Objective:

Many problems in physics can be formulated as operator theory problems, such as in differential or integral equations. To function numerically, the operators must be discretized. One way to achieve discretization is to find (possibly infinite) matrices describing these operators using ONBs. In this project, we will use frames to investigate a way to describe an operator as a matrix.

Method:

The standard matrix description of operators O using an ONB (e_k) involves constructing a matrix M with the entries M_{j,k} = < O e_k, e_j>. In past publications, a concept that described operator R in a very similar way has been presented. However, this description of R used a frame and its canonical dual. Currently, a similar representation is being used for the description of operators using Gabor frames. In this project, we are going to develop and completely generalize this idea for Bessel sequences, frames, and Riesz sequences. We will also look at the dual function that assigns an operator to a matrix.

Application:

This "sampling of operators" is especially important for application areas where frames are heavily used, so that the link between model and discretization is maintained. To facilitate implementations, operator equations can be transformed into a finite, discrete problem with the finite section method (much in the same way as in the ONB case).

Publications:

  • P. Balazs, "Matrix Representation of Operators Using Frames", Sampling Theory in Signal and Image Processing (STSIP) (2007, accepted), preprint
  • P. Balazs, "Hilbert-Schmidt Operators and Frames - Classification, Approximation by Multipliers and Algorithms" , International Journal of Wavelets, Multiresolution and Information Processing, (2007, accepted)  preprint, Codes and Pictures: here