Approximation by exponential sums revisited

We revisit the efficient approximation of functions by sums of exponentials or Gaussians in Beylkin and Monzón (2005) [16] to discuss several new results and applications of these approximations. By using the Poisson summation to discretize integral representations of e.g., power functions r−β, β>0, we obtain approximations with uniform relative error on the whole real line. Our approach is applicable to a class of functions and, in particular, yields a separated representation for the function e−xy. As a result, we obtain sharper error estimates and a simpler method to derive trapezoidal-type quadratures valid on finite intervals. We also introduce a new reduction algorithm for the case where our representation has an excessive number of terms with small exponents.As an application of these new estimates, we simplify and improve previous results on separated representations of operators with radial kernels. For any finite but arbitrary accuracy, we obtain new separated representations of solutions of Laplace's equation satisfying boundary conditions on the half-space or the sphere. These representations inherit a multiresolution structure from the Gaussian approximation leading to fast algorithms for the evaluation of the solutions. In the case of the sphere, our approach provides a foundation for a new multiresolution approach to evaluating and estimating models of gravitational potentials used for satellite orbit computations.