**Pure and Applied Undergraduate Texts**

Volume: 33;
2019;
693 pp;
Hardcover

MSC: Primary 42; 65; 94; 46;

**Print ISBN: 978-1-4704-4191-3
Product Code: AMSTEXT/33**

List Price: $115.00

AMS Member Price: $92.00

MAA Member Price: $103.50

**Electronic ISBN: 978-1-4704-4976-6
Product Code: AMSTEXT/33.E**

List Price: $115.00

AMS Member Price: $92.00

MAA Member Price: $103.50

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#### Supplemental Materials

# Lectures on the Fourier Transform and Its Applications

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*Brad G. Osgood*

This book is derived from lecture notes for a course on Fourier
analysis for engineering and science students at the advanced
undergraduate or beginning graduate level. Beyond teaching specific
topics and techniques—all of which are important in many areas of
engineering and science—the author's goal is to help engineering and
science students cultivate more advanced mathematical know-how and
increase confidence in learning and using mathematics, as well as
appreciate the coherence of the subject. He promises the readers a
little magic on every page.

The section headings are all recognizable to mathematicians, but
the arrangement and emphasis are directed toward students from other
disciplines. The material also serves as a foundation for advanced
courses in signal processing and imaging. There are over 200 problems,
many of which are oriented to applications, and a number use standard
software. An unusual feature for courses meant for engineers is a more
detailed and accessible treatment of distributions and the generalized
Fourier transform. There is also more coverage of higher-dimensional
phenomena than is found in most books at this level.

An instructor's manual for this title is available electronically
to those instructors who have adopted the textbook for classroom use.
Please send email to textbooks@ams.org for more
information.

#### Readership

Undergraduate students and graduate students (engineering students and math majors) and researchers (practicing engineers) interested in Fourier analysis.

#### Reviews & Endorsements

This is a lively introduction to the Fourier Integral...I like this book a lot. I think it's a good choice for students who are interested in signal processing, even though it omits a lot of pure-mathematics topics.

-- Allen Stenger, MAA Reviews

A thoroughly enjoyable yet careful mathematical perspective of the underlying concepts and many applications of modern signal analysis.

-- Les Atlas, University of Washington

Osgood leads his readers from the basics to the more sophisticated parts of applicable Fourier analysis with a lively style, a light touch on the technicalities, and an eye toward communications engineering. This book should be a great resource for students of mathematics, physics, and engineering alike.

-- Gerald B. Folland, University of Washington

Fourier analysis with a swing in its step.

-- Tom Körner, University of Cambridge

#### Table of Contents

# Table of Contents

## Lectures on the Fourier Transform and Its Applications

- Cover Cover11
- Title page iii4
- Preface xi12
- Thanks xv16
- Chapter 1. Fourier Series 118
- 1.1. Choices: Welcome Aboard 118
- 1.2. Periodic Phenomena 219
- 1.3. It All Adds Up 825
- 1.4. Two Examples and a Warning 2037
- 1.5. The Math, Part 1: A Convergence Result 2542
- 1.6. Fourier Series in Action 2845
- 1.7. The Math, Part 2: Orthogonality and Square Integrable Functions 4259
- 1.8. Appendix: Notes on the Convergence of Fourier Series 6077
- 1.9. Appendix: The Cauchy-Schwarz Inequality 7087
- Problems and Further Results 7592

- Chapter 2. Fourier Transform 99116
- Chapter 3. Convolution 159176
- 3.1. A * Is Born 159176
- 3.2. What Is Convolution, Really? 164181
- 3.3. Properties of Convolution: It’s a Lot Like Multiplication 167184
- 3.4. Convolution in Action I: A Little Bit on Filtering 169186
- 3.5. Convolution in Action II: Differential Equations 174191
- 3.6. Convolution in Action III: The Central Limit Theorem 185202
- 3.7. Heisenberg’s Inequality 202219
- Problems and Further Results 205222

- Chapter 4. Distributions and Their Fourier Transforms 229246
- 4.1. The Day of Reckoning 229246
- 4.2. The Best Functions for Fourier Transforms: Rapidly Decreasing Functions 235252
- 4.3. A Very Little on Integrals 244261
- 4.4. Distributions 249266
- 4.5. Defining Distributions 268285
- 4.6. Fluxions Finis: The End of Differential Calculus 285302
- 4.7. Convolutions and the Convolution Theorem 292309
- 4.8. Appendix: Windowing, Convolution, and Smoothing 298315
- 4.9. Epilog and Some References 311328
- Problems and Further Results 312329

- Chapter 5. 𝛿 Hard at Work 321338
- 5.1. Filters, Redux 322339
- 5.2. Diffraction: Sincs Live and in Pure Color 323340
- 5.3. X-Ray Diffraction 333350
- 5.4. The \boldmath\shah-Function on Its Own 335352
- 5.5. Periodic Distributions and Fourier Series 342359
- 5.6. A Formula for 𝛿 Applied to a Function, and a Mention of Pullbacks 344361
- 5.7. Cutting Off a \dl 347364
- 5.8. Appendix: How Special Is \shah? 348365
- Problems and Further Results 349366

- Chapter 6. Sampling and Interpolation 359376
- 6.1. Sampling sines and the Idea of a Bandlimited Signal 359376
- 6.2. Sampling and Interpolation for Bandlimited Signals 362379
- 6.3. Undersampling and Aliasing 371388
- 6.4. Finite Sampling for a Bandlimited Periodic Signal 380397
- 6.5. Appendix: Timelimited vs. Bandlimited Signals 386403
- 6.6. Appendix: Linear Interpolation via Convolution 388405
- 6.7. Appendix: Lagrange Interpolation 390407
- Problems and Further Results 391408

- Chapter 7. Discrete Fourier Transform 411428
- 7.1. The Modern World 411428
- 7.2. From Continuous to Discrete 412429
- 7.3. The Discrete Fourier Transform 414431
- 7.4. Notations and Conventions 1 416433
- 7.5. Two Grids, Reciprocally Related 420437
- 7.6. Getting to Know Your Discrete Fourier Transform 421438
- 7.7. Notations and Conventions 2 430447
- 7.8. Getting to Know Your DFT, Better 436453
- 7.9. The Discrete Rect and Its DFT 444461
- 7.10. Discrete Sampling and Interpolation 446463
- 7.11. The FFT Algorithm 449466
- Problems and Further Results 466483

- Chapter 8. Linear Time-Invariant Systems 483500
- 8.1. We Are All Systemizers Now 483500
- 8.2. Linear Systems 484501
- 8.3. Examples 487504
- 8.4. Cascading Linear Systems 492509
- 8.5. The Impulse Response, or the Deepest Fact in the Theory of Distributions Is Well Known to All Electrical Engineers 494511
- 8.6. Linear Time-Invariant (LTI) Systems 499516
- 8.7. The Fourier Transform and LTI Systems 504521
- 8.8. Causality 509526
- 8.9. The Hilbert Transform 511528
- 8.10. Filters Finis 519536
- 8.11. A Tribute: The Linear Millennium 530547
- Problems and Further Results 532549

- Chapter 9. 𝑛-Dimensional Fourier Transform 549566
- 9.1. Space, the Final Frontier 549566
- 9.2. Getting to Know Your Higher-Dimensional Fourier Transform 559576
- 9.3. A Little \dl Now, More Later 577594
- 9.4. Higher-Dimensional Fourier Series 581598
- 9.5. \boldmath\shah, Lattices, Crystals, and Sampling 592609
- 9.6. The Higher-Dimensional DFT 609626
- 9.7. Naked to the Bone 611628
- 9.8. Appendix: Line Impulses 626643
- 9.9. Appendix: Pullback of a Distribution 637654
- Problems and Further Results 642659

- Appendix A. A List of Mathematical Topics that Are Fair Game 661678
- Appendix B. Complex Numbers and Complex Exponentials 665682
- Appendix C. Geometric Sums 677694
- Index 681698
- Back Cover Back Cover1713