First Course in Fourier Analysis, A, 1/e
David W. Kammler, Southern Illinois University, Carbondale
Coming February, 2000 by Prentice Hall Engineering/Science/Mathematics
Copyright 2000, 700 pp.
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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
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
Develops many fine points of the theory in the exercises
(accompanied by detailed hints). Proofs are given when they support
- 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
Presents an elementary theory of generalized functions
(tempered distributions) using concepts from elementary calculus.
Uses the FT calculus and generalized functions in the
unit on PDEs to study the (univariate) wave equation, diffusion
equation, and diffraction equation.
- Allows the instructor to adjust the level of mathematical
rigor to match the interests and abilities of the students.
Demonstrates real-world applications of Fourier
analysis to the humanities in a chapter on musical tones.
- In each case, a solution is constructed and important
mathematical properties (e.g., maximum principle, conservation of
energy, speed of propagation,...) are derived.
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
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.
9. Partial Differential Equations.
12. Musical Tones.
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
Appendix 7. FORTRAN Code for a Radix 2 FFT.
Appendix 8. The 17 Crystal Symmetries for the Plane.