Fourier Analysis is an important area of mathematics, especially in light of its importance in physics, chemistry, and engineering. Yet it seems that this subject is rarely offered to undergraduates. This book introduces Fourier Analysis in its three most classical settings: The Discrete Fourier Transform for periodic sequences, Fourier Series for periodic functions, and the Fourier Transform for functions on the real line. The presentation is accessible for students with just three or four terms of calculus, but the book is also intended to be suitable for a junior-senior course, for a capstone undergraduate course, or for beginning graduate students. Material needed from real analysis is quoted without proof, and issues of Lebesgue measure theory are treated rather informally. Included are a number of applications of Fourier Series, and Fourier Analysis in higher dimensions is briefly sketched. A student may eventually want to move on to Fourier Analysis discussed in a more advanced way, either by way of more general orthogonal systems, or in the language of Banach spaces, or of locally compact commutative groups, but the experience of the classical setting provides a mental image of what is going on in an abstract setting.
Author(s): Hugh L. Montgomery
Series: Pure and Applied Undergraduate Texts 22
Publisher: American Mathematical Society
Year: 2014
Language: English
Pages: 400
Tags: Математика;Математический анализ;
Contents
Preface ix
Chapter 0. Background 1
0.1. Elementary mathematics 1
0.2. Real analysis 3
0.3. Lebesgue measure theory 7
Chapter 1. Complex Numbers 9
1.1. Basics 9
1.2. Euclidean geometry via complex numbers 13
1.3. Polynomials 17
1.4. Power series 21
Notes 31
Chapter 2. The Discrete Fourier Transform 33
2.1. Sums of roots of unity 33
2.2. The Transform 36
2.3. The Fast Fourier Transform 48
Notes 51
Chapter 3. Fourier Coefficients and First Fourier Series 53
3.1. Definitions and basic properties 53
3.2. Other periods 68
3.3. Convolution 69
3.4. First Convergence Theorems 75
Notes 88
Chapter 4. Summability of Fourier Series 91
4.1. Cesaro summability of Fourier Series 91
4.2. Special coefficients 111
4.3. Summability 120
4.4. Summability kernels 130
Notes 134
Chapter 5. Fourier Series in Mean Square 135
5.1. Vector spaces of functions 135
5.2. Parseval's Identity 138
Notes 148
Chapter 6. Trigonometric Polynomials 149
6.1. Sampling and interpolation 149
6.2. Bernstein's Inequality 158
6.3. Real-valued and nonnegative trigonometric polynomials 162
6.4. Littlewood polynomials 165
6.5. Quantitative approximation of continuous functions 175
Notes 182
Chapter 7. Absolutely Convergent Fourier Series 183
7.1. Convergence 183
7.2. Wiener's theorem 191
Notes 194
Chapter 8. Convergence of Fourier Series 195
8.1. Conditions ensuring convergence 195
8.2. Functions of bounded variation 198
8.3. Examples of divergence 205
Notes 209
Chapter 9. Applications of Fourier Series 211
9.1. The heat equation 211
9.2. The wave equation 213
9.3. Continuous, nowhere differentiable functions 215
9.4. Inequalities 217
9.5. Bernoulli polynomials 220
9.6. Uniform distribution 229
9.7. Positive definite kernels 239
9.8. Norms of polynomials 241
Notes 246
Chapter 10. The Fourier Transform 249
10.1. Definition and basic properties 249
10.2. The inversion formula 255
10.3. Fourier transforms in mean square 263
10.4. The Poisson summation formula 270
10.5. Linear combinations of translates 277
Notes 278
Chapter 11. Higher Dimensions 279
11.1. Multiple Discrete Fourier Transforms 279
11.2. Multiple Fourier Series 280
11.3. Multiple Fourier Transforms 286
Notes 290
Appendix B. The Binomial Theorem 291
B.1. Binomial coefficients 291
B.2. Binomial theorems 293
Appendix C. Chebyshev Polynomials 299
Appendix F. Applications of the Fundamental Theorem of Algebra 309
F.1. Zeros of the derivative of a polynomial 309
F.2. Linear differential equations with constant coefficients 312
F.3. Partial fraction expansions 313
F.4. Linear recurrences 315
Appendix I. Inequalities 319
I.1. The Arithmetic-Geometric Mean Inequality 319
I.2. Hölder's Inequality 325
Notes 338
Appendix L. Topics in Linear Algebra 339
L.1. Familiar vector spaces 339
L.2. Abstract vector spaces 344
L.3. Circulant matrices 347
Notes 348
Appendix O. Orders of Magnitude 349
Appendix T. Trigonometry 351
T.1. Trigonometric functions in plane geometry 351
T.2. Trigonometric functions in calculus 357
T.3. Inverse trigonometric functions 364
T.4. Hyperbolic functions 369
References 377
Notation 383
Index 385
