Based on their research experience, the authors propose a reference textbook in two volumes on the theory of generalized locally Toeplitz sequences and their applications. This first volume focuses on the univariate version of the theory and the related applications in the unidimensional setting, while the second volume, which addresses the multivariate case, is mainly devoted to concrete PDE applications.
This book systematically develops the theory of generalized locally Toeplitz (GLT) sequences and presents some of its main applications, with a particular focus on the numerical discretization of differential equations (DEs). It is the first book to address the relatively new field of GLT sequences, which occur in numerous scientific applications and are especially dominant in the context of DE discretizations. Written for applied mathematicians, engineers, physicists, and scientists who (perhaps unknowingly) encounter GLT sequences in their research, it is also of interest to those working in the fields of Fourier and functional analysis, spectral analysis of DE discretization matrices, matrix analysis, measure and operator theory, numerical analysis and linear algebra. Further, it can be used as a textbook for a graduate or advanced undergraduate course in numerical analysis.
Dr. Carlo Garoni graduated in mathematics at the University of Insubria (Italy) in 2011 and received his Ph.D. in mathematics at the same university in 2015. He has pursued research at the Universities of Insubria and Rome "Tor Vergata", and he has now a Marie-Curie postdoctoral position at the USI University of Lugano (Switzerland). He has published around 20 research papers in different areas of mathematics, most of which are connected with the theory of GLT sequences and its applications.
Prof. Stefano Serra-Capizzano is full professor in numerical analysis at the University of Insubria (Italy), head of the Department of Science and High Technology at the same university, and long-term visiting professor at the Department of Information Technology at Uppsala University (Sweden). He has authored over 170 research papers in different areas of mathematics, with more than 70 collaborators all over the world. He is the founder of the Ph.D. Program "Mathematics of Computation" at the University of Insubria.
1 Introduction.- 2 Mathematical background.- 3 Singular value and eigenvalue distribution of a matrix-sequence.- 4 Spectral distribution of sequences of perturbed Hermitian matrices.- 5 Approximating classes of sequences.- 6 Toeplitz sequences.- 7 Locally Toeplitz sequences.- 8 Generalized locally Toeplitz sequences.- 9 Summary.- 10 Applications.- Future developments.- Solutions to the exercises