Introduction to Fourier Analysis on Euclidean Spaces. (PMS-32)

This book covers those parts of harmonic analysis that genuinely depend on Euclidean space. The Fourier transform of Borel measures, convolution, the Fourier inversion theorem, and Plancherel's theorem, and the relation to the Gelfand theory of Banach algebras are understood most clearly in the category of locally compact abelian groups. In LCA groups a group and its Pontryagin dual are different objects, unlike in Euclidean space where these are identified. But in Euclidean space, beyond translation like we have for LCA groups, we also have derivatives and dilations and rotations, and can use harmonic functions and the residue theorem from complex analysis.The chapter I like the best is Chapter IV, about rotations and functions on the unit sphere in n-dimensions. This involves radial functions, Bessel functions, homogeneous polynomials. There is a proof of Hecke's identity as Theorem 3.4, which is a formula for the Fourier transform of the product of a homogeneous harmonic polynomial and a Gaussian. I also like Chapter VII, about Fourier series in n variables ("multiple Fourier series").

This book deals with the extension of real and complex methods in harmonic analysis to the many-dimensional case. So, its pre-requisites are a strong background in real and complex analysis and some acquaintance with elementary harmonic analysis, that is, this book is intended for graduate students and working mathematicians. Maybe some advanced undergraduates could cover certain parts of the material.This book is one component of the Stein trilogy on harmonic analysis (together with "Singular Integrals and Differentiability Properties of Functions" and "Harmonic Analysis", both also reviewed by myself), and as such it must be regarded as an authoritative reference on the subject since Elias Stein and Guido Weiss are two of the leading experts in the field, and the material they selected was taken from their teaching and research experience.The contents of the book are: The Fourier Transform; Boundary Values of Harmonic Functions; The Theory of H^p Spaces on Tubes; Symmetry Properties of the Fourier Transform; Interpolation of Operators; Singular Integrals and Systems of Conjugate Harmonic Functions; Multiple Fourier Series.Includes motivation and full explanations for each topic, excercises for each chapter, called "further results", and extensive references. Outstanding printing quality and nice clothbound.These three volumes should be present in every analyst's library.Please take a look to the rest of my reviews (just click on my name above).

This is a classic, must-have reference for all students and anyone interested in Fourier analysis. I already own one original copy and I bought a second one for a student. As an unexpected bonus, this particular volume had the index to a totally different book included at the beginning. Caveat emptor.