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Hardcover ISBN:  9781470441913 
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Product Code:  AMSTEXT/33.B 
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Hardcover ISBN:  9781470441913 
Product Code:  AMSTEXT/33 
List Price:  $99.00 
MAA Member Price:  $89.10 
AMS Member Price:  $79.20 
eBook ISBN:  9781470449766 
Product Code:  AMSTEXT/33.E 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $68.00 
Hardcover ISBN:  9781470441913 
eBook ISBN:  9781470449766 
Product Code:  AMSTEXT/33.B 
List Price:  $184.00 $141.50 
MAA Member Price:  $165.60 $127.35 
AMS Member Price:  $147.20 $113.20 

Book DetailsPure and Applied Undergraduate TextsVolume: 33; 2019; 693 ppMSC: Primary 42; 65; 94; 46
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 knowhow 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 higherdimensional phenomena than is found in most books at this level.
Ancillaries:
ReadershipUndergraduate students and graduate students (engineering students and math majors) and researchers (practicing engineers) interested in Fourier analysis.

Table of Contents

Cover

Title page

Preface

Thanks

Chapter 1. Fourier Series

1.1. Choices: Welcome Aboard

1.2. Periodic Phenomena

1.3. It All Adds Up

1.4. Two Examples and a Warning

1.5. The Math, Part 1: A Convergence Result

1.6. Fourier Series in Action

1.7. The Math, Part 2: Orthogonality and Square Integrable Functions

1.8. Appendix: Notes on the Convergence of Fourier Series

1.9. Appendix: The CauchySchwarz Inequality

Problems and Further Results

Chapter 2. Fourier Transform

2.1. A First Look at the Fourier Transform

2.2. Getting to Know Your Fourier Transform

2.3. Getting to Know Your Fourier Transform, Better

2.4. Different Definitions of the Fourier Transform, and What Happens to the Formulas

Problems and Further Results

Chapter 3. Convolution

3.1. A * Is Born

3.2. What Is Convolution, Really?

3.3. Properties of Convolution: It’s a Lot Like Multiplication

3.4. Convolution in Action I: A Little Bit on Filtering

3.5. Convolution in Action II: Differential Equations

3.6. Convolution in Action III: The Central Limit Theorem

3.7. Heisenberg’s Inequality

Problems and Further Results

Chapter 4. Distributions and Their Fourier Transforms

4.1. The Day of Reckoning

4.2. The Best Functions for Fourier Transforms: Rapidly Decreasing Functions

4.3. A Very Little on Integrals

4.4. Distributions

4.5. Defining Distributions

4.6. Fluxions Finis: The End of Differential Calculus

4.7. Convolutions and the Convolution Theorem

4.8. Appendix: Windowing, Convolution, and Smoothing

4.9. Epilog and Some References

Problems and Further Results

Chapter 5. 𝛿 Hard at Work

5.1. Filters, Redux

5.2. Diffraction: Sincs Live and in Pure Color

5.3. XRay Diffraction

5.4. The \boldmath\shahFunction on Its Own

5.5. Periodic Distributions and Fourier Series

5.6. A Formula for 𝛿 Applied to a Function, and a Mention of Pullbacks

5.7. Cutting Off a \dl

5.8. Appendix: How Special Is \shah?

Problems and Further Results

Chapter 6. Sampling and Interpolation

6.1. Sampling sines and the Idea of a Bandlimited Signal

6.2. Sampling and Interpolation for Bandlimited Signals

6.3. Undersampling and Aliasing

6.4. Finite Sampling for a Bandlimited Periodic Signal

6.5. Appendix: Timelimited vs. Bandlimited Signals

6.6. Appendix: Linear Interpolation via Convolution

6.7. Appendix: Lagrange Interpolation

Problems and Further Results

Chapter 7. Discrete Fourier Transform

7.1. The Modern World

7.2. From Continuous to Discrete

7.3. The Discrete Fourier Transform

7.4. Notations and Conventions 1

7.5. Two Grids, Reciprocally Related

7.6. Getting to Know Your Discrete Fourier Transform

7.7. Notations and Conventions 2

7.8. Getting to Know Your DFT, Better

7.9. The Discrete Rect and Its DFT

7.10. Discrete Sampling and Interpolation

7.11. The FFT Algorithm

Problems and Further Results

Chapter 8. Linear TimeInvariant Systems

8.1. We Are All Systemizers Now

8.2. Linear Systems

8.3. Examples

8.4. Cascading Linear Systems

8.5. The Impulse Response, or the Deepest Fact in the Theory of Distributions Is Well Known to All Electrical Engineers

8.6. Linear TimeInvariant (LTI) Systems

8.7. The Fourier Transform and LTI Systems

8.8. Causality

8.9. The Hilbert Transform

8.10. Filters Finis

8.11. A Tribute: The Linear Millennium

Problems and Further Results

Chapter 9. 𝑛Dimensional Fourier Transform

9.1. Space, the Final Frontier

9.2. Getting to Know Your HigherDimensional Fourier Transform

9.3. A Little \dl Now, More Later

9.4. HigherDimensional Fourier Series

9.5. \boldmath\shah, Lattices, Crystals, and Sampling

9.6. The HigherDimensional DFT

9.7. Naked to the Bone

9.8. Appendix: Line Impulses

9.9. Appendix: Pullback of a Distribution

Problems and Further Results

Appendix A. A List of Mathematical Topics that Are Fair Game

Appendix B. Complex Numbers and Complex Exponentials

B.1. Complex Numbers

B.2. The Complex Exponential and Euler’s Formula

B.3. Further Applications of Euler’s Formula

Problems and Further Results

Appendix C. Geometric Sums

Problems and Further Results

Index

Back Cover


Additional Material

Reviews

The author is obviously enjoying the material and wants to transfer this to the reader with some personal anecdotes in footnotes or some witty winking like when he writes as title of a section 'A * is born' when he is about to discuss convolution. He takes the reader along by explicitly formulating the likely appreciation of the student or a question that the student might ask at some point. He also tries to catch the interest of the engineering students by discussing relations with differential equations (heat equation), but also quantum mechanics, diffraction, crystallography, systems theory, tomography, and others, in the text as well as in exercises.
Adhemar François Bultheel, zbMATH 
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 puremathematics 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


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 Book Details
 Table of Contents
 Additional Material
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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 knowhow 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 higherdimensional phenomena than is found in most books at this level.
Ancillaries:
Undergraduate students and graduate students (engineering students and math majors) and researchers (practicing engineers) interested in Fourier analysis.

Cover

Title page

Preface

Thanks

Chapter 1. Fourier Series

1.1. Choices: Welcome Aboard

1.2. Periodic Phenomena

1.3. It All Adds Up

1.4. Two Examples and a Warning

1.5. The Math, Part 1: A Convergence Result

1.6. Fourier Series in Action

1.7. The Math, Part 2: Orthogonality and Square Integrable Functions

1.8. Appendix: Notes on the Convergence of Fourier Series

1.9. Appendix: The CauchySchwarz Inequality

Problems and Further Results

Chapter 2. Fourier Transform

2.1. A First Look at the Fourier Transform

2.2. Getting to Know Your Fourier Transform

2.3. Getting to Know Your Fourier Transform, Better

2.4. Different Definitions of the Fourier Transform, and What Happens to the Formulas

Problems and Further Results

Chapter 3. Convolution

3.1. A * Is Born

3.2. What Is Convolution, Really?

3.3. Properties of Convolution: It’s a Lot Like Multiplication

3.4. Convolution in Action I: A Little Bit on Filtering

3.5. Convolution in Action II: Differential Equations

3.6. Convolution in Action III: The Central Limit Theorem

3.7. Heisenberg’s Inequality

Problems and Further Results

Chapter 4. Distributions and Their Fourier Transforms

4.1. The Day of Reckoning

4.2. The Best Functions for Fourier Transforms: Rapidly Decreasing Functions

4.3. A Very Little on Integrals

4.4. Distributions

4.5. Defining Distributions

4.6. Fluxions Finis: The End of Differential Calculus

4.7. Convolutions and the Convolution Theorem

4.8. Appendix: Windowing, Convolution, and Smoothing

4.9. Epilog and Some References

Problems and Further Results

Chapter 5. 𝛿 Hard at Work

5.1. Filters, Redux

5.2. Diffraction: Sincs Live and in Pure Color

5.3. XRay Diffraction

5.4. The \boldmath\shahFunction on Its Own

5.5. Periodic Distributions and Fourier Series

5.6. A Formula for 𝛿 Applied to a Function, and a Mention of Pullbacks

5.7. Cutting Off a \dl

5.8. Appendix: How Special Is \shah?

Problems and Further Results

Chapter 6. Sampling and Interpolation

6.1. Sampling sines and the Idea of a Bandlimited Signal

6.2. Sampling and Interpolation for Bandlimited Signals

6.3. Undersampling and Aliasing

6.4. Finite Sampling for a Bandlimited Periodic Signal

6.5. Appendix: Timelimited vs. Bandlimited Signals

6.6. Appendix: Linear Interpolation via Convolution

6.7. Appendix: Lagrange Interpolation

Problems and Further Results

Chapter 7. Discrete Fourier Transform

7.1. The Modern World

7.2. From Continuous to Discrete

7.3. The Discrete Fourier Transform

7.4. Notations and Conventions 1

7.5. Two Grids, Reciprocally Related

7.6. Getting to Know Your Discrete Fourier Transform

7.7. Notations and Conventions 2

7.8. Getting to Know Your DFT, Better

7.9. The Discrete Rect and Its DFT

7.10. Discrete Sampling and Interpolation

7.11. The FFT Algorithm

Problems and Further Results

Chapter 8. Linear TimeInvariant Systems

8.1. We Are All Systemizers Now

8.2. Linear Systems

8.3. Examples

8.4. Cascading Linear Systems

8.5. The Impulse Response, or the Deepest Fact in the Theory of Distributions Is Well Known to All Electrical Engineers

8.6. Linear TimeInvariant (LTI) Systems

8.7. The Fourier Transform and LTI Systems

8.8. Causality

8.9. The Hilbert Transform

8.10. Filters Finis

8.11. A Tribute: The Linear Millennium

Problems and Further Results

Chapter 9. 𝑛Dimensional Fourier Transform

9.1. Space, the Final Frontier

9.2. Getting to Know Your HigherDimensional Fourier Transform

9.3. A Little \dl Now, More Later

9.4. HigherDimensional Fourier Series

9.5. \boldmath\shah, Lattices, Crystals, and Sampling

9.6. The HigherDimensional DFT

9.7. Naked to the Bone

9.8. Appendix: Line Impulses

9.9. Appendix: Pullback of a Distribution

Problems and Further Results

Appendix A. A List of Mathematical Topics that Are Fair Game

Appendix B. Complex Numbers and Complex Exponentials

B.1. Complex Numbers

B.2. The Complex Exponential and Euler’s Formula

B.3. Further Applications of Euler’s Formula

Problems and Further Results

Appendix C. Geometric Sums

Problems and Further Results

Index

Back Cover

The author is obviously enjoying the material and wants to transfer this to the reader with some personal anecdotes in footnotes or some witty winking like when he writes as title of a section 'A * is born' when he is about to discuss convolution. He takes the reader along by explicitly formulating the likely appreciation of the student or a question that the student might ask at some point. He also tries to catch the interest of the engineering students by discussing relations with differential equations (heat equation), but also quantum mechanics, diffraction, crystallography, systems theory, tomography, and others, in the text as well as in exercises.
Adhemar François Bultheel, zbMATH 
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 puremathematics 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