Approximating Classes of Sequences

Abstract

In the last 20 years, an asymptotic approximation theory for matrix-sequences has been developed. The aim was to obtain sufficiently powerful tools for computing the asymptotic singular value and eigenvalue distribution of a ‘difficult’ matrix-sequence \(\{A_n\}_n\) from the asymptotic singular value and eigenvalue distributions of ‘simpler’ matrix-sequences \(\{B_{n, m}\}_n\) that ‘converge’ to \(\{A_n\}_n\) in a suitable way as \(m\rightarrow \infty \). The cornerstone of all this approximation theory is the notion of approximating classes of sequences (a.c.s.), which is due to the second author [104], but was actually inspired by Tilli’s pioneering paper on LT sequences [120]. In this chapter, we study the theory of a.c.s., which is absolutely fundamental to the theory of GLT sequences. Throughout this book, we use the abbreviation ‘a.c.s.’ for both the singular ‘approximating class of sequences’ and the plural ‘approximating classes of sequences’; it will be clear from the context whether ‘a.c.s.’ is singular or plural.