Karine Fouchet Isambard

Aix-Marseille University | AMU

Given a finite Blaschke product $B$ we prove asymptotically sharp estimates on the $\ell ^{\infty }$-norm of the sequence of the Fourier coefficients of $B^{n}$ as $n$ tends to $\infty $. This norm decays as $n^{-1/N}$ for some $N\ge 3$. Furthermore, for every $N\ge 3$, we produce explicitly a finite Blaschke product $B$ with decay $n^{-1/N}$. As a...

In 2005, N. Nikolski proved among other things that for any $r\in (0,1)$ and any $K\geq 1$ , the condition number $CN(T)=\Vert T\Vert \cdot \Vert T^{-1}\Vert $ of any invertible n -dimensional complex Banach space operators T satisfying the Kreiss condition, with spectrum contained in $\left \{ r\leq |z|<1\right \}$ , satisfies the inequality $CN(T...

We compute asymptotic formulas for the kth\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k^{\textrm{th}}$$\end{document} Fourier coefficients of bλn\documentclass[12pt...

We compute an asymptotic formula for the supremum of the resolvent norm ||(ζ−T)−1|| over |ζ|≥1 and contractions T acting on an n-dimensional Hilbert space, whose spectral radius does not exceed a given r∈(0,1). We prove that this supremum is achieved on the unit circle by an analytic Toeplitz matrix.

We compute asymptotic formulas for the $k^{\rm th}$ Fourier coefficients of $b_\lambda^n$, where $b_\lambda(z)=\frac{z-\lambda}{1-\lambda z}$ is the Blaschke factor associated to $\lambda\in\mathbb{D}$, $k\in[0,\infty)$ and $n$ is a large integer. We distinguish several regions of different asymptotic behavior of those coefficients in terms of $k$...

This note surveys recent strategies to estimate the condition number CN(T)=‖T‖·‖T-1‖ of complex n×n matrices T with given spectrum. More precisely, we present a proof of the fact that if T acts on the Hilbert space Cn, then the supremum of CN(T) over all contractions T with smallest eigenvalues of modulus r>0, is equal to 1/rn, and is achieved by a...