arXiv subjects
Publications and source records attributed to Karine Fouchet.
We compute an asymptotic formula for the supremum of the resolvent norm ($ζ$ -T ) -1 over |$ζ$| $\ge$ 1 and contractions T acting on an n-dimensional Hilbert space, whose spectral radius does not exceed a given r $\in$ (0, 1). We prove that this supremum is achieved on the unit circle by an analytic Toeplitz matrix.
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 $\infty$. We provide constructive examples which show that our estimates are sharp. As an application we construct a sequence of $n\times n$ invertible matrices $T$ with arbitrary spectrum in the unit disk and such that the quantity $|\det{T}|\cdot\|T^{-1}\|\cdot\|T\|^{1-n}$ grows as a power of $n$. This is motivated by Schäffer's question on norms of inverses.
We compute asymptotic formulas for the $k^{\rm th}$ Fourier coefficients of $b_λ^n$, where $b_λ(z)=\frac{z-λ}{1-λz}$ is the Blaschke factor associated to $λ\in\mathbb{D}$, $k\in[0,\infty)$ and $n$ is a large integer. We distinguish several regions of different asymptotic behavior of those coefficients in terms of $k$ and $n$. Given $β\in((1-λ)/(1+λ),(1+λ)/(1-λ))$ their decay is oscillatory for $k\in[βn,n/β]$. Given $α\in(0,(1-λ)/(1+λ))$ their decay is exponential for $k\in[0,nα]\cup[n/α,\infty).$ Airy-type behavior is happening near the $k$-transition points $n(1-λ)/(1+λ)$ and $n(1+λ)/(1-λ)$. The asymptotic formulas for the $k^{\rm th}$ Fourier coefficients of $b_λ^{n}$ are derived using standard tools of asymptotic analysis of Laplace-type integrals. More precisely, the integral defining the $k^{\rm th}$ Fourier coefficient of $b_λ^n$ is perfectly suited for an application of the method of stationary phase when $k\in\left(n(1-λ)/(1+λ),n(1+λ)/(1-λ)\right)$ and requires the use of the method of the steepest descent when $k\notin[n(1-λ)/(1+λ),n(1+λ)/(1-λ)]$. Uniform versions of those standard methods are required when $k$ approaches one of the boundaries $n(1-λ)/(1+λ),$ $n(1+λ)/(1-λ)$. As an application, we construct strongly annular functions with Taylor coefficients satisfying sharp summation properties.