On the numerical evaluation of fractional Sobolev norms

In several important and active fields of modern applied  mathematics, such as the numerical solution of PDE-constrained  control problems or various applications in image processing and  data fitting, the evaluation of (integer and real) Sobolev norms  constitutes a crucial ingredient.  Different approaches exist for  varying ranges of smoothness indices and with varying properties  concerning exactness, equivalence and the computing time for the  numerical evaluation.  These can usually be expressed in terms of  discrete Riesz operators.  We propose a collection of criteria which allow to compare different  constructions.  Then we develop a unified approach which is valid  for non-negative real smoothness indices for standard finite  elements, and for positive and negative real smoothness for  biorthogonal wavelet bases.  This construction delivers a wider  range of exactness than the currently known constructions and is  computable in linear time.