Simultaneous approximation with generalized exponential sampling series

In the present paper, we focus on the topic of simultaneous approximation by making use of the generalized exponential sampling series, which were originally introduced and rigorously defined by C. Bardaro and his collaborators in the year 2017. The motivation for considering this particular type of sampling series stems from their fundamental construction, which is inherently grounded in the framework of Mellin analysis. Due to this specific analytical setting, it becomes natural and mathematically consistent to employ Mellin derivatives in place of the classical derivatives commonly used in other approximation contexts. By doing so, we are able to formulate a more suitable and accurate approximation scheme that aligns with the structural properties of exponential sampling series within the Mellin framework. After establishing and proving the necessary and foundational main results regarding the behavior and convergence of these series, we proceed further by presenting a kernel example. This example is carefully selected to illustrate and substantiate the theoretical findings, thereby providing tangible instances that both demonstrate and reinforce the validity and applicability of the developed theory.