  We derive analyticity criteria for explicit error bounds and an exponential
rate of convergence of the magic point empirical interpolation method
introduced by Barrault et al. (2004). Furthermore, we investigate its
application to parametric integration. We find that the method is well-suited
to Fourier transforms and has a wide range of applications in such diverse
fields as probability and statistics, signal and image processing, physics,
chemistry and mathematical finance.
  To illustrate the method, we apply it to the evaluation of probability
densities by parametric Fourier inversion. Our numerical experiments display
convergence of exponential order, even in cases where the theoretical results
do not apply.
