arXiv subjects

Abdellatif Seghier

Publications and source records attributed to Abdellatif Seghier.

6 recordsLinked to original sources

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↗

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↗

arXiv2013

Matrices de Toeplitz tronquées sur des polygones convexes. Cas du triangle

We consider the class of positive bounded and semi-continuous functions defined on the two dimensional torus If f belongs to this class, then f will be considered as the symbol of a Toeplitz operator truncated on a triangle parametrised by an integer number . We develop a geometric structure of the inverse of the Toeplitz operator and give an asymptotical development of the trace of its inverse wich brings out the geometry of the triangle. The foundation of this result consists in the possibility of f having a factorisation of type |g|^2 where the spectrum of g will be localised in a given semi-cone. This trace theorem allows in particular to find again the Linnik-Szegö theorem about the asymptotical evaluation of the determinant of the truncated Toeplitz operator (or Toeplitz matrix)

math.FA↗

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↗