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 Szeg\"o limit theorem.