Bernstein's theorem (also called Hausdorff--Bernstein--Widder theorem) enables the integral representation of a completely monotonic function. We introduce a finite completely monotonic function, which is a completely monotonic function with a finite positive integral interval of the integral representation. We consider the exponential sum approximation of a finite completely monotonic function based on the Gaussian quadrature with a variable transformation. If the variable transformation is analytic on an open Bernstein ellipse, the maximum absolute error decreases at least geometrically with respect to the number of exponential functions. The maximization of the decreasing rate of the error bound can be achieved by using a variable transformation represented by Jacobi's delta amplitude function (also called dn function). The error curve is expanded by introducing basis functions, which are eigenfunctions of a fourth order differential operator, satisfy orthogonality conditions, and have the interlacing property of zeros by Kellogg's theorem.

翻译：Bernstein定理（也称为Hausdorff–Bernstein–Widder定理）建立了完全单调函数的积分表示。我们引入有限完全单调函数的概念，该类函数的积分表示具有有限的正积分区间。基于带变量变换的高斯求积，我们研究有限完全单调函数的指数和逼近问题。若变量变换在开Bernstein椭圆上解析，则最大绝对误差至少以关于指数函数个数的几何速率衰减。通过使用由Jacobi delta振幅函数（亦称dn函数）表示的变量变换，可实现误差界衰减率的最大化。通过引入基函数扩展误差曲线，这些基函数是四阶微分算子的本征函数，满足正交性条件，且根据Kellogg定理具有零点的交织性质。