The first volume in this subseries of the Encyclopaedia 1S 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 ofview．Numerous examples illustrate the connections to differential and integral equationS，approximation theory，nutuber 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 Calderon-Zygmund theory．

Introduction
Chapter 1.A Short Course of Fourier Analysis of Periodic Functions
　§1.Translation-Invariant Operators
　 1.1.The Set up
　 1.2.Object ofInvestigation
　 1.3.Convolution
　 1.4.General Form oft.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 ofDistributions
　 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 lsoperimetric Inequality.(Hurwitz’Proof)
　 2.10.Harmonic Analysis on the Torus
Chapter 2.Harmonic Analysis in Rd
　§1.Preliminaries on Distributions in Rd
　 1.1.Distributions in Rd
　§2.From the Circle to the Line.Fourier Transform in Rd(Definition)
　 2.1.Inversion Formula(An Euristic Derivation)
　 2.2.A Proofofthe Inversion Formula
　 2.3.Another Proof
　 2.4.Fourier Transform in Rd(Definition)
　§3.Convolution(Definition).
　 3.1.Difficulties of Harmonic Analysis in Rd
　 3.2.Convolution of Distributions(Construction)
　 3.3.Examples
　 3.4.Convolution Operators
　§4.Convolution Operators as Object of Study(Examples)
　 4.1.Linear Ditierential 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 ofa 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 θ-Function
　 6.9.Evaluation ofthe 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.
　……
Chapter3 Harmonic Analysis on Groups
Chapter4 A Historical Survey
Chapter5 Spectral Analysis and Spectral Synthesis,Intrinsic Problems
Epilogue
Bibliographical Noes
References