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Fourier Analysis is an important area of mathematics, especially in light of its importance in physics, chemistry, and engineering. Yet it seems that this subject is rarely offered to undergraduates. This book introduces Fourier Analysis in its three most classical settings: The Discrete Fourier Transform for periodic sequences, Fourier Series for periodic functions, and the Fourier Transform for functions on the real line.
The presentation is accessible for students with just three or four terms of calculus, but the book is also intended to be suitable for a junior-senior course, for a capstone undergraduate course, or for beginning graduate students. Material needed from real analysis is quoted without proof, and issues of Lebesgue measure theory are treated rather informally. Included are a number of applications of Fourier Series, and Fourier Analysis in higher dimensions is briefly sketched. A student may eventually want to move on to Fourier Analysis discussed in a more advanced way, either by way of more general orthogonal systems, or in the language of Banach spaces, or of locally compact commutative groups, but the experience of the classical setting provides a mental image of what is going on in an abstract setting.
Fourier Analysis is an important area of mathematics, especially in light of its importance in physics, chemistry, and engineering. Yet it seems that this subject is rarely offered to undergraduates. This book introduces Fourier Analysis in its three most classical settings: The Discrete Fourier Transform for periodic sequences, Fourier Series for periodic functions, and the Fourier Transform for functions on the real line.
The presentation is accessible for students with just three or four terms of calculus, but the book is also intended to be suitable for a junior-senior course, for a capstone undergraduate course, or for beginning graduate students. Material needed from real analysis is quoted without proof, and issues of Lebesgue measure theory are treated rather informally. Included are a number of applications of Fourier Series, and Fourier Analysis in higher dimensions is briefly sketched. A student may eventually want to move on to Fourier Analysis discussed in a more advanced way, either by way of more general orthogonal systems, or in the language of Banach spaces, or of locally compact commutative groups, but the experience of the classical setting provides a mental image of what is going on in an abstract setting.
Hugh L. Montgomery, University of Michigan, Ann Arbor, MI, USA.
Hugh Montgomery has written a book which both students and faculty
should appreciate. I wish it had been written 15 years ago so I
could have shared it with students. It is a gem." - Richard Askey,
University of Wisconsin-Madison, USA
"Montgomery has written an exquisite text combining basic material,
exciting examples, advanced topics, wonderful historical notes, and
excellent exercises. It is absolutely compelling and masterful!" -
John Benedetto, University of Maryland, USA
"This nice book is likely to be especially successful. l feel that
the author has managed admirably to bring to light both the beauty
and the usefulness of Fourier's idea, thus making the first
introduction to Fourier analysis a joy for undergraduates. All the
details are included in a way that is both attractive and easy for
students to follow." - Palle Jorgensen, University of Iowa, USA,
Author of Wavelets Through a Looking Glass
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