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

We present new short exponential sum approximations of length $N$ for $f_1(x)=\frac{1}{a+x}$ with $a>0$ on $[0, \infty)$ and for $f_2(x)= {\mathrm e}^{-x^2/2σ}$ with $σ>0$ on ${\mathbb R}$ with geometric error decay $ρ^{-2N}$ for user-defined $N \ge 2$ and $ρ>1$. The approximations are built over