A Geometrical Approach of the Levinson Algorithm for Block Toeplitz Matrices

Abstract

In this paper we obtain a version of the Levinson algorithm for block Toeplitz matrices in an infinite dimensional setting from a geometrical approach. With this methodology we obtain a sequence of operators in the Levinson recurrences whose norms in geometric terms represent angles between subspaces. Additionally, under this geometric framework a block LU decomposition for a block Toeplitz matrix is obtained. 749 this, it is necessary to obtain the LU decomposition of T p . In [9], this decomposition is used to demonstrate that the maximum entropy density subject to the first p + 1 autocovariance matrices is the spectrum of a multivariate autoregressive process of p − th order. Problems involving Toeplitz linear equation arise in several applications, e.g. in the prediction of stationary processes, in inverse scattering problem, in buffer analysis for data communication system, among others (cf.[5], [18] and [27]). In [24] and [25] some previous results are obtained using orthogonal decomposition for the finite dimensional case.The main contribution of this work is to obtain a new version of the Levinson algorithm for block Toeplitz matrices in an infinite dimensional setting using orthogonal decomposition. The parameters obtained in this algorithm are a sequence of contractive operators whose norm can be expressed as angles between subspaces, similar to the finite dimensional case. Also, we obtain a block LU decomposition for block Toeplitz matrices where the triangular matrices can be obtained through Levinson recurrences.Additionally, our methodology could be useful to solve extension problems in statistics and for prediction problems in an infinite dimensional context (cf. [10], [11], [12], [15], [16], [19], [20], [28], [33] and [34]). The scheme to obtain the Levinson algorithm is as follows: the data is a sequence of bounded operators {R k } p k=0 , defined in a separable Hilbert space G. More specifically, based on the sequence of bounded operators {R k } p k=0 , we built a Hilbert space H p and a surjective isometry the Levinson recurrences are obtained as a consequence of the orthogonal decompositions where the subspaces in this decompositions are the defect spaces.The paper is organized as follows: in the second section; we introduce some notations and preliminary results; in the third section, we obtain a block LU decomposition for a block Toeplitz matrix; in fourth section, we state the main result of this work and in the last section, we present the conclusions and discussions.