Mathematical Analysis: Approximation and Discrete Processes - Hardcover

Giaquinta, Mariano; Modica, Giuseppe

Hardcover

ISBN 10: 0817643133 ISBN 13: 9780817643133

Publisher: Birkhäuser, 2004

Synopsis

This volume! aims at introducing some basic ideas for studying approxima­ tion processes and, more generally, discrete processes. The study of discrete processes, which has grown together with the study of infinitesimal calcu­ lus, has become more and more relevant with the use of computers. The volume is suitably divided in two parts. In the first part we illustrate the numerical systems of reals, of integers as a subset of the reals, and of complex numbers. In this context we intro­ duce, in Chapter 2, the notion of sequence which invites also a rethinking of the notions of limit and continuity2 in terms of discrete processes; then, in Chapter 3, we discuss some elements of combinatorial calculus and the mathematical notion of infinity. In Chapter 4 we introduce complex num­ bers and illustrate some of their applications to elementary geometry; in Chapter 5 we prove the fundamental theorem of algebra and present some of the elementary properties of polynomials and rational functions, and of finite sums of harmonic motions. In the second part we deal with discrete processes, first with the process of infinite summation, in the numerical case, i.e., in the case of numerical series in Chapter 6, and in the case of power series in Chapter 7. The last chapter provides an introduction to discrete dynamical systems; it should be regarded as an invitation to further study.

About this title

Review

"This self-contained book aims to introduce the main ideas for studying approximation processes, more generally discrete processes at graduate level. The use of computers induces a growing need for studying discrete processes.... A key feature this lively yet rigorous and systematic treatment is the historical accounts of ideas and methods of the subject. Ideas in mathematics develop in cultural, historical and economical contexts, thus the authors made brief accounts of those aspects and used a large number of beautiful illustrations.... Each chapter has a short summary where the most important facts discussed are collected and described. There is also a large number of exercises inserted at various points into the text....The book is meant principally for graduate students in mathematics, physics, engineering, and computer science, but it can be used at technological and scientific faculties by anyone who wants to approach these topics. It may also be used in graduate seminars and courses or as a reference text by mathematicians, physicists, and engineers."   ―Zentralblatt MATH

"Mathematical Analysis does contain a substantial amount of material that is unusual in terms of an introductory text in real analysis...These are all interesting topics that have gained increasing importance in modern applications of mathematics, albeit outside the traditional area of analysis. It is very nice to have these topics developed outside a specialized textbook, in, e.g., combinatorics, dynamical systems, or number theory. The authors do a very good job presenting this material...Mathematical Analysis includes substantial amounts of historical background...The book also contains a lot of examples...I can happily recommend Mathematical Analysis as a good resource for instructors of introductory analysis courses, especially in terms of providing some unusual applications of analysis and developments of some basic classic topics that are often shortchanged in standard texts."  ―SIAM Review

“This is the second volume of a series on analysis. ... a real hodgepodge that could only be the primary textbook for a course specifically based on it. ... In this series Giaquinta and Modica have set themselves the formidable task of constructing from scratch an analysis sequence of several years length. ... they have more regard for classical topics and arguments than most authors writing analysis books today. ... I enjoyed reading this volume ... .” (Warren Johnson, The Mathematical Association of America, January, 2010)
 

From the Back Cover

This self-contained work on linear and metric structures focuses on studying continuity and its applications to finite- and infinite-dimensional spaces.

The book is divided into three parts. The first part introduces the basic ideas of linear and metric spaces, including the Jordan canonical form of matrices and the spectral theorem for self-adjoint and normal operators. The second part examines the role of general topology in the context of metric spaces and includes the notions of homotopy and degree. The third and final part is a discussion on Banach spaces of continuous functions, Hilbert spaces and the spectral theory of compact operators.

Mathematical Analysis: Linear and Metric Structures and Continuity motivates the study of linear and metric structures with examples, observations, exercises, and illustrations. It may be used in the classroom setting or for self-study by advanced undergraduate and graduate students and as a valuable reference for researchers in mathematics, physics, and engineering.

Other books recently published by the authors include: Mathematical Analysis: Functions of One Variable, and Mathematical Analysis: Approximation and Discrete Processes. This book builds upon the discussion in these books to provide the reader with a strong foundation in modern-day analysis.

Edition details

• Publisher: Birkhäuser
• Publication date: 2004
• Language: English
• ISBN 10: 0817643133
• ISBN 13: 9780817643133
• Binding: Hardcover
• Number of pages: 400