## First Course in Fourier Analysis, A, 1/e

David W. Kammler, Southern Illinois University, Carbondale

Coming February, 2000 by Prentice Hall Engineering/Science/Mathematics

Cloth
ISBN 0-13-578782-3

mailings
on this subject.

Fourier Analysis-Mathematics

For upper-level undergraduate and first-year graduate applied mathematics courses in Mathematics, Physics, Chemistry, Geology, Electrical Engineering, Mechanical Engineering, and other related fields. This unique text provides a meaningful introduction to modern applied mathematics through Fourier analysis and shows how these mathematical ideas can be used to study sampling theory, PDEs, probability, diffraction, musical tones, and wavelets. The purpose of the text is to motivate students, provide historical perspective, develop intuition, and build analytical skills.

Provides unified development of (univariate) Fourier analysis for functions on R, T, Z, and P (i.e., for aperiodic and for periodic functions on R as well as for aperiodic and periodic functions on Z.)
Provides an unusually complete presentation of the Fourier transform calculus.

• Helps students learn to find Fourier transforms and Fourier series with insight and skill, e.g. the Fourier series for a box wave can be written down with no calculation if you know how to use Poisson's formula. Presents an elementary theory of generalized functions (tempered distributions) using concepts from elementary calculus.
Develops many fine points of the theory in the exercises (accompanied by detailed hints). Proofs are given when they support such ends.
• Allows the instructor to adjust the level of mathematical rigor to match the interests and abilities of the students.
Uses the FT calculus and generalized functions in the unit on PDEs to study the (univariate) wave equation, diffusion equation, and diffraction equation.
• In each case, a solution is constructed and important mathematical properties (e.g., maximum principle, conservation of energy, speed of propagation,...) are derived.
Demonstrates real-world applications of Fourier analysis to the humanities in a chapter on musical tones.
Includes more than 400 exercises (half of the text!)—Some provide for routine drill; many lead to the student to explore the history of the discipline; some invite the student to exlore interesting applications; some encourage the student to develop simulations; and some show how to “do the details” associated with informal presentations from the text.

1. Fourier's Representation for Functions on R, Tp, Z, and PN.
2. Convolution of Functions on R, Tp, Z and PN.
3. The Calculus for Finding Fourier Transforms of Functions of R.
4. The Calculus for Finding Fourier Transforms of Functions of Tp, Z, PN.
5. Operator Identities Associated with Fourier Analysis.
6. The Fast Fourier Transform.
7. Generalized Functions on R.
8. Sampling.
9. Partial Differential Equations.
10. Probability.
11. Diffraction.
12. Musical Tones.
13. Wavelets.
Appendix 1. A Table of Functions and Their Fourier Transforms.
Appendix 2. The Fourier Transform Calculus.
Appendix 3. A Table of Operators and Their Fourier Transforms.
Appendix 4. The Standard Normal Probability Density.
Appendix 5. Frequencies of the Piano Keyboard.
Appendix 6. The Whittaker - Robinson Flow Chart for 12 Point Harmonic Analysis.
Appendix 7. FORTRAN Code for a Radix 2 FFT.
Appendix 8. The 17 Crystal Symmetries for the Plane.