From many previous applications, it is known that inverse problems often require a regularization that makes the inversion numerically stable. In this project, sequences that allow a bounded, injective analysis (that is not boundedly invertible) are investigated, .


Even for general sequences, analysis operator and synthesis operator can be defined. The first part of this project will investigate the most general results of these definitions. For example, it can be shown that the analysis operator is always a closed operator. Although it can be shown that the existence of another sequence that allows a perfect reconstruction fit can not be bounded, the question of how to construct such a "dual sequence" will be investigated.


Such sequences have already found applications in wavelet analysis, in which dual sequences were constructed algorithmically. Also, the original system investigated by Gabor with a redundancy of 1 satisfies this condition.


  • M. El-Gebeily, Department of Mathematical Sciences, King Fahd University of Petroleum and Minerals, Saudi Arabia
  • J. P. Antoine, Unité de physique théorique et de physique mathématique – FYMA, Belgium