Higher order corrected trapezoidal rules in Lebesgue and Alexiewicz spaces

Abstract

where φ n is a monic polynomial of degree n and the error is given byThis then gives a quadrature formula for b a f (x) dx. The polynomial φ n is chosen to optimize the error estimate under the assumption that f n) is integrable in the distributional or Henstock-Kurzweil sense. Sharp error estimates are obtained. It is shown that this formula is exact for all such φ n if f is a polynomial of degree at most n − 1. If φ n is a Legendre polynomial then the formula is exact for f a polynomial of degree at most 2n − 1.