This first volume, a three-part introduction to the subject, is
intended for students with a beginning knowledge of mathematical
analysis who are motivated to discover the ideas that shape Fourier
analysis. It begins with the simple conviction that Fourier arrived
at in the early nineteenth century when studying problems in the
physical sciences--that an arbitrary function can be written as an
infinite sum of the most basic trigonometric functions.
The first part implements this idea in terms of notions of
convergence and summability of Fourier series, while highlighting
applications such as the isoperimetric inequality and
equidistribution. The second part deals with the Fourier transform
and its applications to classical partial differential equations
and the Radon transform; a clear introduction to the subject serves
to avoid technical difficulties. The book closes with Fourier
theory for finite abelian groups, which is applied to prime numbers
in arithmetic progression.
In organizing their exposition, the authors have carefully
balanced an emphasis on key conceptual insights against the need to
provide the technical underpinnings of rigorous analysis. Students
of mathematics, physics, engineering and other sciences will find
the theory and applications covered in this volume to be of real
interest.
The Princeton Lectures in Analysis represents a sustained effort
to introduce the core areas of mathematical analysis while also
illustrating the organic unity between them. Numerous examples and
applications throughout its four planned volumes, of which "Fourier
Analysis" is the first, highlight the far-reaching consequences of
certain ideas in analysis to other fields of mathematics and a
variety of sciences. Stein and Shakarchi move from an introduction
addressing Fourier series and integrals to in-depth considerations
of complex analysis; measure and integration theory, and Hilbert
spaces; and, finally, further topics such as functional analysis,
distributions and elements of probability theory.
General
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