This book is primarily about \(H^p\) (Hardy) spaces, that is, the spaces of functions analytic on the open unit disc and such that the \(L^p\) norm of the function on the circle \(|z| = r\) remains bounded as \(r \to 1\). This may seem a specialized subject, but it has close ties to the theory of Fourier series of \(L^p\) functions on the unit circle. The subject was introduced by G. H. Hardy in a 1915 note on mean values and then named and studied further by F. Riesz in 1923. The present volume is a Dover 2007 unaltered reprint of the 1962 Prentice-Hall publication.

The Preface states, “The main purpose of this monograph is to provide an introduction to the segment of mathematics in which functional analysis and analytic function theory merge successfully.” It does provide a very good introduction, but skimps on the connections with Fourier analysis and generally considers the subject of \(H^p\) spaces in isolation. It also only deals with functions on the unit circle and not more general spaces (except a chapter on the real half-plane). However, the proofs are carefully chosen so they will extend to other spaces, even though that extension is not shown here.

About the first quarter of the book is background information on measure spaces, functional analysis, and complex analysis. The book is aimed at second-year graduate students and assumes they have already had some exposure to these subjects. The rest deals with various aspects of \(H^p\) spaces, looking both at properties of member functions, and at the spaces as a whole as Banach spaces and as Banach algebras. The writing is very clear and the focus is concrete rather than abstract. The exercises are excellent; none are easy, many are difficult, and some give additional important results. Several ask to prove a property for a specific function, but even these are not routine.

Having been written in 1962, this book has missed many of the later developments, although it is still very valuable for the “classical” theory on the unit circle. One conspicuous omission is Carleson’s corona theorem, that was proved in the same year. Some good, more modern books that have wider coverage are Peter Duren’s *Theory of \(H^p\) Spaces* (Dover expanded reprint, 2000) and John B. Garnett’s *Bounded Analytic Functions*. The applications to Fourier series are covered well in Zygmund’s *Trigonometric Series*, Chapter VII. There’s a good brief introduction to \(H^p\) spaces in Rudin’s *Real and Complex Analysis*, Chapter 17 (with further developments in later chapters).

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis.