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 v
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