x PREFACE

domain,"6 a claim endorsed and popularized by Hadamard.7 Our aim in this little

book is to illustrate this thesis by bringing together in one volume a variety of

mathematical results whose formulations lie outside complex analysis but whose

proofs employ the theory of analytic functions. The most famous such example

is, of course, the Prime Number Theorem; but, as we show, there are many other

examples as well, some of them basic results.

For whom, then, is this book intended? First of all, for everyone who loves

analysis and enjoys reading pretty proofs. The technical level is relatively

modest. We assume familiarity with basic functional analysis and some elementary

facts about the Fourier transform, as presented, for instance, in the first author’s

Functional Analysis (Wiley-Interscience, 2002), referred to henceforth as [FA]. In

those few instances where we have made use of results not generally covered in

the standard first course in complex variables, we have stated them carefully and

proved them in appendices. Thus the material should be accessible to graduate

students. A second audience consists of instructors of complex variable courses

interested in enriching their lectures with examples which use the theory to solve

problems drawn from outside the field.

Here is a brief summary of the material covered in this volume. We begin with

a short account of how complex variables yields quick and eﬃcient solutions of two

problems which were of great interest in the seventeenth and eighteenth centuries,

viz., the evaluation of

∑

∞

1

1/n2

and related sums and the proof that every algebraic

equation in a single variable (with real or even complex coeﬃcients) is solvable in

the field of complex numbers. Next, we discuss two representative applications of

complex analysis to approximation theory in the real domain: weighted polynomial

approximation on the line and uniform approximation on the unit interval by linear

combinations of the functions

{xnk

}, where nk → ∞ (Müntz’s Theorem). We then

turn to applications of complex variables to operator theory and harmonic analysis.

These chapters form the heart of the book. A first application to operator theory

is Rosenblum’s elegant proof of the Fuglede-Putnam Theorem. We then discuss

Toeplitz operators and their inversion, Beurling’s characterization of the invari-

ant subspaces of the unilateral shift on the Hardy space H2 and the consequent

divisibility theory for the algebra B of bounded analytic functions on the disk or

half-plane, and a celebrated problem in prediction theory (Szegő’s Theorem). We

also prove the Riesz-Thorin Convexity Theorem and use it to deduce the bound-

edness of the Hilbert transform on Lp(R), 1 p ∞. The chapter on applications

to harmonic analysis begins with D.J. Newman’s striking proof of Fourier unique-

ness via complex variables; continues on to a discussion of a curious functional equa-

tion and questions of uniqueness (and nonuniqueness) for the Radon transform; and

then turns to the Paley-Wiener Theorem, which together with the divisibility the-

ory for B referred to above is exploited to provide a simple proof of the Titchmarsh

Convolution Theorem. This chapter concludes with Hardy’s Theorem, which quan-

tifies the fact that a function and its Fourier transform cannot both tend to zero

6“Entre

deux vérités du domain réel, le chemin le plus facile et le plus court passe bien

souvent par le domaine complexe." Paul Painlevé, Analyse des travaux scientifiques, Gauthier-

Villars, 1900, pp.1-2.

7“It

has been written that the shortest and best way between two truths of the real domain

often passes through the imaginary one." Jacques Hadamard, An Essay on the Psychology of

Invention in the Mathematical Field, Princeton University Press, 1945, p. 123.