Real Analysis and Applications

by Davidson, Kenneth R.; Donsig, Allan P.

• ISBN13: 9780387980973
• ISBN10: 0387980970
• Format: Hardcover
• Copyright: 2010-01-30
• Publisher: Springer Verlag

Summary

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.

Author Biography

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

Analysis
Review	p. 3
Calculus	p. 3
Linear Algebra	p. 5
Appendix: Equivalence Relations	p. 7
The Real Numbers	p. 9
An Overview of the Real Numbers	p. 9
The Real Numbers and Their Arithmetic	p. 10
The Least Upper Bound Principle	p. 13
Limits	p. 15
Basic Properties of Limits	p. 19
Monotone Sequences	p. 20
Subsequences	p. 23
Cauchy Sequences	p. 27
Countable Sets	p. 31
Series	p. 35
Convergent Series	p. 35
Convergence Tests for Series	p. 39
Absolute and Conditional Convergence	p. 44
Topology of Rn	p. 48
n-Dimensional Space	p. 48
Convergence and Completeness in Rn	p. 52
Closed and Open Subsets of Rn	p. 56
Compact Sets and the Heine-Borel Theorem	p. 61
Functions	p. 67
Limits and Continuity	p. 67
Discontinuous Functions	p. 72
Properties of Continuous Functions	p. 77
Compactness and Extreme Values	p. 80
Uniform Continuity	p. 82
The Intermediate Value Theorem	p. 88
Monotone Functions	p. 90
Differentiation and Integration	p. 94
Different)able Functions	p. 94
The Mean Value Theorem	p. 99
Riemann Integration	p. 103
The Fundamental Theorem of Calculus	p. 109
Norms and Inner Products	p. 113
Normed Vector Spaces	p. 113
Topology in Normed Spaces	p. 117
Finite-Dimensional Normed Spaces	p. 120
Inner Product Spaces	p. 124
Finite Orthonormal Sets	p. 128
Fourier Series	p. 132
Orthogonal Expansions and Hilbert Spaces	p. 136
Limits orFunctions	p. 142
Limits of Functions	p. 142
Uniform Convergence and Continuity	p. 147
Uniform Convergence and Integration	p. 150
Series of Functions	p. 154
Power Series	p. 161
Compactness and Subsets of C(K)	p. 168
Metric Spaces	p. 175
Definitions and Examples	p. 175
Compact Metric Spaces	p. 180
Complete Metric Spaces	p. 183
Applications
Approximation by Polynomials	p. 189
Taylor Series	p. 189
How Not to Approximate a Function	p. 198
Bernstein's Proof of the Weierstrass Theorem	p. 201
Accuracy of Approximation	p. 204
Existence of Best Approximations	p. 207
Characterizing Best Approximations	p. 211
Expansions Using Chebyshev Polynomials	p. 217
Splines	p. 223
Uniform Approximation by Splines	p. 231
The Stone-Weierstrass Theorem	p. 235
Discrete Dynamical Systems	p. 240
Fixed Points and the Contraction Principle	p. 241
Newton's Method	p. 252
Orbits of a Dynamical System	p. 257
Periodic Points	p. 262
Chaotic Systems	p. 269
Topological Conjugacy	p. 277
Iterated Function Systems	p. 285
Differential Equations	p. 293
Integral Equations and Contractions	p. 293
Calculus of Vector-Valued Functions	p. 297
Differential Equations and Fixed Points	p. 300
Solutions of Differential Equations	p. 304
Local Solutions	p. 309
Linear Differential Equations	p. 316
Perturbation and Stability of DEs	p. 320
Existence Without Uniqueness	p. 324
Fourier Series and Physics	p. 328
The Steady-State Heat Equation	p. 328
Formal Solution	p. 332
Convergence ih the Open Disk	p. 334
The Poisson Formula	p. 337
Poisson's Theorem	p. 341
The Maximum Principle	p. 345
The Vibrating String (Formal Solution)	p. 347
The Vibrating String (Rigorous Solution)	p. 353
Appendix: The Complex Exponential	p. 356
Fourier Series and Approximation	p. 360
The Riemann-Lebesgue Lemma	p. 360
Pointwise Convergence of Fourier Series	p. 364
Gibbs's Phenomenon	p. 372
Cesaro Summation of Fourier Series	p. 376
Least Squares Approximations	p. 383
The Isoperimetric Problem	p. 387
Best Approximation by Trigonometric Polynomials	p. 390
Connections with Polynomial Approximation	p. 393
Jackson's Theorem and Bernstein's Theorem	p. 397
Wavelets	p. 406
Introduction	p. 406
The Haar Wavelet	p. 408
Multiresolution Analysis	p. 412
Recovering the Wavelet	p. 416
Daubechies Wavelets	p. 420
Existence of the Daubechies Wavelet	p. 426
Approximations Using Wavelets	p. 429
The Franklin Wavelet	p. 433
Riesz Multiresolution Analysis	p. 440
Convexity and Optimization	p. 449
Convex Sets	p. 449
Relative Interior	p. 455
Separation Theorems	p. 460
Extreme Points	p. 464
Convex Functions in One Dimension	p. 467
Convex Functions in Higher Dimensions	p. 473
Subdifferentials and Directional Derivatives	p. 477
Tangent and Normal Cones	p. 487
Constrained Minimization	p. 491
The Minimax Theorem	p. 498
References	p. 505
Index	p. 507
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