Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Trace et valeurs propres extr\^emes d'un produit de matrices de
  Toeplitz. Le cas singulier   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.

25
The categories of this paper are: patt-sol
28
The categories of this paper are: alg-geom
31
The categories of this paper are: ao-sci
29
The categories of this paper are: funct-an
13
The categories of this paper are: quant-ph
7
The categories of this paper are: hep-th
35
The categories of this paper are: mtrl-th
6
The categories of this paper are: astro-ph
15
The categories of this paper are: nucl-ex
9
The categories of this paper are: nlin
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The categories of this paper are: dg-ga
33
The categories of this paper are: plasm-ph
36
The categories of this paper are: cmp-lg
4
The categories of this paper are: cond-mat
11
The categories of this paper are: hep-lat
34
The categories of this paper are: chem-ph