Practical experience has shown 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, including sound vibration analysis, are mostly continuous models. Many continuous model problems can be formulated as operator theory problems, as in differential or integral equations. An interesting class of operators is the Hilbert Schmidt class. This project aims to find the best approximation of any matrix by a frame multiplier, using the Hilbert Schmidt norm.
In finite dimensions, every sequence is a frame sequence, so the best approximation of any element can be found only via the frame operator using the dual frame for synthesis. Furthermore, the present best approximation algorithm involves the following steps: 1) The Hilbert-Schmidt inner product of the matrix and the projection operators involved is calculated in an efficient way; 2) Then the pseudo inverse of the Grame matrix is used to avoid the so-called calculation of the dual frames; The pseudo inverse is applied to the coefficients found above to find the lower symbol of the multiplier.
To find the best approximation of matrices via multipliers gives a way to find efficient algorithms to implement such operators. Any time-variant linear system can be modeled by a matrix. Time-invariant systems can be described as circulating matrices. Slowly-time-varying linear systems have a good chance at closely resembling Gabor multipliers. Other matrices can be well approximated by a "diagonalization" using other frames.
- 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
This project ended on 01.01.2009. Its completion allowed the sucessfull application for a 'High Potential'-Project of the WWTF, see MULAC