Ivanenko Y, Nedic M, Gustafsson M, Jonsson B L G, Luger A, Nordebo S
Department of Physics and Electrical Engineering, Linnæus University, 351 95 Växjö, Sweden.
Department of Mathematics, Stockholm University, 106 91 Stockholm, Sweden.
R Soc Open Sci. 2020 Jan 15;7(1):191541. doi: 10.1098/rsos.191541. eCollection 2020 Jan.
We introduce the set of quasi-Herglotz functions and demonstrate that it has properties useful in the modelling of non-passive systems. The linear space of quasi-Herglotz functions constitutes a natural extension of the convex cone of Herglotz functions. It consists of differences of Herglotz functions and we show that several of the important properties and modelling perspectives are inherited by the new set of quasi-Herglotz functions. In particular, this applies to their integral representations, the associated integral identities or sum rules (with adequate additional assumptions), their boundary values on the real axis and the associated approximation theory. Numerical examples are included to demonstrate the modelling of a non-passive gain medium formulated as a convex optimization problem, where the generating measure is modelled by using a finite expansion of B-splines and point masses.
我们引入了拟赫格洛茨函数集，并证明了它具有在非无源系统建模中有用的性质。拟赫格洛茨函数的线性空间构成了赫格洛茨函数凸锥的自然扩展。它由赫格洛茨函数的差组成，并且我们表明新的拟赫格洛茨函数集继承了几个重要的性质和建模观点。特别是，这适用于它们的积分表示、相关的积分恒等式或求和规则（在有适当附加假设的情况下）、它们在实轴上的边界值以及相关的逼近理论。文中包含了数值示例，以展示作为凸优化问题制定的非无源增益介质的建模，其中生成测度是通过使用B样条和点质量的有限展开来建模的。
