Estimates of the Remainder in Taylor's Theorem Using the Henstock-Kurzweil Integral

Abstract

When a real-valued function of one variable is approximated by its nth degree Taylor polynomial, the remainder is estimated using the Alexiewicz and Lebesgue p-norms in cases where f (n) or f (n+1) are Henstock-Kurzweil integrable. When the only assumption is that f (n) is Henstock-Kurzweil integrable then a modified form of the nth degree Taylor polynomial is used. When the only assumption is that f (n) ∈ C 0 then the remainder is estimated by applying the Alexiewicz norm to Schwartz distributions of order 1.