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In the last 200 years, harmonic analysis has been one of the most
influential bodies of mathematical ideas, having been exceptionally
significant both in its theoretical implications and in its
enormous range of applicability throughout mathematics, science,
and engineering. In this book, the authors convey the remarkable
beauty and applicability of the ideas that have grown from Fourier
theory. They present for an advanced undergraduate and beginning
graduate student audience the basics of harmonic analysis, from
Fourier's study of the heat equation, and the decomposition of
functions into sums of cosines and sines (frequency analysis), to
dyadic harmonic analysis, and the decomposition of functions into a
Haar basis (time localization). While concentrating on the Fourier
and Haar cases, the book touches on aspects of the world that lies
between these two different ways of decomposing functions:
time-frequency analysis (wavelets). Both finite and continuous
perspectives are presented, allowing for the introduction of
discrete Fourier and Haar transforms and fast algorithms, such as
the Fast Fourier Transform (FFT) and its wavelet analogues. The
approach combines rigorous proof, inviting motivation, and numerous
applications. Over 250 exercises are included in the text. Each
chapter ends with ideas for projects in harmonic analysis that
students can work on independently. This book is published in
cooperation with IAS/Park City Mathematics Institute.
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