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The eigenvalues of Toeplitz matrices $T_{n}(f)$ with a real-valued symbol $f$, satisfying some conditions and tracing out a simple loop over the interval $[-\pi,\pi]$, are known to admit an asymptotic expansion with the form \[…

It is known that the generating function $f$ of a sequence of Toeplitz matrices $\{T_n(f)\}_n$ may not describe the asymptotic distribution of the eigenvalues of $T_n(f)$ if $f$ is not real. In this paper, we assume as a working hypothesis…

It is known that the generating function $f$ of a sequence of Toeplitz matrices $\{T_n(f)\}_n$ may not describe the asymptotic distribution of the eigenvalues of $T_n(f)$ if $f$ is not real. In a recent paper, we assume as a working…

This paper is devoted to the asymptotic behavior of all eigenvalues of Symmetric (in general non Hermitian) Toeplitz matrices with moderately smooth symbols which trace out a simple loop on the complex plane line as the dimension of the…

The present work is devoted to the eigenvalue asymptotic expansion of the Toeplitz matrix $T_{n}(a)$ whose generating function $a$ is complex valued and has a power singularity at one point. As a consequence, $T_{n}(a)$ is non-Hermitian and…

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…

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…

It was shown in a series of recent publications that the eigenvalues of $n\times n$ Toeplitz matrices generated by so-called simple-loop symbols admit certain regular asymptotic expansions into negative powers of $n+1$. On the other hand,…

In this short article we propose a full large $N$ asymptotic expansion of the probability that the $m^{\text{th}}$ power of a random unitary matrix of size $N$ has all its eigenvalues in a given arc-interval centered in $1$ when $N$ is…

In this article, we study the large $n$ asymptotic expansions of $n\times n$ Toeplitz determinants whose symbols are indicator functions of unions of arc-intervals of the unit circle. In particular, we use an Hermitian matrix model…

Let $T_N$ denote an $N\times N$ Toeplitz matrix with finite, $N$ independent symbol ${\bf a}$. For $E_N$ a noise matrix satisfying mild assumptions (ensuring, in particular, that $N^{-1/2}\|E_N\|_{{\rm HS}}\to_{N\to\infty} 0$ at a…

We study the asymptotic eigenvalue distribution of Toeplitz matrices generated by a singular symbol. It has been conjectured by Widom that, for a generic symbol, the eigenvalues converge to the image of the symbol. In this paper we ask how…

The analysis of the spectral features of a Toeplitz matrix-sequence $\left\{T_{n}(f)\right\}_{n\in\mathbb N}$, generated by a symbol $f\in L^1([-\pi,\pi])$, real-valued almost everywhere (a.e.), has been provided in great detail in the last…

Any sequence of uniformly bounded $N\times N$ Hermitian Toeplitz matrices $\{\boldsymbol{H}_N\}$ is asymptotically equivalent to a certain sequence of $N\times N$ circulant matrices $\{\boldsymbol{C}_N\}$ derived from the Toeplitz matrices…

We study the spectral convergence of compact, self-adjoint operators on a separable Hilbert space under operator norm perturbations, and derive asymptotic expansions for their eigenvalues and eigenprojections. Our analysis focuses on…

Under appropriate technical assumptions, the simple-loop theory allows to deduce various types of asymptotic expansions for the eigenvalues of Toeplitz matrices generated by a function $f$. Independently and under the milder hypothesis that…

This paper studies two structured approximation problems: (1) Recovering a corrupted low-rank Toeplitz matrix and (2) recovering the range of a Fourier matrix from a single observation. Both problems are computationally challenging because…

We study the problem of approximating the eigenspectrum of a symmetric matrix $\mathbf A \in \mathbb{R}^{n \times n}$ with bounded entries (i.e., $\|\mathbf A\|_{\infty} \leq 1$). We present a simple sublinear time algorithm that…

In this thesis, we introduce complex manifolds with local spectral gaps and study their asymptotic behavior using the scaling method. With these asymptotics, we obtain an asymptotic expansion for the Bergman kernel of a Hermitian…

In this article, we determine the spectrum of real-analytic, non self-adjoint Toeplitz operators on compact K{\"a}hler manifolds and on the complex plane, on neighbourhoods of critical values of the symbol. We consider specifically critical…