This book sketches a path for newcomers into the theory of harmonic analysis on the real line. It presents a collection of both basic, well-known and some less known results that may serve as a background for future research around this topic. Many of these results are also a necessary basis for multivariate extensions. An extensive bibliography, as well as hints to open problems are included. The book can be used as a skeleton for designing certain special courses, but it is also suitable for self-study.
Author(s): Elijah Liflyand
Series: Pathways in Mathematics
Edition: 1
Publisher: Birkhäuser
Year: 2021
Language: English
Pages: 197
Tags: Fourier Series, Fourier Transform, Hilbert Transform, Hardy Spaces
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
