x Preface
of functions defined on the real line, matters are similar, but with additional dif-
ficulties. In the two latter situations we encounter points in our arguments where
a detail is needed from advanced calculus or Lebegue measure theory. On such
occasions, we simply quote the needed result and move on.
On the subject of Fourier Series, some authors use cos nx and sin nx, so that
all functions have period 2π. The consequence of this prescription is that most
formulas have a 1/(2π) or 1/π. Our contention is that the subject is more elegant
when one works with functions with period 1, so that the basic building blocks
are cos 2πnx and sin 2πnx. But cos 2πnx + i sin 2πnx =
e2πinx
(a fact that will
be a subject of discussion in Chapter 1), and it is more elegant still to use the
complex exponential rather than sines and cosines. Of course, to proceed in this
way, one must first become more comfortable with complex numbers. Hence that
is the topic of Chapter 1. In general, when we are faced with a function with some
strange period, we make a linear change of variable so that everything is translated
into issues of functions with period 1. If sines and cosines are involved, we may
convert to complex exponentials. When we resolve whatever is at issue, we may
convert back to sines and cosines, if we wish. This is a little reminiscent of a
problem expressed in terms of pounds and feet, which we would convert to grams
and meters, and then convert back after the calculation is done.
Fourier analysis has links to many other branches of mathematics. We occa-
sionally make remarks relating to such topics as linear algebra, probability theory,
or number theory. Such digressions may be safely ignored by readers who are
unfamiliar with the related subject in question.
Among the following chapters, sections, and appendices are found several valu-
able topics that are rarely found in the undergraduate (and sometimes even the
graduate) curriculum. These include linear recurrences (in §F.4), summability the-
ory (in §4.3), Bernoulli polynomials and Euler–Maclaurin summation (in §9.5),
uniform distribution (in §9.6), Chebyshev polynomials (in Appendix C), and in-
equalities (in Appendix I).
The author is indebted to colleagues Al Taylor and Jack Goldberg for initiating
this educational experiment and to the late Curtis Huntington for his unwavering
support. In addition, the author is happy to thank Dick Askey, John Benedetto, Ed-
ward Crane, Peter Duren, Emily Holt, Alex Iosevich, Michael Kelly, Harsh Mehta,
Kristen Moore, Michael Mossinghoff, Chris Nixon, Olivier Ramar´ e, Elmer Rees,
Babar Saffari, and Jeff Vaaler for their valuable contributions. It has been a plea-
sure to work with editor Sergei Gelfand and his competent and attentive support
staff at the AMS. Finally, the author thanks Michele MacFarlane, who cheerfully
accepted a double dose of domestic chores in order that the author would have more
time to write.
Hugh L. Montgomery
Ann Arbor, Michigan
August 31, 2014
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