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Fourier Series, Fourier Transforms, and Function Spaces: A Second Course in Analysis

Tim Hsu San José State University, San José, CA
MAA Press: An Imprint of the American Mathematical Society
Available Formats:
Hardcover ISBN: 978-1-4704-5145-5
Product Code: TEXT/59
List Price: $79.00 MAA Member Price:$59.25
AMS Member Price: $59.25 Electronic ISBN: 978-1-4704-5519-4 Product Code: TEXT/59.E List Price:$79.00
MAA Member Price: $59.25 AMS Member Price:$59.25
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List Price: $118.50 MAA Member Price:$88.88
AMS Member Price: $88.88 Click above image for expanded view Fourier Series, Fourier Transforms, and Function Spaces: A Second Course in Analysis Tim Hsu San José State University, San José, CA MAA Press: An Imprint of the American Mathematical Society Available Formats:  Hardcover ISBN: 978-1-4704-5145-5 Product Code: TEXT/59  List Price:$79.00 MAA Member Price: $59.25 AMS Member Price:$59.25
 Electronic ISBN: 978-1-4704-5519-4 Product Code: TEXT/59.E
 List Price: $79.00 MAA Member Price:$59.25 AMS Member Price: $59.25 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$118.50 MAA Member Price: $88.88 AMS Member Price:$88.88
• Book Details

AMS/MAA Textbooks
Volume: 592020; 354 pp
MSC: 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 self-study or an inquiry-based 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
• 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. Complex-valued 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. Sturm-Liouville 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. Continuous-valued 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

• 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
• Requests

Review Copy – for reviewers who would like to review an AMS book
Desk Copy – for instructors who have adopted an AMS textbook for a course
Examination Copy – for faculty considering an AMS textbook for a course
Accessibility – to request an alternate format of an AMS title
Volume: 592020; 354 pp
MSC: 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 self-study or an inquiry-based 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
• 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. Complex-valued 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. Sturm-Liouville 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. Continuous-valued 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
Review Copy – for reviewers who would like to review an AMS book
Desk Copy – for instructors who have adopted an AMS textbook for a course
Examination Copy – for faculty considering an AMS textbook for a course
Accessibility – to request an alternate format of an AMS title
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