On the $\ell_p$ and $\ell_{p,q}$ norms of Cauchy-Toeplitz matrices
Tserendorj Batbold
TL;DR
This paper analyzes the $\\ell_p$ norms of Cauchy-Toeplitz matrices $T_n=[2/(1+2(i-j))]$, establishing new upper and lower bounds for $1<p<\\infty$ and resolving Bozkurt's conjecture on the $\\ell_{p,q}$ norms. The authors derive asymptotic and $n$-dependent bounds using the Binet-Cauchy identity and power-mean techniques, culminating in the exact limit $\\lim_{m\\to\\infty} m^{-1/p}\\|T_m\\|_p=2^{1/p}[(2^p-1)\\zeta(p)]^{1/p}$ and a refined two-sided bound with correction terms $C'(n,p)$ and $C''(n,p)$. They also provide a complete characterization of the $\\|T_n\\|_{p,q}$ norms, including thresholds $\\delta_p$ and $\\eta_{p,n}$ and edge-case analysis for small $n$, thereby delivering precise norm estimates for these structured matrices. The results have implications for numerical linear algebra and matrix analysis where sharp norm bounds of structured matrices are essential.
Abstract
New upper and lower bounds for the $\ell_p (1<p<\infty)$ norms of Cauchy-Toeplitz matrices in the form $T_n=[2/(1+2(i-j))]_{i,j=1}^n$ are derived. Moreover, we give a complete answer to a conjecture proposed by D. Bozkurt.