The set of quasi-Herglotz functions is introduced as a natural extension of the convex cone of Herglotz functions. The new class of functions consists of differences of Herglotz functions and we demonstrate that it has properties that are useful in the modeling of non-passive systems. The linear space of quasi-Herglotz functions constitutes a natural extension of the convex cone of Herglotz functions and we will illustrate that several of the important properties and modeling perspectives are inherited by the new set of quasi-Herglotz functions. In this presentation, we will focus on the approximation theory and the formulation as a convex optimization problem where the generating measure is modeled by using a finite expansion of B-splines and point masses. Numerical examples are included to demonstrate the modeling of a non-passive gain media.