Numerical Approximation of Generalized Func- tions: Aliasing, the Gibbs Phenomenon and a Numerical Uncertainty Principle

Abstract. A general recipe for high order approximation of generalized func-tions is introduced which is based on the use of L2-orthonormal bases consist-ing of C∞-functions and the appropriate choice of a discrete quadrature rule. Particular attention is paid to maintaining the distinction between pointwise functions (that is, which can be evaluated pointwise) and linear functionals defined on spaces of smooth functions (that is, distributions). It turns out that “best ” pointwise approximation and “best ” distributional approxima-tion 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. In spite of this, high order accurate pointwise approximations can be obtained in special cases from a high accuracy distributional approximation when more information is available concerning the function which is to be approximated. A few special cases with application to PDEs are considered in detail