Toeplitz Matrices: Theme and Variations
Toeplitz matrices (constant on diagonals) describe the action of linear time-invariant operations, as well as the correlations among the variables of a stationary stochastic process. They are therefore often met in applications, where one wants to invert the operation or predict the outcome of the process. In addition, they are central in the theory of the trigonometric moment problem, a source of major developments in analysis, which asks for the set of positive measures for which an initial block of Fourier coefficients is prescribed. In each of these three contexts the same element plays a strong role, suggested in prediction by combinatorial ideas of information, and in inversion by orthogonal decomposition. We will try to explain these results and extend them to higher dimensions.