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

We study the asymptotic invertibility as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$n \to+ \infty $$ \end{document} of matrices of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\alpha _{kj}^{(n)}= a(k/n,j/n,k - j)$$ \end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\beta _{kj}^{(n)}= b(k/E(n),j/E(n),k - j)$$ \end{document}, where a and b are functions defined on the sets \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$[0,1] \times [0,1] \times \mathbb{Z}{\text{ and [0, + }}\infty {\text{)}} \times {\text{[0, + }}\infty {\text{)}} \times \mathbb{Z}{\text{, respectively, }}E(n) \to+ \infty ,{\text{ and }}n/E(n) \to+ \infty $$ \end{document}. The joint asymptotic behavior of the spectrum of these matrices is analyzed.