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Here, we consider a more general class of matrix-sequences and we prove that they belong to the maximal $*$-algebra of generalized locally Toeplitz (GLT) matrix-sequences. Then, we identify the associated GLT symbols and GLT momentary…

The ensemble covariance matrix of a wide sense stationary signal spatially sampled by a full linear array is positive semi-definite and Toeplitz. However, the direct augmented covariance matrix of an augmentable sparse array is Toeplitz but…

Under appropriate technical assumptions, the simple-loop theory allows to deduce various types of asymptotic expansions for the eigenvalues of Toeplitz matrices $T_{n}(f)$ generated by a function $f$, unfortunately, such a theory is not…

We consider the problem of estimating the spectrum of a symmetric bounded entry (not necessarily PSD) matrix via entrywise sampling. This problem was introduced by [Bhattacharjee, Dexter, Drineas, Musco, Ray '22], where it was shown that…

Sparse grids are popular tools for high-dimensional function approximation. In this work, we study the use of sparse grids for interpolation using separable Mat\'ern kernels…

Synchronized measurements of a large power grid enable an unprecedented opportunity to study the spatialtemporal correlations. Statistical analytics for those massive datasets start with high-dimensional data matrices. Uncertainty is…

Consider random symmetric Toeplitz matrices $T_{n}=(a_{i-j})_{i,j=1}^{n}$ with matrix entries $a_{j}, j=0,1,2,...,$ being independent real random variables such that \be \mathbb{E}[a_{j}]=0, \ \ \mathbb{E}[|a_{j}|^{2}]=1 \ \ \textrm{for}\,\…

This note starts from work done by Dai, Geary, and Kadanoff (Hui Dai, Zachary Geary, and Leo P. Kadanoff, H. Dai, Z. Geary and L. P. Kadanoff, Journal of Statistical Mechanics, P05012 (2009)) on exact eigenfunctions for Toeplitz operators.…

The spectral symbols are useful tools to analyse the eigenvalue distribution when dealing with high dimensional linear systems. Given a matrix sequence with an asymptotic symbol, the last one depends only on the spectra of the individual…

The limiting behavior of the eigenvalues of the Toeplitz matrices $T_{n}[\sigma]=(\hat{\sigma}(i-j))$, where $0\leq i,j \leq n$, as $n \to \infty$, is investigated in the case of complex valued functions $\sigma$ defined on the unit circle…

This article is about finding the limit set for banded Toeplitz matrices. Our main result is a new approach to approximate the limit set $\Lambda(b)$ where $b$ is the symbol of the banded Toeplitz matrix. The new approach is geometrical and…

We describe the asymptotics of the spectral norm of finite Toeplitz matrices generated by functions with Fisher-Hartwig singularities as the matrix dimension goes to infinity. In the case of positive generating functions, our result…

The self-consistent expansion (SCE) is a powerful technique for obtaining perturbative solutions to problems in statistical physics but it suffers from a subtle problem - too much freedom! The SCE can be used to generate an enormous number…

In a mixed generalized linear model, the goal is to learn multiple signals from unlabeled observations: each sample comes from exactly one signal, but it is not known which one. We consider the prototypical problem of estimating two…

The spectrum of a network or graph $G=(V,E)$ with adjacency matrix $A$, consists of the eigenvalues of the normalized Laplacian $L= I - D^{-1/2} A D^{-1/2}$. This set of eigenvalues encapsulates many aspects of the structure of the graph,…

This paper is concerned with the asymptotic expansions of the amplitude of the solution of the Helmholtz equation. The original expansions were obtained using a pseudo-differential decomposition of the Dirichlet to Neumann operator. This…

This paper is essentially devoted to the study of the minimal eigenvalue $\lambda_{N,\alpha}$ of the Toepllitz matrice $T_N(\varphi_{\alpha})$ where $\varphi_{\alpha}(e^{i \theta})=|1- e^{i \theta} |^{2\alpha} c_{1}(e^{i \theta})$ with…

This paper uses an incremental matrix expansion approach to derive asymptotic eigenvalue distributions (a.e.d.'s) of sums and products of large random matrices. We show that the result can be derived directly as a consequence of two common…

Let $A$ be an infinite Toeplitz matrix with a real symbol $f$ defined on $[-\pi, \pi]$. It is well known that the sequence of spectra of finite truncations $A_N$ of $A$ converges to the convex hull of the range of $f$. Recently, Levitin and…

We consider the family of arithmetical matrices given explicitly by $$E(\sigma,\tau)=\left\{\frac{n^\sigma m^\sigma}{[n,m]^\tau}\right\}_{n,m=1}^\infty,$$ where $[n,m]$ is the least common multiple of $n$ and $m$ and the real parameters…