**AMS/MAA Textbooks**

Volume: 59;
2020;
354 pp;
Hardcover

MSC: Primary 26; 42;

**Print ISBN: 978-1-4704-5145-5
Product Code: TEXT/59**

List Price: $79.00

AMS Member Price: $59.25

MAA Member Price: $59.25

**Electronic ISBN: 978-1-4704-5519-4
Product Code: TEXT/59.E**

List Price: $79.00

AMS Member Price: $59.25

MAA Member Price: $59.25

#### You may also like

#### Supplemental Materials

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

Share this page
*Tim Hsu*

MAA Press: An Imprint of the American Mathematical Society

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.

#### Readership

Undergraduate and graduate students and researchers interested in analysis, differential equations, and applied math.

#### Reviews & Endorsements

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

#### Table of Contents

# Table of Contents

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

- Cover Cover11
- Title page iii5
- Copyright iv6
- Contents vii9
- Introduction xi13
- Chapter 1. Overture 117
- Part 1 Complex functions of a real variable 723
- Part 2 Fourier series and Hilbert spaces 113129
- Part 3 Operators and differential equations 201217
- Chapter 9. PDEs and diagonalization 203219
- Chapter 10. Operators on Hilbert spaces 213229
- Chapter 11. Eigenbases and differential equations 229245
- 11.1. The heat equation on the circle 230246
- 11.2. The eigenbasis method 235251
- 11.3. The wave equation on the circle 237253
- 11.4. Boundary value problems 244260
- 11.5. Legendre polynomials 250266
- 11.6. Hermite functions 254270
- 11.7. The quantum harmonic oscillator 257273
- 11.8. Sturm-Liouville theory 259275

- Part 4 The Fourier transform and beyond 261277
- Chapter 12. The Fourier transform 263279
- Chapter 13. Applications of the Fourier transform 281297
- 13.1. A table of Fourier transforms 281297
- 13.2. Linear differential equations with constant coefficients 283299
- 13.3. The heat and wave equations on R 285301
- 13.4. An eigenbasis for the Fourier transform 289305
- 13.5. Continuous-valued quantum observables 291307
- 13.6. Poisson summation and theta functions 296312
- 13.7. Miscellaneous applications of the Fourier transform 301317

- Chapter 14. What’s next? 305321

- Appendices 317333