Asymptotic Invertibility and the Collective Asymptotic Spectral Behavior of Generalized One-Dimensional Discrete Convolutions

We study the asymptotic invertibility as $n\to+\infty$ of matrices of the form $\alpha_{kj}^{(n)}=a(k/n,j/n,k-j)$ and $\beta_{kj}^{(n)}=b(k/E(n),j/E(n),k-j)$, where $a$ and $b$ are functions defined on the sets $[0,1]\times[0,1]\times\mathbb{Z}$ and $[0,+\infty)\times[0,+\infty)\times\mathbb{Z}$, respectively, $E(n)\to+\infty$, and $n/E(n)\to+\infty$. The joint asymptotic behavior of the spectrum of these matrices is analyzed.