TY - JOUR

T1 - From convergence in measure to convergence of matrix-sequences through concave functions and singular values

AU - Barbarino, Giovanni

AU - Garoni, Carlo

N1 - Funding Information:
Acknowledgment. Carlo Garoni is a Marie-Curie fellow of the Italian INdAM (Istituto Nazionale di Alta Matematica) under grant agreement PCOFUND-GA-2012-600198.
Publisher Copyright:
© 2017, International Linear Algebra Society. All rights reserved.

PY - 2017

Y1 - 2017

N2 - Sequences of matrices with increasing size naturally arise in several areas of science, such as, for example, the numerical discretization of differential and integral equations. An approximation theory for sequences of this kind has recently been developed, with the aim of providing tools for computing their asymptotic singular value and eigenvalue distributions. The cornerstone of this theory is the notion of approximating classes of sequences (a.c.s.), which is also fundamental to the theory of generalized locally Toeplitz (GLT) sequences, and hence to the spectral analysis of PDE discretization matrices. Drawing inspiration from measure theory, here it is introduced a class of functions which are proved to be complete pseudometrics inducing the a.c.s. convergence. It is also shown that each of these pseudometrics gives rise to a natural isometry between the spaces of GLT sequences and measurable functions. Furthermore, it is highlighted that the a.c.s. convergence is an asymptotic matrix version of the convergence in measure, thus suggesting a way to obtain matrix theory results from measure theory results.

AB - Sequences of matrices with increasing size naturally arise in several areas of science, such as, for example, the numerical discretization of differential and integral equations. An approximation theory for sequences of this kind has recently been developed, with the aim of providing tools for computing their asymptotic singular value and eigenvalue distributions. The cornerstone of this theory is the notion of approximating classes of sequences (a.c.s.), which is also fundamental to the theory of generalized locally Toeplitz (GLT) sequences, and hence to the spectral analysis of PDE discretization matrices. Drawing inspiration from measure theory, here it is introduced a class of functions which are proved to be complete pseudometrics inducing the a.c.s. convergence. It is also shown that each of these pseudometrics gives rise to a natural isometry between the spaces of GLT sequences and measurable functions. Furthermore, it is highlighted that the a.c.s. convergence is an asymptotic matrix version of the convergence in measure, thus suggesting a way to obtain matrix theory results from measure theory results.

KW - Concave functions

KW - Convergence in measure

KW - Generalized locally Toeplitz sequences

KW - Matrix-sequences

KW - PDE discretizations

KW - Singular value and eigenvalue asymptotics

UR - http://www.scopus.com/inward/record.url?scp=85041598841&partnerID=8YFLogxK

U2 - 10.13001/1081-3810.3663

DO - 10.13001/1081-3810.3663

M3 - Article

AN - SCOPUS:85041598841

VL - 32

SP - 500

EP - 513

JO - ELECTRONIC JOURNAL OF LINEAR ALGEBRA

JF - ELECTRONIC JOURNAL OF LINEAR ALGEBRA

SN - 1537-9582

M1 - 37

ER -