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.

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