Quadrature rules derived from linear convergence accelerations schemes

We shall develop efficient schemes for approximating the integral over the entire real line with a finite linear combination of functional values. To do this we first approximate the integral with a trapezoidal sum, then apply a linear convergence acceleration scheme to approximate this infinite sum with a linear combination of a finite number of terms. We will study two important cases in detail, namely when the integrand is oscillating or decays exponentially. If we also require the integrand to be analytic on a strip along the real axis, the trapezoidal approximation can be shown to be very accurate, provided some simple regularity assumptions are satisfied.