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The principal aim in writing this book has been to provide an intro-
duction, barely more, to some aspects of Fourier series and related topics
in which a liberal use is made of modem techniques and which guides the
reader toward some of the problems of current interest in harmonic analysis
generally. The use of modem concepts and techniques is, in fact, as wide-
spread as is deemed to be compatible with the desire that the book shall
be useful to senior undergraduates and beginning graduate students, for
whom it may perhaps serve as preparation for Rudin's Harmonic Analysis
on Groups and the promised second volume of Hewitt and Ross's Abstract
Harmonic Analysis.
The emphasis on modem techniques and outlook has affected not only
the type of arguments favored, but also to a considerable extent the choice
of material. Above all, it has led to a minimal treatment of pointwise con-
vergence and summability: as is argued in Chapter 1, Fourier series are not
necessarily seen in their best or most natural role through pointwise-tinted
spectacles. Moreover, the famous treatises by Zygmund and by Baryon
trigonometric series cover these aspects in great detail, while leaving some
gaps in the presentation of the modern viewpoint; the same is true of the
more elementary account given by Tolstov. Likewise, and again for reasons
discussed in Chapter 1, trigonometric series in general form no part of the
program attempted