This book is designed to be an introduction to harmonic analysis in Euclidean spaces. The subject has seen a considerable flowering during the past twenty years. We have not tried to cover all phases of this development. Rather, our chief concern was to illustrate various methods used in this aspect of Fourier analysis that exploit the structure of Euclidean spaces. In particular, we try to show the role played by the action of translations, dilations, and rotations. Another concern, not independent of this chief one, is to motivate the study of harmonic analysis on more general spaces having an analogous structure (such as arises in symmetric spaces). It is our feeling that the study of Fourier analysis in that context and, also, in other general settings, is more meaningful once the special Euclidean case is understood.

Preface

CHAPTER Ⅰ The Fourier Transform

 1. The basic L1 theory of the Fourier transform

　2.The L2 theory and the Plancherel theorem

　3.The class of tempered distributions

　4.Further results

CHAPTER Ⅱ Boundary Values of Harmonic Functions

　1.Basic properties of harmonic functions

　2.The characterization of Poisson integrals

　3.The Hardy-Littlewood maximal function and nontangential convergence of harmonic functions

　4.Subharmonic functions and majorization by harmonic functions　

　5.Further results

CHAPTER Ⅲ The Theory of Hp Spaces on Tubes

　1.Introductory remarks

　2.The H2 theory

　3.Tubes over cones

　4.The Paley-Wiener theorem

　5.The Hp theory

　6.Further results

CHAPTER Ⅳ.Symmetry Properties of the Fourier Transform

　1.Decomposition of L2(Ez) into'sub, paces invariant under the Fourier transform

　2.Spherical harmonics

　3.The action of the Fourier transform on the spaces

　4.Some applications

　5.Further results

CHAPTER Ⅴ Interpolation of Operators

　1.The M. Riesz convexity theorem and interpolation of operators defined on Lp spaces

 2.The Marcinkiewicz interpolation theorem

 3.L(p, q) spaces

 4.Interpolation of analytic families of operators

 5.Further results

CHAPTER Ⅵ Singular Integrals and Systems of Conjugate Harmonic Functions

　1.The Hilbert transform

 2.Singular integral operators with odd kernels

 3.Singular integral operators with even kernels

 4.Hp spaces of conjugate harmonic functions

　5.Further results

CHAPTER Ⅶ Multiple Fourier Series

　1.Elementary properties

　2.The Poisson summation formula

　3.Multiplier transformations

　4.Summability below the critical index (negative results)

　5.Summability below the critical index

　6.Further results

Bibliography

Index