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Zeitschrift für Analysis und ihre Anwendungen


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Volume 24, Issue 2, 2005, pp. 251–275
DOI: 10.4171/ZAA/1239

Published online: 2005-06-30

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

Igor B. Simonenko[1] and Olga N. Zabroda[2]

(1) State University, Rostov-On-Don, Russian Federation
(2) Technische Universität Chemnitz, Germany

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 $$ A_N(a) \sim \left(a\left(\frac{n}{E(N)}, \frac{k}{E(N)}, n-k\right)\right)_{n,k=1,\ldots,N}, $$ 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.

Keywords: Szegö limit theorem, convolution operator, eigenvalues, Toeplitz operator

Simonenko Igor, Zabroda Olga: Finite Truncations of Generalized One-Dimensional Discrete Convolution Operators and Asymptotic Behavior of the Spectrum. The Matrix Case. Z. Anal. Anwend. 24 (2005), 251-275. doi: 10.4171/ZAA/1239