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

Abstract

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 $$[0,1] \times [0,1] \times \mathbb{Z}{\text{ and [0, + }}\infty {\text{)}} \times {\text{[0, + }}\infty {\text{)}} \times \mathbb{Z}{\text{, respectively, }}E(n) \to+ \infty ,{\text{ and }}n/E(n) \to+ \infty $$ . The joint asymptotic behavior of the spectrum of these matrices is analyzed.