MULTIDIMENSIONAL GAUGE THEORY VIA SUMMABILITY METHODS

Abstract

Kurzweil and Henstock presented the notion of Gauge integral, independently.
Using their definition Savas and Patterson examined the relationship between
Gauge integral and Summability theory. Because of the esoteric of both Gauge
and Summability theory, the body of literature is limited. As such the
only accessible notion to both theories is Pringsheim limits. The goal of this paper is to present a natural multidimensional extension of Gauge theory
via Summability methods. To accomplish this we examine double measurable
real-valued functions of the type of $f(x,y)$ in the Gauge sense on $\left(
1,\infty \right) \times \left( 1,\infty \right) $. Additionally, we
introduce the definition of double $\overline{\gamma }_{2}-$strongly
summable to $L$ with respect to Gauge and present inclusion theorems.