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The first volume in this subseries of the Encyclopaedia is meant to familiarize the reader with the discipline Commutative Harmonic Analysis.
The first article is a thorough introduction, moving from Fourier series to the Fourier transform, and on to the group theoretic point of view. Numerous examples illustrate the connections to differential and integral equations, approximation theory, number theory,probability theory and physics. The development of Fourier analysis is discussed in a brief historical essay.
The second article focuses on some of the classical problems of Fourier series; it\\\\\\\'s a\\\\\\\"mini-Zygmund\\\\\\\"for the beginner. The third article is the most modern of the three, concentrating on singular integral operators. It also contains an introduction to Calderón-Zygmund theory.

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• I. Methods and Structure of Commutative Harmonic Analysis
  Introduction
  Chapter 1 A Short Course of Fourier Analysis of Periodic Functions
  §1. Translation-Invariant Operators
  1.1. The Set up
  1.2. Object of Investigation
  1.3. Convolution
  1.4. General Form of t.i. Operators
  §2. Harmonics. Basic Principles of Harmonic Analysis on the Circle
  2.1. Eigenvectors and Eigenfunctions of t.i. Operators
  2.2. Basic Principles of Harmonic Analysis on the Circle T
  2.3. Smoothing of Distributions
  2.4. Weierstrass' Theorem
  2.5. Fourier Coefficients. The Main Theorem of Harmonic Analysis on the Circle
  2.6. Spectral Characteristics of the Classes * and *'
  2.7. L²-Theory of Fourier Series
  2.8. Wirtinger's Inequality
  2.9. The Isoperimetric Inequality. (Hurwitz' Proof)
  2.10. Harmonic Analysis on the Torus
  Chapter 2 Harmonic Analysis in R^d
  §1. Preliminaries on Distributions in R^d
  1.1. Distributions in R^d
  §2. From the Circle to the Line. Fourier Transform in Rd (Definition)
  2.1. Inversion Formula (An Euristic Derivation)
  2.2. A Proof of the Inversion Formula
  2.3. Another Proof
  2.4. Fourier Transform in Rd (Definition)
  §3. Convolution (Definition)
  3.1. Difficulties of Harmonic Analysis in R^d
  3.2. Convolution of Distributions (Construction)
  3.3. Examples
  3.4. Convolution Operators
  §4. Convolution Operators as Object of Study (Examples)
  4.1. Linear Differential and Difference Operators
  4.2. Integral Operators with a Kernel Depending on Difference of Arguments
  4.3. Integration and Differentiation of a Fractional Order
  4.4. Hilbert Transform
  4.5. Cauchy's Problem and Convolution Operators
  4.6. Fundamental Solutions. The Newtonian Potential
  4.7. Distribution of the Sum of Independent Random Variables
  4.8. Convolution Operators in Approximation Theory
  4.9. The Impulse Response Function of a System
  §5. Means of Investigation-Fourier Transform (S-Theory and L²-Theory)
  5.1. Spaces S and S'
  5.2. S'-Theory of Fourier Transform. Preliminary Discussion
  5.3. S'-Theory of Fourier Transform (Basic Facts)
  5.4. L²-Theory
  5.5. "x-Representation" and "ξ-Representation"
  §6. Fourier Transform in Examples
  6.1. Some Formulae
  6.2. Fourier Transform and a Linear Change of Variable
  6.3. Digression: Heisenberg Uncertainty Principle
  6.4. Radially-Symmetric Distributions
  6.5. Harmonic Analysis of Periodic Functions
  6.6. The Poisson Summation Formula
  6.7. Minkowski's Theorem on Integral Solutions of Systems of Linear Inequalities
  6.8. Jacobi's Identity for the &Function
  6.9. Evaluation of the Gaussian Sum
  §7. Fourier Transform in Action. Spectral Analysis of Convolution Operators
  7.1. Symbol
  7.2. Construction of Fundamental Solutions
  7.3. Hypoellipticity
  7.4. Singular Integral Operators and PDO
  7.5. The Law of Large Numbers and Central Limit Theorem
  7.6. δ-Families and Summation of Diverging Integrals
  7.7. Tauberian Theorems
  7.8. Spectral Characteristic of a System
  7.9. More on Summation Methods
  §8. Additional Remarks
  8.1. Fourier Transform in Ultra-Distributions
  8.2. Certain Generalizations of the L²-Theory
  8.3. Radon Transform
  Chapter 3 Harmonic Analysis on Groups
  §1. An Outline of Harmonic Analysis on a Compact Group
  1.1. A New Set Up
  1.2. Harmonics
  1.3. Representations
  1.4. The Peter-H. Weyl Theorem
  §2. Commutative Harmonic Analysis
  2.1. Simplifications Implied by Commutativity
  2.2. Fourier Transform of Measures and Summable Functions
  2.3. Convolution
  2.4. Uniqueness Theorem. The Inversion Formula
  2.5. Classical Harmonic Analysis from a General Point of View
  2.6. Fast Multiplication of Large Numbers
  2.7. Plancherel's Theorem.
  2.8. The Theorem of Bochner and A. Weil
  §3. Examples
  3.1. Pontryagin's Duality Theorem
  3.2. Almost Periodic Functions
  3.3. Quadratic Reciprocity Law
  §4. Unitary Representations of the Group R
  4.1. Stone's Theorem
  4.2. Infinitesimal Generator
  4.3. Examples
  Chapter 4 A Historical Survey
  Chapter 5 Spectral Analysis and Spectral Synthesis. Intrinsic Problems of Harmonic Analysis
  §1. Harmonic Analysis "For Itself"
  §2. Spectral Analysis
  2.1. Linear Combination of Exponentials
  2.2. Generalizations
  2.3. Spectrum
  §3. Spectral Synthesis
  3.1. Methods of Synthesis
  3.2. Spectral Analysis-Synthesis of t.i. Operators According to L. Schwartz
  3.3. Continuation, Periodicity in the Mean and Stability
  3.4. Problems of Translation
  3.5. Exceptional Sets
  §4. Translation-Invariant Operators. Singular Integrals，Multipliers
  §5. Complex-Analytic Methods
  Epilogue
  Bibliographical Notes
  References
  II. Classical Themes of Fourier Analysis
  Introduction
  Chapter 1 Fourier Series: Convergence and Summability
  §1. Convergence at a Point
  1.1. The Trigonometric System
  1.2. The Riemann-Lebesgue Theorem
  1.3. Uniqueness Theorem. Fourier Series
  1.4. The Dirichlet Kernel
  1.5. Convolutions
  1.6. The Lebesgue Constants. Theorem of du Bois-Reymond
  1.7. The Littlewood Conjecture
  1.8. The Localization Principle
  1.9. The Dini Test
  1.10. The Jordan Test
  1.11. Other Tests
  1.12. Uniform Convergence
  §2. Summation of Fourier Series
  2.1. Convolution Summation Methods
  2.2. Approximate Identity
  2.3. Convergence in L^P
  2.4. Classes of Functions and Convolutions with an Approximate Identity
  2.5. The FejCr Kernels
  2.6. The Poisson Kernel
  2.7. The Hardy-Littlewood Maximal Function and its Estimates
  2.8. Estimates of Convolutions in Terms of the Maximal Function
  2.9. Summability Almost Everywhere
  Chapter 2 The Harmonic Conjugation Operator
  §1. The Definition. HP-Spaces
  1.1. Preliminary Remarks
  1.2. Fourier Series and the Class L²
  1.3. The Riesz Projection and the Harmonic Conjugation Operator
  1.4. The Spaces h^P and H^P
  1.5. Theorem of the Brothirs Riesz
  §2. Continuity of the Harmonic Conjugation Operator. Singular Integrals
  2.1. The Weak Type (1，1) Inequality
  2.2. Continuity in L

  ，1 2.3. Commentary
  2.4. Explicit Formulae for Hf
  2.5. Singular Integral Operators
  2.6. Multipliers. The Mikhlin-Hormander Theorem
  2.7. Quadratic Functions
  2.8. The Harmonic Conjugation in Lipschitz Classes
  2.9. The Conjugate Fourier Series
  §3. How Large is the Divergence Set?
  3.1. The Carleson-Hunt Theorem
  3.2. Kolmogorov's Example
  3.3. An Application of the Carleson-Hunt Theorem
  3.4. The Set of Divergence of the Fourier Series of a Continuous Function
  3.5. Capacity of Divergence Sets
  Chapter 3 Fourier Coefficients
  §1. The Rate of Decrease
  1.1. Smooth Functions
  1.2. Lipschitz Functions
  1.3. The Hausdorff-Young Inequality
  1.4. Hardy's Theorem
  1.5. The Khinchin-Kahane Inequality
  1.6. Fourier Coefficients of L^P-Functions，p>2
  §2. Lacunary Series
  2.1. Lacunary Sets
  2.2. Relations with the Khinchin Inequality
  2.3. The Sets of Type A_p
  2.4. Sidon Sets
  2.5. The Paley-Rudin Theorem
  §3. Fourier Coefficients of Bounded Functions
  3.1. General Remarks
  3.2. A₂-Sets
  3.3. Sidon's Problem
  §4. Fourier Coefficients of Measures
  4.1. Positive Definite Sequences
  4.2. The Class R. The Rajchman Theorem.
  4.3. Relations between Properties of the Functions μ[O，+∞) and μ|(-∞，0)
  4.4. Wiener'sTheorem
  4.5. RieszProducts
  4.6. Ivashev-Musatov's Theorem
  4.7. Perfect Sets of Constant Ratio
  4.8. Sets Defined in Terms of Asymptotic Distribution
  4.9. W-Sets and A-Sets
  4.10. H-Sets
  4.11. W*-Sets
  Chapter 4 Absolutely Convergent Fourier Series. Pseudomeasures. Pseudofunctions
  §1. The Class A and Absolute Convergence
  1.1. Definitions and General Remarks
  1.2. The Sets of Absolute Convergence
  §2. Sufficient Conditions for a Function to Belong to A
  2.1. Bernstein's Theorem
  2.2. The Zygmund-Bochkarev Theorem
  2.3. Two Results Valid for More General Groups than T
  2.4. Stechkin's Criterion
  §3. A Comparison of the Classes A and U
  3.1. Introductory Remarks
  3.2. A Correction Theorem
  3.3. Incorrigible Functions
  3.4. Substitutions：The Class U
  3.5. Substitutions：The Class A
  3.6. 1nterpolation：TheClass U
  3.7. Interpolation：The Class A
  §4. Pseudomeasures，Pseudofunctions，Uniqueness
  4.1. Definitions
  4.2. The Problem of Uniqueness
  4.3. Riemann's Theory of Trigonometric Series
  4.4. A Condition Sufficient for Uniqueness
  4.5. Perfect Sets of Constant Ratio
  4.6. The Classes Uoa nd Mo
  Chapter 5 Fourier Integrals
  §1. Definitions and General Remarks
  1.1. The Fourier Transform and the Fourier Integral
  1.2. Remarks on Relations with Fourier Series
  1.3. Properties of Fourier Transforms
  §2. Recovering a Function. The Poisson Kernel. The Plancherel Theorem
  2.1. Approximate Identities
  2.2. The Poisson Kernel
  2.3. Recovering a Function from its Fourier Transform
  2.4. The Plancherel Theorem
  2.5. The Hausdorff-Young Theorem.
  §3. The Spaces H^p in the Half-plane. The Hilbert Transform
  3.1. The Spaces H^p
  3.2 The Paley-Wiener Theorem
  3.3. The Hilbert Transform
  References
  III. Methods of the Theory of Singular Integrals：Hilbert Transform and Calderón-Zygmund Theory
  Introduction
  Chapter 1 Preliminaries
  §1. Notations
  1.1. Geometry
  1.2. Lipschitz Domains
  1.3. Function Spaces
  1.4. Weak Type (1-1)
  1.5. Fourier Transform
  1.6. Probability Theory
  §2. Maximal Functions
  2.1. The Hardy-Littlewood Maximal Function
  2.2. Kolmogorov's Inequality
  2.3. Carleson's Imbedding Theorem
  §3. CoveringTheorems
  3.1. Whitney Theorem
  3.2. Calderón-Zygmund Decomposition
  §4. Weight Norms
  4.1. Muckenhoupt Condition
  4.2. The Muckenhoupt Maximal Theorem
  §5. Distribution Function Inequalities
  §6. Rademacher Functions
  §7. Harmonic Functions
  7.1. Poisson Integral
  7.2. Maximal Functions
  7.3. Green's Formula
  7.4. Harmonic Vector Fields
  §8. Hardy Classes
  8.1. Hardy Classes of Analytic Functions
  8.2. Real Hardy Classes
  8.3. Atoms
  §9. Bounded Mean Oscillation
  9.1. The Space BMO (Rⁿ)
  9.2. Ch. Fefferman's Duality Theorem
  Chapter 2 Hilbert Transform
  §1. Definition and Elementary Properties
  1.1. Hilbert Transform on the Real Line
  1.2. Hilbert Transform on the Circle
  1.3. Fourier Transform
  1.4. Maximal Operator
  1.5. Privalov's Proof
  1.6. N.N. Luzin on Singular Integrals
  §2. Hilbert Transform in L²
  2.1. Boundedness of Operator H
  2.2. Hilbert Transform and Fourier Series
  2.3. The Helson-Szegö Theorem
  §3. Hilbert Transform in L¹
  3.1. Kolmogorov's Theorem
  3.2. Harmonic Estimates
  3.3. Explicit Formulae
  3.4. Further Estimates in L¹
  3.5. A-Integral
  §4. Hilbert Transform in L^P
  4.1. Theorem of M. Riesz
  4.2. Other Proofs of M. Riesz' Theorem
  4.3. The Case p=∞
  4.4. Boyd's Theorem
  §5. Applications of Brownian Motion
  5.1. Preliminaries
  5.2. Weak Type (1-1)
  5.3. Brownian Maximal Functions
  §6. Cauchy Type Integral
  6.1. Definition and Basic Properties
  6.2. Existence of Boundary Values. L^P-Estimates
  §7. Hilbert Transform in Holder Classes.
  7.1. The Plemelj-Privalov Theorem
  7.2. Other Methods of Proof.
  7.3. Smoothness of Cauchy Type Integrals
  Chapter 3 Calderón-Zygmund Theory
  §1. Calderón-Zymund Operators
  1.1. Definition
  1.2. Connections Between an Operator and a Kernel. Maximal Operator
  1.3. The Calderón-Zygmund-Cotlar Theorem
  §2. Examples of Calderbn-Zygmund Operators
  2.1. The M. Riesz Transform
  2.2. Homogeneous Kernels
  2.3. Integrals of Imaginary Order
  2.4. Calderón Commutators
  2.5. Pseudodifferential Operators
  2.6. Anisotropic Analogues
  §3. L²-Estimates
  3.1. Estimates of the Fourier Transform
  3.2. The Cotlar-Stein Lemma
  3.3. L²-Estimates and the Carleson-Hunt Theorem
  §4. Method of Rotations
  4.1. Formulae of the Rotation Method
  4.2. Results of the Rotation Method
  4.3. Sjolin'sTheorem
  §5. Estimates in L^P
  5.1. The Main Inequality
  5.2. The Weak Type (1-1) Estimate
  5.3. L^P-Estimates
  5.4. Atoms and the Space H¹
  5.5. Marcinkiewicz' Integral
  5.6. The Case p=∞
  5.7. Hypersingular Integrals
  §6. The Maximal Operator
  6.1. Decomposition of a Function
  6.2. Estimates of the Maximal Operator
  6.3. Estimates of the Distribution Function
  6.4. The Fefferman-Stein Function f^#
  §7. Weighted and Vector Analogues
  7.1. Weighted Estimates of Calderbn-Zygmund Operators
  7.2. Vector Analogues of Calderbn-Zygmund Operators
  7.3. Connections Between Weighted and Vector Estimates
  Bibliographical Notes
  References