Least Squares Approximation of Completely Monotonic Functions by Sums of Exponentials

We consider the approximation of a given completely monotonic function $F(t)$ by an exponential sum Y(t) = a₁ exp ( - λ t) + ⋯ + aₙ exp ( - λ ₙ t) using the L₂-norm associated with a suitable measure dμ (t). A best approximation exists and is characterized by a generalized version of the Aigrain–Williams equations. Examples are given to show that a given F may have several best approximations. We apply Rice’s dual algorithm for nonlinear L₂-approximation to this problem and show that it reduces the norm of the error at each step. The algorithm calls for the repeated solution of an associated nonlinear Hermite interpolation problem which involves simple rational functions when dμ (t) = dt,0 < t < ∞. The rate of convergence can be accelerated from first to second order by an inexpensive extrapolation. We discuss the numerical implementation of the algorithm and present numerical results for F(t) = 1 /(1 + t),dμ (t) = dt,0 < t < ∞.