Rational interpolation: II. Quadrature and convergence

Consider an n th rational interpolatory quadrature rule J n σ ( f ) = ∑ j = 1 n λ j f ( x j ) to approximate integrals of the form J σ ( f ) = ∫ − 1 1 f ( x ) d σ ( x ) , where σ is a (possibly complex) bounded measure with infinite support on the interval [ − 1 , 1 ] . First, we discuss the connection of J n σ ( f ) with certain rational interpolatory quadratures on the complex unit circle to approximate integrals of the form ∫ − π π f ̊ ( e i θ ) d σ c ( θ ) , where σ c is a (possibly complex) bounded measure with infinite support on [ − π , π ] . Next, we provide conditions to ensure the convergence of J n σ ( f ) to J σ ( f ) for n tending to infinity. Finally, an upper bound for the error on the n th approximation and an estimate for the rate of convergence is provided.