Applications of Operator Theory to Maximum Entropy Problems
Annual rept. 15 Jun 1987-14 Jun 1988
MARYLAND UNIV COLLEGE PARK
Pagination or Media Count:
This project is focusing on problems in operator theory and matrix theory that underlie the maximum entropy principle in signal processing and system theory. We have found generalizations of this principle for finite-dimensional problems to certain broad classes of hermitian band matrices, including Toeplitz matrices, and we have shown why a similar principle cannot exist for all hermitian band matrices. Zeros of orthogonal polynomials are studied in a setting that generalizes the usual minimum phase theorem for the errror prediction filters related to stationary time series. In another paper, we analyze the number of negative eigenvalues of an extension of a hermitian band matrix in terms of the entries in the band matrix. From this we obtain results on maximum entropy and on singular values of extensions of triangular matrices. The latter results are related to the finite-dimensional model reduction problem for linear systems. Keywords Operator theory, Matrix theory, Maximum entropy principle, Band matrices, Toeplitz matrix, Orgothogonal polynomials, Negative eigenvalues, Signal processing, Model reduction, Linear systems.