On the Dini calculus (Q1084605)

This paper is concerned with Dini integrals, i.e. integrals of the form \(\int_{| t| >\epsilon}\frac{e^{itX}}{t^ n}f(t,\epsilon)dt.\) It is proved that such an integral has a singular part (a polynomial in \(\epsilon^{-1})\), and a resular part which, as \(\epsilon\to 0\), behaves near \(X=-\infty\) as a polynomial in X. A calculus is developed for computing this polynomial, and conjectures are made regarding higher dimensional Dini integrals.