Fourier transformation can improve quadrature efficiency of Laplace distribution (Q1861606)

Let \(X\) and \(Y\) be random variables with probability density functions (PDFs) \(f\) and \(g\) respectively, let \(F[g](x)\) be the characteristic function of the PDF \(g\). If \(f\) and \(g\in L^1(R)\cap L^2(R)\), then \(E[F[g](X)]=E[F[f](Y)]\). The authors propose to use this fact for numerical calculation of integrals of the form \(E[F[g](x)]=\int F[g](x)f(x) dx\). It is shown by numerical comparison that this transformation can improve the efficiency of Gauss-Hermite quadrature formulas for finding \[ {1\over\pi}\int_{-\infty}^\infty{1\over \vartheta^2+x^2}e^{-x^2}dx= {1\over 2\vartheta}\int_{-\infty}^\infty e^{-\vartheta|y|}e^{-y^2/2}dy. \] (Cauchy and Gaussian distributions are used here).