Truncation Errors in Exponential Fitting for Oscillatory Problems

Abstract

A generalization of Peano's kernel theorem due to Ghizzetti and Ossicini [Quadrature Formulae, Birkhauser, Basel, Switzerland, 1970] provides expressions, in the form of integrals, for the truncation errors in a variety of exponential-fitting formulae for oscillatory problems. In some circumstances this leads to an expression analogous to the Lagrange form of remainder; more generally the error can be expressed as a sum of two terms of Lagrange type. Our examples include formulae for quadrature and numerical differentiation, and linear multistep methods for ordinary differential equations. Two families of exponential-fitting quadrature formulae are investigated, one with evenly spaced abscissas and the other based on the philosophy of Gaussian quadrature. In particular, the integral representation can be used to determine the asymptotic rate of decay of the error with increasing frequency for a class of oscillatory integrands.