A mollifier useful for approximations in Sobolev spaces and some applications to approximating solutions of differential equations

Abstract

For a given uniform grid of E N {E^N} (N-dimensional Euclidean space) with mesh h, a class of smoothing functions (mollifiers) is constructed. If a function is an element of the Sobolev space H 2 m H_2^m , then the error made by replacing the given function by a smoother ( C ∞ ) ({C^\infty }) function (which is the given function convolved with one of the mollifiers) is bounded by a constant times h m {h^m} . This result is used to construct approximations for functions using Hermite or spline interpolation, even though the function to be approximated need not satisfy the continuity conditions necessary for the existence of a Hermite or spline interpolate. These techniques are used to find approximations to the generalized solution of a second order elliptic Neumann problem.