Error bounds for interpolatory quadrature rules on the unit circle

Abstract:

For the construction of an interpolatory integration rule on the unit circle $T$ with $n$ nodes by means of the Laurent polynomials as basis functions for the approximation, we have at our disposal two nonnegative integers $p_n$ and $q_n,$ $p_n+q_n=n-1,$ which determine the subspace of basis functions. The quadrature rule will integrate correctly any function from this subspace. In this paper upper bounds for the remainder term of interpolatory integration rules on $T$ are obtained. These bounds apply to analytic functions up to a finite number of isolated poles outside $T.$ In addition, if the integrand function has no poles in the closed unit disc or is a rational function with poles outside $T$, we propose a simple rule to determine the value of $p_n$ and hence $q_n$ in order to minimize the quadrature error term. Several numerical examples are given to illustrate the theoretical results.