, we define the space $C_{*}(\mathrm{\Omega })$ of $*$-continuous functions and state that it forms a vector space with respect to the non-Newtonian addition and scalar multiplication and we prove that $C_{*}(\mathrm{\Omega })$ is a Banach space. Finally, Multiplicative calculus (MC), which is one of the most popular non-Newtonian calculus and created by the famous exp function, is applied to complex numbers and functions to investigate some advance inner product properties and give inclusion relationship between $C_{*}(\mathrm{\Omega })$ and the set of $C_{*}^{\mathrm{\prime }}(\mathrm{\Omega })$ $*$-differentiable functions.