SIAM AG on Orthogonal Polynomials and Special Functions


Extract from OP-SF NET


Topic #5  ------------   OP-SF NET 8.1  --------------  January 15, 2001
From: OP-SF NET Editor (
Subject: New book on Fourier Analysis

[From the AMS web page]
Fourier Analysis
Javier Duoandikoetxea, Universidad del País Vasco/Euskal Herriko Unibertsitatea,
Bilbao, Spain 
Expected publication date is January 11, 2001

Fourier analysis encompasses a variety of perspectives and techniques. This
volume presents the real variable methods of Fourier analysis introduced by
Calderón and Zygmund. The text was born from a graduate course taught at the
Universidad Autónoma de Madrid and incorporates lecture notes from a course
taught by José Luis Rubio de Francia at the same university.

Motivated by the study of Fourier series and integrals, classical topics are
introduced, such as the Hardy-Littlewood maximal function and the Hilbert
transform. The remaining portions of the text are devoted to the study of
singular integral operators and multipliers. Both classical aspects of the theory
and more recent developments, such as weighted inequalities, H^1, BMO spaces,
and the T1 theorem, are discussed.

Chapter 1 presents a review of Fourier series and integrals; Chapters 2 and 3
introduce two operators that are basic to the field: the Hardy-Littlewood maximal
function and the Hilbert transform. Chapters 4 and 5 discuss singular integrals,
including modern generalizations. Chapter 6 studies the relationship between
H^1, BMO, and singular integrals; Chapter 7 presents the elementary theory of
weighted norm inequalities. Chapter 8 discusses Littlewood-Paley theory, which
had developments that resulted in a number of applications. The final chapter
concludes with an important result, the T1 theorem, which has been of crucial
importance in the field.

This volume has been updated and translated from the Spanish edition that was
published in 1995. Minor changes have been made to the core of the book; however,
the sections, "Notes and Further Results" have been considerably expanded and
incorporate new topics, results, and references. It is geared toward graduate
students seeking a concise introduction to the main aspects of the classical
theory of singular operators and multipliers. Prerequisites include basic
knowledge in Lebesgue integrals and functional analysis.

    Fourier series and integrals 
    The Hardy-Littlewood maximal function 
    The Hilbert transform 
    Singular integrals (I) 
    Singular integrals (II) 
    H^1 and BMO 
    Weighted inequalities 
    Littlewood-Paley theory and multipliers 
    The T1 theorem 

                 Publisher: American Mathematical Society 
                 Distributor: American Mathematical Society 
                 Series: Graduate Studies in Mathematics, ISSN: 1065-7339 
                 Volume: 29 
                 Publication Year: 2001 
                 ISBN: 0-8218-2172-5 
                 Paging: 222 pp. 
                 Binding: Hardcover 
                 List Price: $35 
                 Institutional Member Price: $28 
                 Individual Member Price: $28 
                 Order Code: GSM/29 

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