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JOURNALS // Matematicheskii Sbornik // Archive

Mat. Sb., 2017 Volume 208, Number 11, Pages 29–55 (Mi sm8823)

This article is cited in 16 papers

Exploration of Toeplitz-like matrices with unbounded symbols is not a purely academic journey

A. Böttchera, C. Garonibc, S. Serra-Capizzanobd

a Fakultät für Mathematik, Technische Universität Chemnitz, Chemnitz, Germany
b Department of Science and High Technology, University of Insubria, Como, Italy
c Institute of Computational Science, Università della Svizzera Italiana, Lugano, Switzerland
d Department of Information Technology, Uppsala University, Uppsala, Sweden

Abstract: It is often asked why Toeplitz-like matrices with unbounded symbols are worth studying. This paper gives an answer by presenting several concrete problems that motivate such studies. It surveys the central results of the theory of Generalized Locally Toeplitz (GLT) sequences in a self-contained tool-kit fashion, and gives a new extension from bounded Riemann integrable functions to unbounded almost everywhere continuous functions. The emergence of unbounded symbols is illustrated by local grid refinements in finite difference and finite element discretizations and also by preconditioning strategies.
Bibliography: 40 titles.

Keywords: Toeplitz-like matrices, eigenvalue distribution, singular value distribution, GLT-sequences, local grid refinement.

UDC: 517.983.3+512.643.8+519.62

MSC: Primary 47B35; Secondary 15B05, 65F08, 65F15, 65L50

Received: 02.07.2016

DOI: 10.4213/sm8823


 English version:
Sbornik: Mathematics, 2017, 208:11, 1602–1627

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© Steklov Math. Inst. of RAS, 2025