Harmonic Analysis on the Real Line - A Path in the Theory

Contents
1 Introduction
	1.1 Motivation and Background
	1.2 Structure
	1.3 Before Reading the Book
2 Classes of Functions
	2.1 Continuous Functions and Lebesgue Spaces
		2.1.1 Continuous Functions
		2.1.2 Lebesgue Spaces
		2.1.3 The Hardy–Littlewood Maximal Function
		2.1.4 Calderón-Zygmund Decomposition
		2.1.5 Absolute Continuity
	2.2 Functions of Bounded Variation
3 Fourier Series
	3.1 Definition and Basic Properties
	3.2 Convergence
	3.3 Absolute Convergence
	3.4 Lebesgue Constants
	3.5 Summability
	3.6 Trigonometric Series Versus Fourier Series
4 Fourier Transform
	4.1 Definitions and Around
	4.2 From Discussion to Calculations
	4.3 Poisson Summation Formula
	4.4 Amalgam Type Spaces
	4.5 Summability
		4.5.1 Summability and Poisson Summation
		4.5.2 Wiener Algebras and Bounded Variation
5 Hilbert Transform
	5.1 Definitions and Calculations
	5.2 The Hilbert Transform Comes into Play
	5.3 Existence Almost Everywhere
		5.3.1 Weak Estimate
		5.3.2 Extension to L1
	5.4 Integrability of the Hilbert Transform
	5.5 Special Cases of the Hilbert Transform
		5.5.1 Conditions for the Integrability of the Hilbert Transform
		5.5.2 General Conditions
	5.6 Summability to the Hilbert Transform
6 Hardy Spaces and their Subspaces
	6.1 Some Starting Points
	6.2 Atomic Characterization
		6.2.1 Atoms
		6.2.2 Atomic Proof of the Fourier-Hardy Inequality
		6.2.3 A Postponed Proof
		6.2.4 More About Atomic Characterization
	6.3 Molecular Characterization
	6.4 Subspaces
	6.5 A Paley–Wiener Theorem
	6.6 Discrete Hardy Spaces
	6.7 Back to Trigonometric Series
7 Hardy Inequalities
	7.1 Discrete Hardy Inequality
	7.2 Hardy Inequalities for Hausdorff Operators
8 Certain Applications
	8.1 Interpolation Properties of a Scale of Spaces
		8.1.1 Results
		8.1.2 Proofs
	8.2 Fourier Re-expansions
	8.3 Absolute Convergence
		8.3.1 Proof of the Main Theorem
		8.3.2 Proof of the Corollary
		8.3.3 Proof of the Extended Theorem
	8.4 Boas' Conjecture
	8.5 Salem Type Conditions
		8.5.1 Non-periodic Salem Conditions
		8.5.2 Applications
	8.6 L1 Convergence of Fourier Transforms
		8.6.1 L1 Convergence
		8.6.2 Application to Trigonometric Series
	8.7 More About Applications
Basic Notations
Bibliography
	Textbooks and Monographs
	Papers
Index