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.
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