Abstract
The theory of spectral symbols links sequences of matrices with measurable functions expressing their asymptotic eigenvalue distributions. Usually, a sequence admits several spectral symbols, and it is not clear if a canonical one exists. Here we present a way to connect the sequences with the space of probability measure, so that each sequence admits a uniquely determined measure. The methods used are similar to those employed in the theory of generalized locally Toeplitz (GLT) sequences: a goal of this present contribution is in fact that of explaining how the two concepts are connected.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Ambrosio, L., Da Prato, G., Mennucci, A.: Introduction to Measure Theory and Integration. Edizioni della Normale, Pisa (2011)
Barbarino, G.: Equivalence between GLT sequences and measurable functions. Linear Algebra Appl. 529, 397–412 (2017)
Barbarino, G.: Diagonal Matrix Sequences and Their Spectral Symbols. http://arxiv.org/abs/1710.00810 (2017)
Barbarino, G., Garoni, C.: From convergence in measure to convergence of matrix-sequences through concave functions and singular values. Electron. J. Linear Algebra 32, 500–513 (2017)
Bhatia, R.: Matrix Analysis. Springer, New York (1997)
Böttcher, A., Garoni, C., Serra-Capizzano, S.: Exploration of Toeplitz-like matrices with unbounded symbols: not a purely academic journey. Sb. Math. 208(11), 29–55 (2017)
Brualdi, R.A.: Introductory Combinatorics. Pearson/Prentice Hall, Upper Saddle River (2010)
Garoni, C.: Topological foundations of an asymptotic approximation theory for sequences of matrices with increasing size. Linear Algebra Appl. 513, 324–341 (2017)
Garoni, C., Serra-Capizzano, S.: Generalized Locally Toeplitz Sequences: Theory and Applications, vol I. Springer, Cham (2017)
Prokhorov, Y.V.: Convergence of random processes and limit theorems in probability theory. Theory Probab. Appl. 1(2), 157–214 (1956)
Serra-Capizzano, S.: Distribution results on the algebra generated by Toeplitz sequences: a finite-dimensional approach. Linear Algebra Appl. 328, 121–130 (2001)
Tilli, P.: Locally Toeplitz sequences: spectral properties and applications. Linear Algebra Appl. 97, 91–120 (1998)
Tsfasman M.A., Vlăduţ S.G.: Asymptotic properties of zeta-functions. J. Math. Sci. 84(5), 1445–1467 (1997)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Barbarino, G. (2019). Spectral Measures. In: Bini, D., Di Benedetto, F., Tyrtyshnikov, E., Van Barel, M. (eds) Structured Matrices in Numerical Linear Algebra. Springer INdAM Series, vol 30. Springer, Cham. https://doi.org/10.1007/978-3-030-04088-8_1
Download citation
DOI: https://doi.org/10.1007/978-3-030-04088-8_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-04087-1
Online ISBN: 978-3-030-04088-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)