Trace et valeurs propres extrêmes d'un produit de matrices de Toeplitz. Le cas singulier

Details Trace et valeurs propres extrêmes d'un produit de matrices de Toeplitz. Le cas singulier Trace et valeurs propres extrêmes d'un produit de matrices de Toeplitz. Le cas singulier

2011-06-10

arxiv.org/abs/1106.2072v1

Mathematics

Functional Analysis

Scientific paper

Trace and extreme eigenvalues of a product of truncated Toeplitz matrices. The singular case. In a first theorem we give an asymptotic expansion of Tr (T_N (f_1) T_N^{-1}(f_2)) where f1 ({\theta}) = |1 - e^{i {\theta}} | ^{2{{\alpha}1}c1 (ei{\theta}{\theta}) and f2 ({\theta}) = |1 - e ^{i{\theta}}| ^{2{\alpha}2}c2 (ei{\theta}), with c1 and c2 are two regular functions of the torus and - 1/2 < {\alpha}1, {\alpha}2 < 1/2 . In a second part of this work we study the particular case where {\alpha}1 > 0 and {\alpha}2 < 0. Then we obtain the asymptotic of the trace of the powers of Tr (T_N (f_1) T_N^{-1}(f_2)) for s {\in} N* that provides us the limits when N goes to the infinity of the extreme eigenvalues of this matrix. This last result allows us to give a large deviation principle for a family of quadratic forms of stationnary process.

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