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

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. In spite of this, high-order accurate point-wise 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.