Approximation by short exponential sums with geometric error decay based on Gauss quadrature

We present new short explicit exponential sum approximations of length [Formula: see text] for [Formula: see text] with [Formula: see text] on [Formula: see text] and for [Formula: see text] with [Formula: see text] on [Formula: see text] with geometric error decay [Formula: see text] for user-defined [Formula: see text] and [Formula: see text]. The approximations are built over consecutive intervals [Formula: see text], [Formula: see text], with interval lengths that depend on [Formula: see text] and grow exponentially for [Formula: see text] and are equidistant for [Formula: see text]. All parameters determining the exponential sum approximations on [Formula: see text] are explicitly given and are closely related to initial parameters on [Formula: see text]. Our method is based on Gauss-Laguerre and Gauss-Hermite quadrature, respectively, applied to suitable parametric integral representations of [Formula: see text] and [Formula: see text]. This technique ensures consistent relative errors across all intervals. Since all parameters of the exponential sum are given explicitly in closed form, in terms of [Formula: see text], [Formula: see text], and the nodes and weights of the underlying Gauss quadrature rule, our method avoids the non-linear optimization and the numerical instabilities inherent in Prony-type or Remez-based constructions. Using the obtained exponential sum approximations, we achieve highly accurate approximations of [Formula: see text] on [Formula: see text] and of the error function [Formula: see text] with predictable geometric error decay. Numerical examples for [Formula: see text] and [Formula: see text] clearly illustrate the theoretical error estimates.