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The subject of this text is the relation between the properties of infinite Toeplitz matrices ao a_I a_2 al ao a_I a2 al ao and their large finite sections This is very big and even inexhaustible subject, and therefore we must limit ourselves to a few concrete problems here. We will focus our attention on singular values. The singular values of An are the eigenvalues of (A~An)I/2. The properties of the singular values of An for fixed n (or, as in so-called interlacing theorems, for some consecutive n) are studied in linear algebra. The problem of determining the singular values of An for large n (say n = 700) is a business of numerical linear algebra. The behavior of the singular 23 values of An for n --+ 00 (or, say, for n = 10 ) is a concern of asymptotic linear algebra. Finally, the investigation of the properties of the infinite matrix A is a task of functional analysis. To get an idea of what this text is about, we cite a few questions we will consider. Preface viii Question 1. Does the smallest singular value 81 (An) stay away from zero as n -t oo? Because this is the question whether the norms IIA;111 are uniformly bounded for all sufficiently large n.