Error bounds for interpolatory quadrature rules on the unit circle

Abstract

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 pn and qn, pn + qn = 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 pn and hence qn in order to minimize the quadrature error term. Several numerical examples are given to illustrate the theoretical results.

Untitled

“…It has been observed experimentally [7,[11][12][13] that this formula, the interpolatory quadrature formula with uniformly distributed nodes on the unit circle, gives as good results as the Szegö formula. As a consequence, these properties make this formula suitable for practical computations.…”

Rational quadrature formulae on the unit circle with arbitrary poles

“…In that case, set P(z) = (z − β 1 ) · · · (z − β ν ) and suppose that |E n (z k )| ≤ M n , k ∈ Z. Then (see [30], Corollary 1)…”

“…Next, one can apply the quadrature rule to each function separately. It is shown in [30] that such a way of proceeding does not impair the accuracy of the quadrature rule if p and q are properly chosen. So, for numerical purposes, we have considered the functions f 2 (z) = exp z z − 2 and f 7 (z) = exp z z − 1/5 as representatives of each type.…”

“…In [30] an upper bound of the remainder term E n for interpolatory quadrature rules ( p+q = n−1) is given. The error bound is valid for rules which approximate integrals with arbitrary weight functions.…”

“…However, in the case of the trigonometric moment problem it is very natural to consider Laurent polynomials instead motivated by the fact that a continuous function on T can be uniformly approximated by Laurent polynomials. As a consequence, quadrature formulae with nodes on T and exactly integrating certain subspaces of Laurent polynomials arose (see e.g., [24,23,15,30]). …”