Exponential sum approximations of finite completely monotonic functions

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.