150 research outputs found
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
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