arXiv subjects

Philippe Rambour

Publications and source records attributed to Philippe Rambour.

12 recordsLinked to original sources

arXiv2021

Asymptotic expansion of the eigenvalues of a Toeplitz matrix with a real symbol

Asymptotic expansion of the eigenvalues of a Toeplitz matrix with real symbol. This work provides two results obtained as a consequence of an inversion formula for Toeplitz matrices with real symbol. First we obtain an symptotic expression for the minimal eigenvalues of a Toeplitz matrix with a symbol which is periodic, even and derivable on $[0, 2π[$. Next we prove that a Toeplitz band matrix with a symbol without zeros on the united circle is invertible with an inverse which is essentially a band matrix. As a consequence of this last statement we give an asymptotic estimation for the entries of the inverse of a Toplitz matrix with a regular symbol.

math.CA↗

arXiv2021

Toeplitz matrices for the study of the fractional Laplacian on a bounded interval

Toeplitz matrices for the study of the fractional Laplacian on a bounded interval. In this work we get a deep link between (--$Δ$) $α$ ]0,1[ the fractional Laplacian on the interval ]0, 1[ and T N ($Φ$ $α$) the Toeplitz matrices of symbol $Φ$ $α$ : $θ$ $\rightarrow$ |1 -- e i$θ$ | 2$α$ when N goes to the infinity and for $α$ $\in$]0, 1 2 [$\cup$] 1 2 , 1[. In the second part of the paper we provide a Green function for the fractional equation (--$Δ$) $α$ ]0,1[ ($ψ$) = f for $α$ $\in$]0, 1 2 [ and f a sufficiently smooth function on [0, 1]. The interest is that this Green's function is the same as the Laplacian operator of order 2n, n $\in$ N. Mathematical Subject Classification (2000) Primary 35S05, 35S10,35S11 ; Secondary 47G30.

math.CA↗

arXiv2021

Asmptotic of the eigenvalues of Toeplitz matrices with even symbol

In this paper we consider an interval $[\theta\_{1}, \theta\_{2}] \subset [0, \pi]$ and $f$ a differentiable, periodic and even function sufficiently smooth such that $f(\theta) \in [f(\theta\_{1}, f(\theta\_{2})] \iff \theta \in [\theta\_{1}, \theta\_{2}]$. Then we obtain an higher order asymptotic formula for all the eigenvalues of the Toeplitz matrix $T\_N(f)$ as $N \to + \infty$ which belong to $[f(\theta\_{1}, f(\theta\_{2})]$ (resp. $[f(\theta\_{2}, f(\theta\_1)]$).

math.CA↗

arXiv2018

Fractional differential operators and Toeplitz matrices

In this work we generalize to few fractional differential operators the method used to reverse differential operators $\frac{d^{2n}}{dx^{2n}}$ by inverting a Toeplitz matrix. The interest of this work is to show that the method provides by the classical analytic methods of the analysis are easily founded by this means. (Titre: Opérateurs différentiels fractionnaires et matrices de Toeplitz) Opérateurs différentiels fractionnaire et matrices de Toeplitz.) Dans ce travail on généralise à certains opérateurs fractionnaires la méthode utlisée pour pour inverser les opérateurs différentiels $\frac{d^{2n}}{dx^{2n}}$ en inversant une matrice de Toeplitz. L'intérêt de ce travail est de montrer que l'on retrouve facilement par ce moyen les résultats fournis par les méthodes classiques d'analyse.

math.CA↗

arXiv2017

Expression asymptotique des valeurs propres d'une matrice de Toeplitz à symbole réel

This work provides two results obtained as a consequence of an inversion formula for Toeplitz matrices with real symbol. First we obtain an asymptotic expression for the minimal eigenvalues of a Toeplitz matrix with a symbolwhich is periodic, even and derivable on $[0, 2π[$. Next we prove that a Toeplitz band matrix with a symbol without zeros on the united circle is invertible with an inverse which is essentially a band matrix. As a consequence of this last statement we give an asymptotic estimation for the entries of the inverse of a Toeplitz matrix with a regular symbol.

math.FA↗

arXiv2014

Orthogonal polynomials with respect of a class of Fisher-Hartwig symbols

In this paper we give an asymptotic of the coefficients of the orthogonal polynomials on the unit circle, with respect of a weight of type $\displaystyle{ f : θ\mapsto \prod_{1\le j \le M} \vert 1 - e^{i(θ_{j}-θ)}\vert ^{2α_{j}} c}$ with $θ_{j}\in ]-π,π]$, $-\frac{1}{2} < α_{j}<\frac{1}{2}$ and $c$ a sufficiently smooth function.

math.CA↗

arXiv2014

Asymptotic of the terms of the Gegenbauer polynomial on the unit circle and applications to the inverse of Toeplitz matrices

The first part of this paper is devoted to the study of the orthogonal polynomial on the circle, with respect of a weight of type $f_α(θ) = (2\cos θ- 2\cos θ_0)^{2α} c_1$ with $θ_0 \in ]0,π[$, -1/2 <α<1/2 and c_1 a sufficiently smooth function. In a second part of the paper we obtain an asymptotic of the entries $(T_N f_α)^{-1}_{k+1,l+1}$ for sufficiently large values of $k,l$, that provides a lower bound on the eigenvalues of this matrix.

math.CA↗

arXiv2013

Valeur propre minimale d'une matrice de Toeplitz et d'un produit de matrices de Toeplitz

This paper is essentially devoted to the study of the minimal eigenvalue $λ_{N,α}$ of the Toepllitz matrice $T_N(φ_α)$ where $φ_α(e^{i θ})=|1- e^{i θ} |^{2α} c_{1}(e^{i θ})$ with $c_{1}$ a positive sufficiently smooth function and $0<α<\frac{1}{2}$. We obtain $λ_{N,α}\sim c_αN^{-2α}c_{1}(1)$ when $N$ goes to the infinity and we have the bounds of $c_α$. To obtain the asymptotic of $λ_{N,α}$ we give a theorem which suggests that the entries of $T_N^{-1}(φ_α)$ and $T_N (φ^{-1}_α)$ are closely related. If $α_1 + α_2 > \frac{1}{2}$ we obtain the asymptotic of the minimal eigenvalue of $T_N (φ_{α_1}) T_N (φ_{α_2}).$

math.SP↗

arXiv2012

Maximal eigenvalue and norm of the product of Toeplitz matrices. Study of a particular case

In this paper we describe the asymptotic behaviour of the spectral norm of the product of two finite Toeplitz matrices as the matrix dimension goes to the infinity. These Toeplitz matrices are generated by positive functions with Fisher-Hartwig singularities of negative order. Since we have positive operators it is known that the spectral norm is also the largest eigenvalue of this product.

math.SP↗

arXiv2011

Trace et valeurs propres extrêmes 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 (θ) = |1 - e^{i θ} | ^{2{α1}c1 (eiθθ) and f2 (θ) = |1 - e ^{iθ}| ^{2α2}c2 (eiθ), with c1 and c2 are two regular functions of the torus and - 1/2 < α1, α2 < 1/2 . In a second part of this work we study the particular case where α1 > 0 and α2 < 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.

math.FA↗

arXiv2010

Inversion des matrices de Toeplitz dont le symbole admet un zéro d'ordre fractionnaire positif, valeur propre minimale

Inversion of Toeplitz matrices with singular symbol. Minimal eigenvalues. Three results are stated in this paper. The first one is devoted to the study of the orthogonal polynomial with respect of the weight $φ_α (θ)=\vert 1- e^{i θ} \vert ^{2α} f_{1}(e^{i θ})$, with $α> \demi$ and $α\in \rr \setminus \nn $, and $f_{1}$ a regular function. We obtain an asymptotic expansion of the coefficients of these polynomials, and we deduce an asymptotic of the entries of $\left( T_{N} (φ_α)\right)^{-1}$ where $T_{N} (φ_α)$ is a Toeplitz matrix with symbol $φ_α$. Then we extend a result of A. Böttcher and H. Widom result related to the minimal eigenvalue of the Toeplitz matrix $T_{N}(φ_α)$. For $N$ goes to the infinity it is well known that this minimal eigenvalue admit as asymptotic $\frac{c_α}{N^{2α}} f_{1}(1)$. When $α\in \nn$ the previous authors obtain an asymptotic of $c_α$ for $α$ going to the infinity, and they have the bounds of $c_α$ for the other cases. Here we obtain the same type of results but for $α$ a positive real.

math.FA↗

arXiv2009

Comportement asymptotique des polynômes orthogonaux associés à un poids ayant un zéro d'ordre fractionnaire sur le cercle. Applications aux valeurs propres d'une classe de matrices aléatoires unitaires

Asymptotic behavior of orthogonal polynomials on the circle, with respect to a weight having a fractional zero on the torus. Applications to the eigenvalues of certain unitary random matrices. This paper is devoted to the orthogonal polynomial on the circle, with respect to a weight of type $ f=(1-\cos θ)^αc$ where $c$ is a sufficiently smooth function and $α\in ]-{1/2}, {1/2}[$. We obtain an asymptotic expansion of the coefficients of this polynomial and of $Φ^{(p)}_{N}(1)$ for all integer $p$. These results allow us to obtain an asymptotic expansion of the associated Christofel-Darboux kernel, and to compute the distribution of the eigenvalues of a family of random unitary matrices. The proof of the resuts related with the orthogonal polynomials are essentialy based on the inversion of Toeplitz matice associated to the symbol $f$.

math.FA↗