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1answer
44 views

On the limit set of eigenvalues of banded Toeplitz Hessenberg matrices

Let $T_{n}(b)$ be the $n\times n$ Toeplitz matrix determined by the symbol $$ b(z)=\frac{1}{z}+\sum_{j=0}^{k}a_{j}z^{j} $$ where $k\in\mathbb{N}$ and $a_{0},\dots,a_{k}\in\mathbb{R}$, $a_{k}\neq0$. ...
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0answers
48 views

Riemann theta function with asymptotically large Toeplitz Matrix

As a follow up to How to compute $\sum_{x \in \mathbb{Z}^n} e^{-x^TMx}$ efficiently Suppose that $M$ is a large Toeplitz matrix. With a suitable scaling $K^{-n}$ for some $K$, what will the Riemann ...
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51 views

inverse of asymptotic Toeplitz matrix with band limited associated function

I am reviewing a controversial paper, and the main result, a revolution within my field, comes down to whether or not the following is true. I strongly believe it is not, but would need confirmation. ...
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52 views

Asymptotic determinant of $2\times 2$ Toeplitz matrix

The problem that I am dealing with is to compute the determinant of a $2\times 2$ Toeplitz matrix[1] (in general I would like to generalize to a more general case, but let's consider the easiest case ...
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0answers
99 views

On triangular Toeplitz matrices

Let $R(x)$ be the upper triangular Toeplitz matrix with first row $x$, so that $R_{ik}=x_{k-i+1}$ if $i\le k$ and $R_{ik}=0$ otherwise. Let $N(n)$ be the smallest number $N$ such that there exist $u_j,...
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44 views

Is there a possibility to find the toeplitz matrix that has the nearest column space to a given matrix?

As is said, if we have a known matrix $D \in \mathbb{R}^{N \times k}$ with $k<N$, is there a way that we can find a toeplitz matrix $D_T \in \mathbb{R}^{N \times k}$, which satisfies $$ D_T=\arg\...
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609 views

The maximal eigenvalue of a symmetric Toeplitz matrix

Let $0\le x\le 1$ be a real number. Denote by $A_n(x)=(a_{ij})$ the $n$ by $n$ matrix such that $a_{ij}=x^{|i-j|}$ and let $\lambda_n(x)$ be the maximal eigenvalue of $A_n(x)$. Is there any ...
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1answer
115 views

Infinite Real Symmetric Toeplitz Matrix Reference

I am looking for a good starting point (book or articles) for studying Toeplitz matrices. Specifically as mentioned in the title, I am most interested in the case where they are of the form $$A = \{\...
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0answers
224 views

L1-regularized Least Squares on a matrix with Toeplitz Blocks (not block-Toeplitz)

I am trying to speed up a sparse signal recovery algorithms. My sensing matrix is a set of Toeplitz Blocks, M = [T1,T2,T3,...,Tk] The objective is min ||Mx - b||_2^2 + ||x||1 What I'm actually ...
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1answer
254 views

Invertibility of frame/sampling operator on Bargmann-Fock spaces

Let $F_\alpha ^p (\mathbb{C}^n)$ for $1 < p < \infty$ and $\alpha > 0$ be the Bargmann-Fock space defined as the Banach space of entire functions $f$ such that $f(\cdot) e^{- \frac{\alpha}{2} ...
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1answer
921 views

annihilator/common left multiple of matrix polynomials

Let $A_{n,d}$ be the space of polynomials of degree $d$ or less whose coefficients are real $n\times n$ matrices --- or, if you prefer, the space of matrices whose entries are degree-$d$ polynomials. ...
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0answers
239 views

Can one pose a Toeplitz index problem associated to a discrete group?

Before posing my question, let me provide a little background since the Wikipedia page on this stuff is sorely lacking. Let's start with the classical case of the Toeplitz index problem on the circle....
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0answers
405 views

Convolutions and Toeplitz Operators

Let be $d>0$ an integer number and consider the Cartesian product $\mathbb Z^d$ as metric space, with the distance between $x,y\in\mathbb Z^d$ given by $\|x-y\|_1=\sum_{j=0}^d|x_j-y_j|$. Let be $...
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2answers
354 views

Where can I learn about (the asymptotics of) Toeplitz operators?

Toeplitz operators provide a natural language with which to do geometric quantization. I don't want to really understand them, and I don't need them in full generality. I'm looking for some ...