Numerical Approximation of Generalized Functions: Aliasing, the Gibbs Phenomenon and a Numerical Uncertainty Principle

Abstract

A general recipe for high-order approximation of generalized functions is introduced which is based on the use of L2-orthonormal bases consisting of C ∞ -functions and the appropriate choice of a discrete quadrature rule. Particular attention is paid to maintaining the distinction between point-wise functions (that is, which can be evaluated point-wise) and linear functionals defined on spaces of smooth functions (that is, distributions). It turns out that ``best'' point-wise approximation and ``best'' distributional approximation cannot be achieved simultaneously. This entails the validity of a kind of ``numerical uncertainty principle'': The local value of a function and its action as a linear functional on test functions cannot be known at the same time with high accuracy, in general.