Finite Truncations of Generalized One-Dimensional Discrete Convolution Operators and Asymptotic Behavior of the Spectrum. The Matrix Case

Abstract

We study the sequence $\{A_N(a){\}}_{N\in \mathbb{N}}$ of finite truncations of a generalized discrete convolution operator, which have matrices of the form

where $a$ is some function defined on $[0,+\infty )\times [0,+\infty )$, $E(\cdot )$ is defined on $\mathbb{N}$ and $E(N)\to \infty$, $\frac{N}{E(N)}\to \infty$ as $N\to \infty$. For this sequence we get a generalization of the Szegő limit theorem.