The Effect of Quadrature Errors in the Computation of $L^2 $ Piecewise Polynomial Approximations

Abstract

In this paper we investigate the $L^2 $ piecewise polynomial approximation problem. $L^2 $ bounds for the derivatives of the error in approximating sufficiently smooth functions by polynomial splines follow immediately from the analogous results for polynomial spline interpolation. We derive $L^2 $ bounds for the errors introduced by the use of two types of quadrature rules for the numerical computation of $L^2 $ piecewise polynomial approximations. These bounds enable us to present some asymptotic results and to examine the consistent convergence of appropriately chosen sequences of such approximations. Some numerical results are also included.