Generalized inversion of Toeplitz-plus-Hankel matrices

In many applications, e.g. digital signal processing, discrete inverse scattering, linear prediction etc., Toeplitz-plus-Hankel (T + H) matrices need to be inverted. (For further applications see [1] and references therein). Firstly the T +H matrix inversion problem has been solved in [2] where it was reduced to the inversion problem of the block Toeplitz matrix (the so-called mosaic matrix). The drawback of the method is that it does not work for any invertible T +H matrix since it requires also invertibility of the corresponding T −H matrix. This drawback appeared also in [3], [4]. Later on the drawback was put out (see, e.g. [4], [5]), moreover, the inversion problem was solved for the block T +H matrix [6],[7]. The generalized inversion of Toeplitz-plus-Hankel matrices is of great interest to, e.g., Pade-Chebyshev approximations. The generalized inversion for matrix A is meant to be the matrix A such that AAA = A. Our goal is to obtain the generalized inversion of block T +H matrices with help of the well-known method of reducing the block-diagonal matrix formed from the T+H and T−H matrices to the mosaic matrix (as in the work [2]). We will need the generalized inversion for the block Toeplitz matrix which has been already found in, e.g. [8]. It is shown in the present paper that there is no need for T −H matrix to be inverted: if the T +H matrix is invertible than the obtained generalized inverse matrix proves to be its inverse matrix. The paper is organized as follows. At first the basic definitions and the main results of the paper [8] are given, then our main theorem is formulated and proved. This theorem is demonstrated with an example in the end of the paper. This work was supported by Russian Foundation for Basic Research (RFFI), grant N 04-04-96006.