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 error of the Gaussian quadrature is also expanded by basis functions associated with the variable transformation. The basis functions form a Chebyshev system on the positive real axis. The maximization of the decreasing rate of the error bound can be achieved by constructing a one-to-one mapping of an open Bernstein ellipse onto the right half-plane. The mapping is realized by the composition of Jacobi's delta amplitude function (also called dn function) and the multivalued inverse cosine function. The function is single-valued, meromorphic, and strictly absolutely monotonic function. The corresponding basis functions are eigenfunctions of a fourth order differential operator, satisfy orthogonality conditions, and have the interlacing property of zeros by Kellogg's theorem. We also analyze the initialization method of the Remez algorithm based on a Gaussian quadrature to compute the best exponential sum approximation of a finite completely monotonic function. The numerical experiments are conducted by using finite completely monotonic functions related to the inverse power function.

翻译：伯恩斯坦定理（亦称豪斯多夫-伯恩斯坦-维德定理）实现了完全单调函数的积分表示。我们引入有限完全单调函数，即积分表示中具有有限正积分区间的完全单调函数。基于带变量变换的高斯求积，我们考虑有限完全单调函数的指数和逼近。若变量变换在开伯恩斯坦椭圆上解析，则最大绝对误差相对于指数函数个数至少呈几何级数递减。高斯求积的误差亦可由与变量变换关联的基函数展开。这些基函数在正实轴上构成切比雪夫系统。通过构造开伯恩斯坦椭圆到右半平面的一一映射，可实现误差界衰减率的最大化。该映射由雅可比德尔塔振幅函数（亦称dn函数）与多值反余弦函数的复合实现，所得函数为单值、亚纯且严格绝对单调函数。相应基函数是四阶微分算子的本征函数，满足正交性条件，并由凯洛格定理具有零点的交错性质。我们还分析了基于高斯求积的雷梅兹算法初始化方法，用于计算有限完全单调函数的最优指数和逼近。数值实验采用与反幂函数相关的有限完全单调函数进行。