Harmonic Analysis on the Real Line

1. Front Matter

   Pages i-ix

   PDF

2. Introduction

   • Elijah Liflyand

   Pages 1-4

3. Classes of Functions

   • Elijah Liflyand

   Pages 5-21

4. Fourier Series

   • Elijah Liflyand

   Pages 23-45

5. Fourier Transform

   • Elijah Liflyand

   Pages 47-69

6. Hilbert Transform

   • Elijah Liflyand

   Pages 71-99

7. Hardy Spaces and their Subspaces

   • Elijah Liflyand

   Pages 101-129

8. Hardy Inequalities

   • Elijah Liflyand

   Pages 131-140

9. Certain Applications

   • Elijah Liflyand

   Pages 141-181

10. Back Matter

    Pages 183-197

    PDF

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.

• Bounded Variation
• Absolute Continuity
• Integrability
• Fourier Series
• Fourier Transform
• Summability
• Hilbert Transform
• Hardy Space
• Interpolation

“The book under review takes the reader on a journey along a particular path through the vast landscape of modern harmonic analysis in one real variable. From beginning to end, the text is uniquely flavored by the author’s mathematical interests which provides the reader with a good sense of direction. … The book should be accessible to beginning graduate students in analysis and advanced undergraduates with basic knowledge in real analysis … .” (Joris Roos, zbMATH 1514.42001, 2023)

“This book is very accurately described by its subtitle ‘a path in the theory’. The book is at times a textbook, an introduction to harmonic analysis, an essay, or a survey, or some combination of these. … Some theorems are stated and proved, some are discussed, and others are quickly mentioned. It's not a standard path, but an engaging one, offering insights and connections that arenew or not well known.” (‪Charles N. Moore, Mathematical Reviews, September, 2022)

• Department of Mathematics, Bar-Ilan University, Ramat-Gan, Israel;, Regional Mathematical Center of Southern Federal University, Rostov-on-Don, Russia

  Elijah Liflyand

Elijah Liflyand has recently retired from his position at the Department of Mathematics at Bar-Ilan University in Israel, but not from mathematics. He also holds a position at the at the Regional Mathematical Center of Southern Federal University in Rostov-on-Don, Russia. His areas of expertise are in Fourier Analysis, Complex Analysis, and Approximation Theory, among others. With Birkhäuser/Springer, he has published two books: "Decay of the Fourier Transform" (with Alex Iosevich, 2014), and "Functions of Bounded Variation and Their Fourier Transforms" (in the Applied Numerical and Harmonic Analysis series, 2019).

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