Schur Parameters, Factorization and Dilation Problems by Tiberiu Constantinescu

By Tiberiu Constantinescu

This ebook is dedicated to the ubiquity of the Schur parameters. A dilation theoretic view results in a unified standpoint on numerous subject matters the place Schur parameters look as easy cells. including the transmission line, their actual counter- half, they seem in scattering concept, in modeling, prediction and filtering of nonstationary tactics, in sign processing, geophysics and process thought. Modeling difficulties are thought of for sure periods of operators, interpolation difficulties, determinental formulae, in addition to connections with sure periods of graphs the place, back, the Schur parameters may play a job. a few common algorithms that discover the transmission line also are offered during this booklet. As a complete, the textual content is self-contained and it's addressed to humans drawn to the formerly pointed out subject matters or connections among them.

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Example text

Therefore, Tn_lH~2) C H~221 and, in fact, Tn_lH~2) = H~221. Obviously, Tn_dH~2) is unitary. Finally, it is readily checked that if {H~}nEZ is another family of subspaces such that for all nEZ, H~ ~ Hn and Tn-dH~ (2) is unitary, then H~ ~ H~2). 1) are all trivial. 1. 1) reduce to the residual subspaces of the family.

10, it follows that the Cholesky operator Fon of A is also invertible and we can define the vectors fJ i = the i-th column of FO-;,l, i = 0, 1, ... 12) which constitute an orthonormal basis of (cn+1 )A. 13) where {aj;}j=o are the non-zero elements of the i-th column of FO-;,l. 13) represent the Gram-Schmidt procedure of orthonormalizing the family of vectors {Ei} in (C n + 1) A. This connection with the (inverse of) Cholesky operators justifies why the Gram-Schmidt procedure can be interpreted as a procedure for inverting positive matrices.

I=O Then, it follows by the definition of the Cholesky operator FOn that n-l det FOn = det FO,n-l x II det D rkn · k=O This equality leads immediately to the required formula for det A. 6 CHAPTER 1. SCHUR PARAMETERS AND POSITIVE BLOCK MATRICES Kolmogorov Decompositions. II In this section we use the Schur parameters in order to describe the Kolmogorov decomposition of a positive definite kernel. 1. Besides, the Naimark dilation of a positive definite Toeplitz kernel appears to be essentially the elementary rotation of a well defined row contraction of infinite length.

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