Abstract
Sparsity promoting regularization is an important technique for signal reconstruction and several other ill-posed problems. Theoretical investigation typically bases on the assumption that the unknown solution has a sparse representation with respect to a fixed basis. We drop this sparsity assumption and provide error estimates for nonsparse solutions. After discussing a result in this direction published earlier by one of the authors and co-authors, we prove a similar error estimate under weaker assumptions. Two examples illustrate that this set of weaker assumptions indeed covers additional situations which appear in applications.
Acknowledgments
The authors thank Bernd Hofmann for many valuable comments on a draft of this article and for fruitful discussions on the subject.
Notes
J. Flemming was supported by the German Science Foundation (DFG) under grant FL 832/1-1. M. Hegland was partially supported by the Technische Universität München Institute of Advanced Study, funded by the German Excellence Initiative. Work on this article was partially conducted during a stay of M. Hegland at TU Chemnitz, supported by the German Science Foundation (DFG) under grant HO 1454/8-1.
Dedicated to Professor Bernd Hofmann on the occasion of his 60th birthday