Rational interpolation: II. Quadrature and convergence

Consider an nth rational interpolatory quadrature rule Jnσ(f)=∑j=1nλjf(xj) to approximate integrals of the form Jσ(f)=∫−11f(x)dσ(x), where σ is a (possibly complex) bounded measure with infinite support on the interval [−1,1]. First, we discuss the connection of Jnσ(f) with certain rational interpolatory quadratures on the complex unit circle to approximate integrals of the form ∫−ππf̊(eiθ)dσc(θ), where σc is a (possibly complex) bounded measure with infinite support on [−π,π]. Next, we provide conditions to ensure the convergence of Jnσ(f) to Jσ(f) for n tending to infinity. Finally, an upper bound for the error on the nth approximation and an estimate for the rate of convergence is provided.