目录
 前辅文
 Preface
 Chapter 0. Background
  0.1. Elementary mathematics
  0.2. Real analysis
  0.3. Lebesgue measure theory
 Chapter 1. Complex Numbers
  1.1. Basics
  1.2. Euclidean geometry via complex numbers
  1.3. Polynomials
  1.4. Power series
  Notes
 Chapter 2. The Discrete Fourier Transform
  2.1. Sums of roots of unity
  2.2. The Transform
  2.3. The Fast Fourier Transform
  Notes
 Chapter 3. Fourier Coefficients and First Fourier Series
  3.1. Definitions and basic properties
  3.2. Other periods
  3.3. Convolution
  3.4. First Convergence Theorems
  Notes
 Chapter 4. Summability of Fourier Series
  4.1. Cesàro summability of Fourier Series
  4.2. Special coefficients
  4.3. Summability
  4.4. Summability kernels
  Notes
 Chapter 5. Fourier Series in Mean Square
  5.1. Vector spaces of functions
  5.2. Parseval’s Identity
  Notes
 Chapter 6. Trigonometric Polynomials
  6.1. Sampling and interpolation
  6.2. Bernstein’s Inequality
  6.3. Real-valued and nonnegative trigonometric polynomials
  6.4. Littlewood polynomials
  6.5. Quantitative approximation of continuous functions
  Notes
 Chapter 7. Absolutely Convergent Fourier Series
  7.1. Convergence
  7.2. Wiener’s theorem
  Notes
 Chapter 8. Convergence of Fourier Series
  8.1. Conditions ensuring convergence
  8.2. Functions of bounded variation
  8.3. Examples of divergence
  Notes
 Chapter 9. Applications of Fourier Series
  9.1. The heat equation
  9.2. The wave equation
  9.3. Continuous, nowhere differentiable functions
  9.4. Inequalities
  9.5. Bernoulli polynomials
  9.6. Uniform distribution
  9.7. Positive definite kernels
  9.8. Norms of polynomials
  Notes
 Chapter 10. The Fourier Transform
  10.1. Definition and basic properties
  10.2. The inversion formula
  10.3. Fourier transforms in mean square
  10.4. The Poisson summation formula
  10.5. Linear combinations of translates
  Notes
 Chapter 11. Higher Dimensions
  11.1. Multiple Discrete Fourier Transforms
  11.2. Multiple Fourier Series
  11.3. Multiple Fourier Transforms
  Notes
 Appendix B. The Binomial Theorem
  B.1. Binomial coefficients
  B.2. Binomial theorems
 Appendix C. Chebyshev Polynomials
 Appendix F. Applications of the Fundamental Theorem of Algebra
  F.1. Zeros of the derivative of a polynomial
  F.2. Linear differential equations with constant coefficients
  F.3. Partial fraction expansions
  F.4. Linear recurrences
 Appendix I. Inequalities
  I.1. The Arithmetic–Geometric Mean Inequality
  I.2. Hölder’s Inequality
  Notes
 Appendix L. Topics in Linear Algebra
  L.1. Familiar vector spaces
  L.2. Abstract vector spaces
  L.3. Circulant matrices
  Notes
 Appendix O. Orders of Magnitude
 Appendix T. Trigonometry
  T.1. Trigonometric functions in plane geometry
  T.2. Trigonometric functions in calculus
  T.3. Inverse trigonometric functions
  T.4. Hyperbolic functions
 References
 Notation
 Index
 前辅文
 Preface
 Chapter 0. Background
  0.1. Elementary mathematics
  0.2. Real analysis
  0.3. Lebesgue measure theory
 Chapter 1. Complex Numbers
  1.1. Basics
  1.2. Euclidean geometry via complex numbers
  1.3. Polynomials
  1.4. Power series
  Notes
 Chapter 2. The Discrete Fourier Transform
  2.1. Sums of roots of unity
  2.2. The Transform
  2.3. The Fast Fourier Transform
  Notes
 Chapter 3. Fourier Coefficients and First Fourier Series
  3.1. Definitions and basic properties
  3.2. Other periods
  3.3. Convolution
  3.4. First Convergence Theorems
  Notes
 Chapter 4. Summability of Fourier Series
  4.1. Cesàro summability of Fourier Series
  4.2. Special coefficients
  4.3. Summability
  4.4. Summability kernels
  Notes
 Chapter 5. Fourier Series in Mean Square
  5.1. Vector spaces of functions
  5.2. Parseval’s Identity
  Notes
 Chapter 6. Trigonometric Polynomials
  6.1. Sampling and interpolation
  6.2. Bernstein’s Inequality
  6.3. Real-valued and nonnegative trigonometric polynomials
  6.4. Littlewood polynomials
  6.5. Quantitative approximation of continuous functions
  Notes
 Chapter 7. Absolutely Convergent Fourier Series
  7.1. Convergence
  7.2. Wiener’s theorem
  Notes
 Chapter 8. Convergence of Fourier Series
  8.1. Conditions ensuring convergence
  8.2. Functions of bounded variation
  8.3. Examples of divergence
  Notes
 Chapter 9. Applications of Fourier Series
  9.1. The heat equation
  9.2. The wave equation
  9.3. Continuous, nowhere differentiable functions
  9.4. Inequalities
  9.5. Bernoulli polynomials
  9.6. Uniform distribution
  9.7. Positive definite kernels
  9.8. Norms of polynomials
  Notes
 Chapter 10. The Fourier Transform
  10.1. Definition and basic properties
  10.2. The inversion formula
  10.3. Fourier transforms in mean square
  10.4. The Poisson summation formula
  10.5. Linear combinations of translates
  Notes
 Chapter 11. Higher Dimensions
  11.1. Multiple Discrete Fourier Transforms
  11.2. Multiple Fourier Series
  11.3. Multiple Fourier Transforms
  Notes
 Appendix B. The Binomial Theorem
  B.1. Binomial coefficients
  B.2. Binomial theorems
 Appendix C. Chebyshev Polynomials
 Appendix F. Applications of the Fundamental Theorem of Algebra
  F.1. Zeros of the derivative of a polynomial
  F.2. Linear differential equations with constant coefficients
  F.3. Partial fraction expansions
  F.4. Linear recurrences
 Appendix I. Inequalities
  I.1. The Arithmetic–Geometric Mean Inequality
  I.2. Hölder’s Inequality
  Notes
 Appendix L. Topics in Linear Algebra
  L.1. Familiar vector spaces
  L.2. Abstract vector spaces
  L.3. Circulant matrices
  Notes
 Appendix O. Orders of Magnitude
 Appendix T. Trigonometry
  T.1. Trigonometric functions in plane geometry
  T.2. Trigonometric functions in calculus
  T.3. Inverse trigonometric functions
  T.4. Hyperbolic functions
 References
 Notation
 Index