Real Analysis and Applications: Theory in Practice

Kenneth R. Davidson, Allan P. Donsig (auth.)0387980970, 978-0-387-98097-3, 978-0-387-98098-0

This new approach to real analysis stresses the use of the subject in applications, showing how the principles and theory of real analysis can be applied in various settings. Applications cover approximation by polynomials, discrete dynamical systems, differential equations, Fourier series and physics, Fourier series and approximation, wavelets, and convexity and optimization. Each chapter has many useful exercises.

The treatment of the basic theory covers the real numbers, functions, and calculus, while emphasizing the role of normed vector spaces, and particularly of Rn. The applied chapters are mostly independent, giving the reader a choice of topics. This book is appropriate for students with a prior knowledge of both calculus and linear algebra who want a careful development of both analysis and its use in applications.

Review of the previous version of this book, Real Analysis with Real Applications:

“A well balanced book! The first solid analysis course, with proofs, is central in the offerings of any math.-dept.; —and yet, the new books that hit the market don’t always hit the mark: the balance between theory and applications, —between technical proofs and intuitive ideas, —between classical and modern subjects, and between real life exercises vs. the ones that drill a new concept. The Davidson-Donsig book is outstanding, and it does hit the mark.”

Palle E. T. Jorgenson, Review from Amazon.com

Kenneth R. Davidson is University Professor of Mathematics at the University of Waterloo. Allan P. Donsig is Associate Professor of Mathematics at the University of Nebraska-Lincoln.

Table of contents :
Front Matter….Pages 1-10
Front Matter….Pages 1-1
Review….Pages 3-8
The Real Numbers….Pages 9-34
Series….Pages 35-47
Topology of $${mathbb{R}^n}$$ ….Pages 48-66
Functions….Pages 67-93
Differentiation and Integration….Pages 94-112
Norms and Inner Products….Pages 113-141
Limits of Functions….Pages 142-174
Metric Spaces….Pages 175-186
Front Matter….Pages 188-188
Approximation by Polynomials….Pages 189-239
Discrete Dynamical Systems….Pages 240-292
Differential Equations….Pages 293-327
Fourier Series and Physics….Pages 328-359
Fourier Series and Approximation….Pages 360-405
Wavelets….Pages 406-448
Convexity and Optimization….Pages 449-504
Back Matter….Pages 1-9