JournalszaaVol. 24, No. 2pp. 251–275

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

  • Igor B. Simonenko

    State University, Rostov-On-Don, Russian Federation
  • Olga N. Zabroda

    Technische Universität Chemnitz, Germany
Finite Truncations of Generalized One-Dimensional Discrete Convolution Operators and Asymptotic Behavior of the Spectrum. The Matrix Case cover
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Abstract

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

AN(a)(a(nE(N),kE(N),nk))n,k=1,,N,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 aa is some function defined on [0,+)×[0,+)[0,+\infty) \times [0,+\infty), E()E(\cdot) is defined on N\mathbb{N} and E(N)E(N) \to \infty, NE(N)\frac{N}{E(N)} \to \infty as NN \to \infty. For this sequence we get a generalization of the Szeg\"o limit theorem.

Cite this article

Igor B. Simonenko, Olga N. Zabroda, Finite Truncations of Generalized One-Dimensional Discrete Convolution Operators and Asymptotic Behavior of the Spectrum. The Matrix Case. Z. Anal. Anwend. 24 (2005), no. 2, pp. 251–275

DOI 10.4171/ZAA/1239