A generalized Peano Kernel Theorem for distributions of exponential decay

Abstract

The Peano Kernel Theorem (PKT) is a classical representation theorem in numerical integration. The idea is that if T is a quadrature rule that exactly integrates polynomials up to degree n−1 on a bounded interval [a,b], then there exists a kernel K, depending only on T, such thatE(f)=T(f)−∫abf(t)dt=∫abK(t)f(n)(t)dt, whenever f∈Cn([a,b]). In this work, we generalize the PKT from the class of linear functionals on Cn([a,b]) to the class of Laplace transformable tempered distributions of exponential decay. In particular, it is not necessary that the functionals being approximated have compact support. The generalized result is proven using an approach that provides a formula for computing the kernel K in the Fourier domain, which can be more computationally tractable and efficient in many cases. We conclude with examples of how the generalized Peano Kernel Theorem can be used for error analysis in signal processing.