Hardcover ISBN:  9781470451455 
Product Code:  TEXT/59 
List Price:  $79.00 
MAA Member Price:  $59.25 
AMS Member Price:  $59.25 
Electronic ISBN:  9781470455194 
Product Code:  TEXT/59.E 
List Price:  $79.00 
MAA Member Price:  $59.25 
AMS Member Price:  $59.25 

Book DetailsAMS/MAA TextbooksVolume: 59; 2020; 354 ppMSC: Primary 26; 42;
Fourier Series, Fourier Transforms, and Function Spaces is designed as a textbook for a second course or capstone course in analysis for advanced undergraduate or beginning graduate students. By assuming the existence and properties of the Lebesgue integral, this book makes it possible for students who have previously taken only one course in real analysis to learn Fourier analysis in terms of Hilbert spaces, allowing for both a deeper and more elegant approach. This approach also allows junior and senior undergraduates to study topics like PDEs, quantum mechanics, and signal processing in a rigorous manner.
Students interested in statistics (time series), machine learning (kernel methods), mathematical physics (quantum mechanics), or electrical engineering (signal processing) will find this book useful. With 400 problems, many of which guide readers in developing key theoretical concepts themselves, this text can also be adapted to selfstudy or an inquirybased approach. Finally, of course, this text can also serve as motivation and preparation for students going on to further study in analysis.ReadershipUndergraduate and graduate students and researchers interested in analysis, differential equations, and applied math.

Table of Contents

Cover

Title page

Copyright

Contents

Introduction

Chapter 1. Overture

1.1. Mathematical motivation: Series of functions

1.2. Physical motivation: Acoustics

Part 1 Complex functions of a real variable

Chapter 2. Real and complex numbers

2.1. Axioms for the real numbers

2.2. Complex numbers

2.3. Metrics and metric spaces

2.4. Sequences in C and other metric spaces

2.5. Completeness in metric spaces

2.6. The topology of metric spaces

Chapter 3. Complexvalued calculus

3.1. Continuity and limits

3.2. Differentiation

3.3. The Riemann integral: Definition

3.4. The Riemann integral: Properties

3.5. The Fundamental Theorem of Calculus

3.6. Other results from calculus

Chapter 4. Series of functions

4.1. Infinite series

4.2. Sequences and series of functions

4.3. Uniform convergence

4.4. Power series

4.5. Exponential and trigonometric functions

4.6. More about exponential functions

4.7. The Schwartz space

4.8. Integration on R

Part 2 Fourier series and Hilbert spaces

Chapter 5. The idea of a function space

5.1. Which clock keeps better time?

5.2. Function spaces and metrics

5.3. Dot products

Chapter 6. Fourier series

6.1. Fourier polynomials

6.2. Fourier series

6.3. Real Fourier series

6.4. Convergence of Fourier series* of differentiable functions

Chapter 7. Hilbert spaces

7.1. Inner product spaces

7.2. Normed spaces

7.3. Orthogonal sets and bases

7.4. The Lebesgue integral: Measure zero

7.5. The Lebesgue integral: Axioms

7.6. Hilbert spaces

Chapter 8. Convergence of Fourier series

8.1. Fourier series in 𝐿²(𝑆¹)

8.2. Convolutions

8.3. Dirac kernels

8.4. Proof of the Inversion Theorem

8.5. Applications of Fourier series

Part 3 Operators and differential equations

Chapter 9. PDEs and diagonalization

9.1. Some PDEs from classical physics

9.2. Schrödinger’s equation

9.3. Diagonalization

Chapter 10. Operators on Hilbert spaces

10.1. Operators on Hilbert spaces

10.2. Hermitian and positive operators

10.3. Eigenvectors and eigenvalues

10.4. Eigenbases

Chapter 11. Eigenbases and differential equations

11.1. The heat equation on the circle

11.2. The eigenbasis method

11.3. The wave equation on the circle

11.4. Boundary value problems

11.5. Legendre polynomials

11.6. Hermite functions

11.7. The quantum harmonic oscillator

11.8. SturmLiouville theory

Part 4 The Fourier transform and beyond

Chapter 12. The Fourier transform

12.1. The big picture

12.2. Convolutions, Dirac kernels, and calculus on R

12.3. The Fourier transform on schwartz

12.4. Inversion and the Plancherel theorem

12.5. The 𝐿² Fourier transform

Chapter 13. Applications of the Fourier transform

13.1. A table of Fourier transforms

13.2. Linear differential equations with constant coefficients

13.3. The heat and wave equations on R

13.4. An eigenbasis for the Fourier transform

13.5. Continuousvalued quantum observables

13.6. Poisson summation and theta functions

13.7. Miscellaneous applications of the Fourier transform

Chapter 14. What’s next?

14.1. What’s next: More analysis

14.2. What’s next: Signal processing and distributions

14.3. What’s next: Wavelets

14.4. What’s next: Quantum mechanics

14.5. What’s next: Spectra and number theory

14.6. What’s next: Harmonic analysis on groups

Appendices

Appendix A. Rearrangements of series

Appendix B. Linear algebra

Appendix C. Bump functions

Appendix D. Suggestions for problems

Bibliography

Index

Index

Back Cover


Additional Material

Reviews

This is an interesting take on the second course in analysis: rather than the Lebesgue integral, we study Fourier analysis and applications. The book is well done and makes a strong case for this approach. The Introduction (which is the Introduction for the Instructor) is one of the best I.ve read, and you should definitely study if you are considering adopting the book. It explains very clearly the goals of the book, the limitations of this approach, and some other unusual features of the book.
Allen Stenger, MAA Reviews


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Fourier Series, Fourier Transforms, and Function Spaces is designed as a textbook for a second course or capstone course in analysis for advanced undergraduate or beginning graduate students. By assuming the existence and properties of the Lebesgue integral, this book makes it possible for students who have previously taken only one course in real analysis to learn Fourier analysis in terms of Hilbert spaces, allowing for both a deeper and more elegant approach. This approach also allows junior and senior undergraduates to study topics like PDEs, quantum mechanics, and signal processing in a rigorous manner.
Students interested in statistics (time series), machine learning (kernel methods), mathematical physics (quantum mechanics), or electrical engineering (signal processing) will find this book useful. With 400 problems, many of which guide readers in developing key theoretical concepts themselves, this text can also be adapted to selfstudy or an inquirybased approach. Finally, of course, this text can also serve as motivation and preparation for students going on to further study in analysis.
Undergraduate and graduate students and researchers interested in analysis, differential equations, and applied math.

Cover

Title page

Copyright

Contents

Introduction

Chapter 1. Overture

1.1. Mathematical motivation: Series of functions

1.2. Physical motivation: Acoustics

Part 1 Complex functions of a real variable

Chapter 2. Real and complex numbers

2.1. Axioms for the real numbers

2.2. Complex numbers

2.3. Metrics and metric spaces

2.4. Sequences in C and other metric spaces

2.5. Completeness in metric spaces

2.6. The topology of metric spaces

Chapter 3. Complexvalued calculus

3.1. Continuity and limits

3.2. Differentiation

3.3. The Riemann integral: Definition

3.4. The Riemann integral: Properties

3.5. The Fundamental Theorem of Calculus

3.6. Other results from calculus

Chapter 4. Series of functions

4.1. Infinite series

4.2. Sequences and series of functions

4.3. Uniform convergence

4.4. Power series

4.5. Exponential and trigonometric functions

4.6. More about exponential functions

4.7. The Schwartz space

4.8. Integration on R

Part 2 Fourier series and Hilbert spaces

Chapter 5. The idea of a function space

5.1. Which clock keeps better time?

5.2. Function spaces and metrics

5.3. Dot products

Chapter 6. Fourier series

6.1. Fourier polynomials

6.2. Fourier series

6.3. Real Fourier series

6.4. Convergence of Fourier series* of differentiable functions

Chapter 7. Hilbert spaces

7.1. Inner product spaces

7.2. Normed spaces

7.3. Orthogonal sets and bases

7.4. The Lebesgue integral: Measure zero

7.5. The Lebesgue integral: Axioms

7.6. Hilbert spaces

Chapter 8. Convergence of Fourier series

8.1. Fourier series in 𝐿²(𝑆¹)

8.2. Convolutions

8.3. Dirac kernels

8.4. Proof of the Inversion Theorem

8.5. Applications of Fourier series

Part 3 Operators and differential equations

Chapter 9. PDEs and diagonalization

9.1. Some PDEs from classical physics

9.2. Schrödinger’s equation

9.3. Diagonalization

Chapter 10. Operators on Hilbert spaces

10.1. Operators on Hilbert spaces

10.2. Hermitian and positive operators

10.3. Eigenvectors and eigenvalues

10.4. Eigenbases

Chapter 11. Eigenbases and differential equations

11.1. The heat equation on the circle

11.2. The eigenbasis method

11.3. The wave equation on the circle

11.4. Boundary value problems

11.5. Legendre polynomials

11.6. Hermite functions

11.7. The quantum harmonic oscillator

11.8. SturmLiouville theory

Part 4 The Fourier transform and beyond

Chapter 12. The Fourier transform

12.1. The big picture

12.2. Convolutions, Dirac kernels, and calculus on R

12.3. The Fourier transform on schwartz

12.4. Inversion and the Plancherel theorem

12.5. The 𝐿² Fourier transform

Chapter 13. Applications of the Fourier transform

13.1. A table of Fourier transforms

13.2. Linear differential equations with constant coefficients

13.3. The heat and wave equations on R

13.4. An eigenbasis for the Fourier transform

13.5. Continuousvalued quantum observables

13.6. Poisson summation and theta functions

13.7. Miscellaneous applications of the Fourier transform

Chapter 14. What’s next?

14.1. What’s next: More analysis

14.2. What’s next: Signal processing and distributions

14.3. What’s next: Wavelets

14.4. What’s next: Quantum mechanics

14.5. What’s next: Spectra and number theory

14.6. What’s next: Harmonic analysis on groups

Appendices

Appendix A. Rearrangements of series

Appendix B. Linear algebra

Appendix C. Bump functions

Appendix D. Suggestions for problems

Bibliography

Index

Index

Back Cover

This is an interesting take on the second course in analysis: rather than the Lebesgue integral, we study Fourier analysis and applications. The book is well done and makes a strong case for this approach. The Introduction (which is the Introduction for the Instructor) is one of the best I.ve read, and you should definitely study if you are considering adopting the book. It explains very clearly the goals of the book, the limitations of this approach, and some other unusual features of the book.
Allen Stenger, MAA Reviews