Course Description:

(i) Fourier series and their properties;
(ii) inner products and orthogonality of functions;
(iii) separation of variations of boundary value problems;
(iv) Bessel functions;
(v) Orthogonal polynomials;
(vi) Fourier transforms;
(vii) Generalized functions (if schedule allows);
(viii) Green's functions (if schedule allows)

Instructor:

Aaron Nung Kwan Yip

Department of Mathematics

Purdue University

Contact Information:

Office: MATH 432

Contact info: click here

Lecture Time and Place:

52000-001 (16863) T, Th 12:00pm - 01:15pm, SCHM 123

I have also created a zoom session (click here) for our class. You can also access this link by going to Brightspace/MA520 Main page/Content (upper left corner)/Course Materials. (Occasionally, it is convenient to have an alternative method of delivering lectures due to my travels or unexpected events. Of course, this will be kept to a minimum.)

Office Hours:

Tuesdays and Wednesdays, 5:00pm-6:00pm (in MATH 432)
Unless otherwise stated, the above office hours will be in person. Additional hours can also be arranged/requested/by appointment, either in person or in zoom (same link as previous).

Textbook:

Main Text:
[F] Fourier Analysis and Its Applications, by G. B. Folland
(Note that there is only one version of the book. The one published by Brooks/Cole in 1992 and the one reprinted by the American Mathematical Society in 2009 are exactly the same.)

Additional References:
The following resources might give you some more background or different perspectives:
[S] Partial Differential Equations, an Introduction, W. Strauss;
[K] Fourier Analysis, Korner;
[DM] Fourier Series and Integrals, H. Dym and H. P. McKean;
[SS] Fourier Analysis: An Introduction, E. M. Stein and R. Shakarchi

There are also a tremendous amount of other references and web-resources on Fourier series, Fourier analysis, and BVPs.
Feel free to pick your choice. (But beware of their (lack of) quality control.)

Prerequisites:

Linear algebra (e.g. MA265, 351), differential equations (ODEs) (e.g. MA266, 366), mathematical analysis (e.g. MA341) (concepts of convergence) and mathematical maturity.

Homework:

Weekly homework will be gradually assigned as the course progress. Normally it is due on Thursday, in class.
Please refer to the course announcement below.

Steps must be shown to explain your answers. No credit will be given for just writing down the answers, even if it is correct.

As a rule of thumb, you should only use those methods that have been covered in class. If you use some other methods ``for the sake of convenience'', at our discretion, we might not give you any credit. You have the right to contest. In that event, you will be asked to explain your answer using only what has been covered in class up to the point of time of the assignment.

Please staple all loose sheets of your homework to prevent 5% penalty.

Please resolve any error in the grading (hws and tests) WINTHIN ONE WEEK after the return of each homework and exam.

No late homeworks are accepted (in principle).

You are encouraged to discuss the homework problems with your classmates but all your handed-in homeworks must be your own work.

Examinations:

Midterm Tests: Feb 15 and Apr 4, both on Thursday, in class

Final Exam: During Final Exam Week

Grading Policy:

Homeworks (30%)

Midterm Test (40%=20%+20%)

Final Exam (30%)

The following is departmental policy for the grade cut-offs:
97% of the total points in this course are guaranteed an A+,
93% an A,
90% an A-,
87% a B+,
83% a B
80% a B-,
77% a C+,
73% a C,
70% a C-,
67% a D+,
63% a D, and
60% a D-.
For each of these grades, it's possible that at the end of the semester a lower percentage will be enough to achieve that grade.

You are expected to observe academic honesty to the highest standard. Any form of cheating will automatically lead to an F grade, plus any other disciplinary action, deemed appropriate.

Nondiscrimination Statement:

This class, as part of Purdue University's educational endeavor, is committed to maintaining a community which recognizes and values the inherent worth and dignity of every person; fosters tolerance, sensitivity, understanding, and mutual respect among its members; and encourages each individual to strive to reach his or her own potential.

Student Rights:

Any student who has substantial reason to believe that another person is threatening the safety of others by not complying with Protect Purdue protocols is encouraged to report the behavior to and discuss the next steps with their instructor. Students also have the option of reporting the behavior to the Office of the Student Rights and Responsibilities. See also Purdue University Bill of Student Rights and the Violent Behavior Policy under University Resources in Brightspace.

Accommodations for Students with Disabilities and Academic Adjustment:

Purdue University strives to make learning experiences accessible to all participants. If you anticipate or experience physical or academic barriers based on disability, you are also encouraged to contact the Disability Resource Center (DRC) at: drc@purdue.edu or by phone at 765-494-1247.

If you have been certified by the DRC as eligible for accommodations, you should contact me to discuss your accommodations as soon as possible. Click here for instructions for sending your Course Accessibility Letter to me. See also Courses: ADA Information for further information from the Department of Mathematics.

Campus Emergency:

In the event of a major campus emergency or circumstances beyond the instructor's control, course requirements, deadlines and grading percentages are subject to change. Check your email and this course web page for such information.

See also Emergency Preparedness and Planning for campus wide updates.

More information on University Policies:

See your MA520 course homepage in Brightspace.
Content (tab at upper left corner): Student Support and Resources, and University Policies and Statements.

Course Progress and Announcement:

(You should consult this section regularly, for homework assignments, additional materials and announcements.)

NOTATION MATTERS!
A clear understanding of notations is one of the keys to full appreciation of any mathematical subject.

READ THE TEXTBOOK!
Get used to how mathematics are formulated and presented.

My MOTTO on the use of technology (which I use often):
IF TECHNOLOGY HELPS YOU UNDERSTRAND, BY ALL MEANS USE IT. OTHERWISE, USE IT AT YOUR OWN RISK!
For the homework, I believe all the problems should be and can be done by hand. In order to get full credit, sufficient steps must be shown.
You are welcome to use technology to check your answers.

BEWARE THAT DURING THE TESTS AND EXAM, NO TECHNOLOGY WILL BE ALLOWED.

---

WEEK 1: Jan 9, 11

[F Chapter 1]
Examples of PDEs: heat/diffusion equation, wave equation, Laplace and Poisson equations;
Separation of variables, expansion using eigenvectors/eigenfunctions.

Note: Examples of PDEs
Note: (Review) Solutions of first and second order linear differential equations
(This material can be found in any first course or book in ODEs, for example, Boyce-DiPrima.)
Note: Separation of variables: matrix example

Homework 1: due Thursday, Jan. 18th, in class.

---

WEEK 2: Jan 16, 18

[F Chapter 2]
Cosine and sine functions:
Orthogonality property and eigenfunctions of (one-dimensional) Laplacian;
Fourier series of (2pi-) periodic functions, computation of the coefficients.
Properties of Fourier series coefficients: Riemann-Lebesgue Lemma, rate of decay, and smoothness of function;
Bessel's inequality and orthogonality property of trigonometric functions.

Note: Examples of Fourier Series
Note (some history): The Analytical Theory of Heat (Original treatise by Fourier)
Note (some history): J. B. Fourier - On the Occasion of His Two Hundred Birthday
Note (some history): Fourier Series - The Genesis and Evolution of a Theory

Homework 2: due Thursday, Jan. 25th, in class.

---

WEEK 3: Jan 23, 25

[F Chapter 2]
Fourier series using complex exponents;
[F Theorem 2.1, p.35] Pointwise convergence of Fourier series: Dirichlet kernel;
[F Theorem 2.5, p.41] Uniform convergence of Fourier series.

Note: Fourier series using complex numbers
Note: Bessel Inequality using complex numbers
Note: Convergence of Fourier series

Homework 3: due Thursday, Feb. 1st, in class.

---

WEEK 4: Jan 30, Feb 1

[F, Chapter 3]
Inner product space,
Cauchy-Schwarz Inequality and Equality,
Triangle Inequality and Equality
(When do the above inequalities become equalities?)
Orthogonal vectors and their applications:

Pythagoras Theorem,

Linear combinations using orthogonal vectors,

(Orthogonal) projection and least square approximation,

Bessel's Inequality vs Parseval's Identity (Equality)

Note: Inner Product Space

Homework 4: due Thursday, Feb 8th, in class.
Section 3.2 (p.71): #1, 2, 3;
Section 3.3 (p.79): #1, 2, 9, 10;
Section 3.4 (p.85): #2, 3, 7.

---

WEEK 5: Feb 6, 8

[F, Chapter 3]
Proof of Cauchy-Schwarz and Triangle Inequality
(cont.) Linear combinations using orthogonal vectors,
(cont.) (Orthogonal) projection and least square approximation,
(cont.) Bessel's Inequality vs Parseval's Identity (Equality)
Different types of Convergence:

[F, Theorem 2.1, p.35] Pointwise convergence: \text{when $f'(x)$ exist at $x$.}"border="0" align="middle">

[F, Theorem 2.5, p.41] Uniform convergence:

[F, Theorem 3.5, p.78] L^2 convergence:
[F, Theorem 3.4, p.77] Completeness of orthogonal set.

---

WEEK 6: Feb 13, 15
Midterm One, Feb 15, in class
(Review on Tuesday, Feb 13.)

Materials covered: Chapters 1-3.4
No electronic devices and no formula sheet are allowed.
At my discretion, some integration formulas might be provided inside the exam but you should remember all the `conceptual' formulas.
(My rule of thumb is: if you are debating whether to remember a formula, then remember it.)

Past Exam 1, Past Exam 2, Past Exam 3
Selected Homework Solutions
(Going over the textbook, homework problems, and lecture materials are the best way to prepare for this (and any) exam.)

Midterm One Solution

---

WEEK 7: Feb 20, 22



Note: Solution of heat equation - homogenenous equation
Note: Solution of heat equation - inhomogenenous equation

---

WEEK 8: Feb 27, 29



Note: Laplace equation in circular domain - I
Note: Poisson kernel

---

WEEK 9: Mar 5, 7



Note: Laplace equation in circular domain - II (4.4) #7

---

(Spring Break: Mar 11 - 15)
WEEK 10: Mar 19, 21

[F Chapter 5.1, 5.2, 5.3]
Bessel equations, power series expansion, linear independence,
Bessel functions of first and second kinds
Properties, asymptotics of Bessel functions

Note: Bessel equations: derivation
Note: Bessel functions: properties

Homework 8: due Thursday, Mar 28th, in 11:59pm, in Gradescope.
Section 5.1 (p.132): #3, 4, 5;
Section 5.2 (p.137): #1, 2;
Section 5.3 (p.143): #2;
Section 5.4 (p.148): #1, 4, 9;
Additional problem: explicitly write down the first five non-zero terms of .

---

WEEK 11: Mar 26, 28

[F Chapter 5.3, 5.4, 5.5]
Zeros of Bessel functions and their connection to eigenvalues of Laplacian in circular domains
Bessel functions and eigenfunctions of Laplacian in circular domains
Applications of Bessel functions in solving PDEs.

Note: Bessel functions: zeros, eigenvalues
Note: Bessel functions: eigenfunctions

---

WEEK 12: Apr 2, 4
Midterm One, Apr 4, in class
(Review on Tuesday, Apr 2.)

Materials covered: Chapters 3.5, 3.6, 4, 5.
No electronic devices and no formula sheet are allowed.
At my discretion, some formulas might be provided inside the exam but you should remember all the `conceptual' formulas.
(My rule of thumb is: if you are debating whether to remember a formula, then remember it.)

Past Exam A, Past Exam B, Past Exam C
Selected Homework (5-8) Solutions
(Going over the textbook, homework problems, and lecture materials are the best way to prepare for this (and any) exam.)

Office hours this week (online):
Tuesday, Apr. 2, 10:30am - 11:30am, Wednesday, Apr 3, 10:00am-12:00pm
(Additional times can be requested.)

Midterm Two Solution

---

WEEK 13: Apr 9, 11

[F Chapter 6] Orthogonal polynomials: Rodrigues formula and differential equations;
[6.2, 6.3] Legendre polynomial, spherical harmonics:
Application in solving PDE in spherical coordinates;
[6.4] Hermite polynomials:
Application in solving Schrodinger equation for one dimensional quantum harmonic oscillator;
[6.5] Laguerre polynomials:
Application in solving Schrodinger equation for the hydrogen atom.

Mathematics of musical instruments (Hall and Josic)
Acoustics of ancient Chinese bells (Shen)
Physics of Kettledrums (Rossing)

Note: Orthogonal polynomials

Homework 9: due Sunday, Apr. 21st, 11:59pm, submit in Gradescope.

---

WEEK 14: Apr 16, 18

[F, 7.1, 7.2] Fourier transform and its properties.
Analogy with Fourier series: spatial vs frequency variables;
Solving heat equation using Fourier series and Fourier transforms;
Convolution in space vs multiplication in frequency
Duality and Plancherel's Theorem;
Scaling, shift, and differentiation in spatial and frequency variables;
Examples of Fourier Transforms;
Heat kernel and its properties.

Note: Basic properties and examples of Fourier transforms

---

WEEK 15: Apr 23, 25

---

WEEK 16: Apr 29 - May 4
Final Exam Week