Preface | ix | |
0 | Background | 1 |
0.1 | Elementary mathematics | 1 |
0.2 | Real analysis | 3 |
0.3 | Lebesgue measure theory | 7 |
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 | |
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 | |
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 | |
4 | Summability of Fourier Series | 91 |
4.1 | Cesàro summability of Fourier Series | 91 |
4.2 | Special coefficients | 111 |
4.3 | Summability | 120 |
4.4 | Summability kernels | 130 |
Notes | 134 | |
5 | Fourier Series in Mean Square | 135 |
5.1 | Vector spaces of functions | 135 |
5.2 | Parseval's Identity | 138 |
Notes | 148 | |
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 | |
7 | Absolutely Convergent Fourier Series | 183 |
7.1 | Convergence | 183 |
7.2 | Wiener's theorem | 191 |
Notes | 194 | |
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 | |
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 | |
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 | |
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 | |
Appendices | ||
B | The Binomial Theorem | 291 |
B.1 | Binomial coefficients | 291 |
B.2 | Binomial theorems | 293 |
C | Chebyshev polynomials | 299 |
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 |
I | Inequalities | 319 |
I.1 | The Arithmetic-Geometric Mean Inequality | 319 |
I.2 | Hölder's Inequality | 325 |
Notes | 338 | |
293 | ||
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 | |
O | Orders of Magnitude | 349 |
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 | |