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Early Fourier Analysis
Pure and Applied Undergraduate Texts

Rating
Format
Hardback, 390 pages
Published
United States, 6 December 2019
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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.


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Product Description

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.

Product Details
EAN
9781470415600
ISBN
1470415607
Dimensions
26.6 x 18.9 x 2.5 centimeters (0.84 kg)

Table of Contents

  • Background
  • Complex numbers
  • The discrete Fourier transform
  • Fourier coefficients and first Fourier series
  • Summability of Fourier series
  • Fourier series in mean square
  • Trigonometric polynomials
  • Absolutely convergent Fourier series
  • Convergence of Fourier series
  • Applications of Fourier series
  • The Fourier transform
  • Higher dimensions
  • Appendix B. The binomial theorem
  • Appendix C. Chebyshev polynomials
  • Appendix F. Applications of the fundamental theorem of algebra
  • Appendix I. Inequalities
  • Appendix L. Topics in linear algebra
  • Appendix O. Orders of magnitude
  • Appendix T. Trigonometry
  • References
  • Index

About the Author

Hugh L. Montgomery, University of Michigan, Ann Arbor, MI, USA.

Reviews

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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