Basic Description:

Practical experience quickly revealed that the concept of an orthonormal basis is not always useful. This led to the concept of frames. Models in physics and other application areas (for example sound vibration analysis) are mostly continuous models. Many continuous model problems can be formulated as operator theory problems, such as in differential or integral equations. Operators provide an opportunity to describe scientific models, and frames provide a way to discretize them.

Sequences are often used in physical models, allowing numerically unstable re- synthesis. This can be called an "unbounded frame". How this inversion can be regularized is being investigated. For many applications, a certain frame is very useful in describing the model. Therefore, it is also beneficial to use the same sequence to find a discretization of involved operators.

Subprojects:

Frames in Finite Dimensional Spaces:

In this project, the theory of frames in the finite discrete case is investigated further.

Matrix Representation of Operators using Frames:

The standard matrix description of operators using orthonormal bases is extended to the more general case of frames.

Weighted and Controlled Frames:

Weighted and controlled frames were introduced to speed up the inversion algorithm for the frame matrix of a wavelet frame. In this project, these kinds of frames are investigated further.

Basic Properties of Unbounded Frames

Irregular Frames of Translates:

In this project, one function's sequences of irregular shifts are investigated.

Partners:

  • S. Heineken, Research Group on Real and Harmonic Analysis, University of Buenos Aires
  • J. P. Antoine, Unité de physique théorique et de physique mathématique – FYMA
  • M. El-Gebeily,  Department of Mathematical Sciences, King Fahd University of Petroleum and Minerals, Saudi Arabia