On Functions of quasi Toeplitz matrices

Abstract

Let $a(z)={\sum }_{i\in \mathbb{Z}}{a}_i{z}^i$ be a complex valued continuous function, defined for $|z|=1$, such that ${\sum }_{i=-\infty }^{+\infty }|i{a}_i|<\infty$. Consider the semi-infinite Toeplitz matrix $T(a)=(t_{i,j}{)}_{i,j\in {\mathbb{Z}}^{+}}$ associated with the symbol $a(z)$ such that $t_{i,j}=a_{j-i}$. A quasi-Toeplitz matrix associated with the continuous symbol $a(z)$ is a matrix of the form $A=T(a)+E$ where $E=(e_{i,j})$, ${\sum }_{i,j\in {\mathbb{Z}}^{+}}|e_{i,j}|<\infty$, and is called a CQT-matrix. Given a function $f(x)$ and a CQT matrix $M$, we provide conditions under which $f(M)$ is well defined and is a CQT matrix. Moreover, we introduce a parametrization of CQT matrices and algorithms for the computation of $f(M)$. We treat the case where $f(x)$ is assigned in terms of power series and the case where $f(x)$ is defined in terms of a Cauchy integral. This analysis is applied also to finite matrices which can be written as the sum of a Toeplitz matrix and of a low rank correction.

References in corpus (3)

• Semi-Infinite Quasi-Toeplitz Matrices with Applications to QBD Stochastic Processes
• Natural Boundary for a Sum Involving Toeplitz Determinants
• On the exponential of semi-infinite quasi-Toeplitz matrices

Cited by in corpus (2)

• Semi-Infinite Quasi-Toeplitz Matrices with Applications to QBD Stochastic Processes
• On the exponential of semi-infinite quasi-Toeplitz matrices