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

Authors:	O N Zabroda I B Simonenko

Institution:	(1) Department of Mechanics and Mathematics, Rostov State University, Russia

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.

Keywords:	asymptotic invertibility matrix operator spectrum
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