Contents

 

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