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From the Preface (1955): The first part of the present exposition is devoted to the theory of Toeplitz forms. The second part deals with applications, in particular to the calculus of probability and mathematical statistics. Neither part claims completeness in any way. Our purpose has been to elucidate the principal ideas of this remarkable chapter of modern analysis and to help the interested student of mathematical statistics to acquire a working knowledge of the subject. The somewhat protracted Chapter 1 explains not only the notation employed but contains also the definition of important auxiliary concepts and the exposition of basic results which will be used later. This arrangement avoids interruptions in the main text ... [It is assumed] that the reader is in possession of the fundamental facts of the theory of functions. ``In chapters 2 and 3 certain topics appear which were treated in the book on orthogonal polynomials by G. Szego. In view of the progress made in this subject since the publication of that book (1939) it was possible to bring some details in an improved setting. The other chapters contain partly old and partly more recent results, some older facts in a new setting, and finally some completely new results. Chapter 1-6 and Chapter 9 have been prepared by Szego, the other chapters by Grenander ... ''

From the Preface (1955): The first part of the present exposition is devoted to the theory of Toeplitz forms. The second part deals with applications, in particular to the calculus of probability and mathematical statistics. Neither part claims completeness in any way. Our purpose has been to elucidate the principal ideas of this remarkable chapter of modern analysis and to help the interested student of mathematical statistics to acquire a working knowledge of the subject. The somewhat protracted Chapter 1 explains not only the notation employed but contains also the definition of important auxiliary concepts and the exposition of basic results which will be used later. This arrangement avoids interruptions in the main text ... [It is assumed] that the reader is in possession of the fundamental facts of the theory of functions. ``In chapters 2 and 3 certain topics appear which were treated in the book on orthogonal polynomials by G. Szego. In view of the progress made in this subject since the publication of that book (1939) it was possible to bring some details in an improved setting. The other chapters contain partly old and partly more recent results, some older facts in a new setting, and finally some completely new results. Chapter 1-6 and Chapter 9 have been prepared by Szego, the other chapters by Grenander ... ''

Applying functional analysis and operator theory to some concrete asymptotic problems of linear algebra, this book contains results on the stability of projection methods, deals with asymptotic inverses and Moore-Penrose inversion of large Toeplitz matrices, and embarks on the asymptotic behaviour of the norms of inverses, the pseudospectra, the singular values, and the eigenvalues of large Toeplitz matrices. The approach is heavily based on Banach algebra techniques and nicely demonstrates the usefulness of C*-algebras and local principles in numerical analysis, including classical topics as well as results and methods from the last few years. Though employing modern tools, the exposition is elementary and points out the mathematical background behind some interesting phenomena encountered with large Toeplitz matrices. Accessible to readers with basic knowledge in functional analysis, the book addresses graduates, teachers, and researchers and should be of interest to everyone who has to deal with infinite matrices (Toeplitz or not) and their large truncations.

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When first published posthumously in 1963, this bookpresented a radically different approach to the teaching of calculus.  In sharp contrast to the methods of his time, Otto Toeplitz did not teach calculus as a static system of techniques and facts to be memorized. Instead, he drew on his knowledge of the history of mathematics and presented calculus as an organic evolution of ideas beginning with the discoveries of Greek scholars, such as Archimedes, Pythagoras, and Euclid, and developing through the centuries in the work of Kepler, Galileo, Fermat, Newton, and Leibniz. Through this unique approach, Toeplitz summarized and elucidated the major mathematical advances that contributed to modern calculus.Reissued for the first time since 1981 and updated with a new foreword, this classic text in the field of mathematics is experiencing a resurgence of interest among students and educators of calculus today.

When first published posthumously in 1963, this bookpresented a radically different approach to the teaching of calculus. In sharp contrast to the methods of his time, Otto Toeplitz did not teach calculus as a static system of techniques and facts to be memorized. Instead, he drew on his knowledge of the history of mathematics and presented calculus as an organic evolution of ideas beginning with the discoveries of Greek scholars, such as Archimedes, Pythagoras, and Euclid, and developing through the centuries in the work of Kepler, Galileo, Fermat, Newton, and Leibniz. Through this unique approach, Toeplitz summarized and elucidated the major mathematical advances that contributed to modern calculus. Reissued for the first time since 1981 and updated with a new foreword, this classic text in the field of mathematics is experiencing a resurgence of interest among students and educators of calculus today.

When first published posthumously in 1963, this bookpresented a radically different approach to the teaching of calculus. In sharp contrast to the methods of his time, Otto Toeplitz did not teach calculus as a static system of techniques and facts to be memorized. Instead, he drew on his knowledge of the history of mathematics and presented calculus as an organic evolution of ideas beginning with the discoveries of Greek scholars, such as Archimedes, Pythagoras, and Euclid, and developing through the centuries in the work of Kepler, Galileo, Fermat, Newton, and Leibniz. Through this unique approach, Toeplitz summarized and elucidated the major mathematical advances that contributed to modern calculus. Reissued for the first time since 1981 and updated with a new foreword, this classic text in the field of mathematics is experiencing a resurgence of interest among students and educators of calculus today.