## Math 480 Fourier Analysis - Introduction to the course

Prerequisites: Ideally, you should have at least a 2.5 in both Math 309 and Math 328, or at least a 2.5 in Math 335. An introduction to Fourier series and their use in solving boundary value problems may substitute for Math 309. If you have not taken 328, you will have to fill in some prerequisite ideas at a few points during the course. If you have grades below 3.0 in Math 300 and/or 327, you may find this course extremely challenging.

Course description. Solving boundary value problems (BVPs) for certain partial differential equations (PDEs) serves as the organizing theme for this course. The main PDEs we will study are the heat equation and the wave equation; considering steady state solutions to these equations also leads to Laplace's equation. A BVP is a PDE together with boundary conditions and possibly initial conditions that imply existence and uniqueness of solutions. Applying the separation of variables technique to BVPs leads us to use of Fourier series, or more generally series of "orthogonal sets" of functions, including Bessel functions and Legendre polynomials. These topics are studied at the end of Math 309, but in 480 we will consider a greater variety of boundary conditions, higher dimensions, and geometries that lead to the Bessel and Legendre variations.

Mathematical features of the BVPs and their solutions typically will have physical interpretations. For example, the wave equation models the vibration of a string or a membrane, and boundary conditions model how the ends of the string or the edges of the membrane are constrained. Different boundary conditions give rise to different kinds of Fourier Series, and properties of these series determine how waves "bounce" at the ends or edges. Series solutions to the wave and heat equations converge differently. This in turn implies differing continuity properties for solutions to the two equations, and these continuity properties have interesting physical interpretations.

What we will cover. Our goal in this course is to see some of the "big picture" of using infinite series of orthogonal functions to solve boundary value problems in partial differential equations (PDEs). To see this big picture, we will have to cover enough variations on the theme so we can see what features are aspects of the theme, and what features are variations. The text has a summary of this big picture in section 4.1; you may find it helpful to read this section several times during the quarter.

Section 1.1 in the text includes some discussion of how the PDEs model the physics, which I will leave it to you to read. Sections 1.2 and 1.3 should be review from Math 309, but we will discuss them quickly in class. We will study essentially everything in Chapters 2, 3 and 4, which present Fourier Series, our first variation on the theme. Bessel Functions in Chapter 5 are variation number 2, and we will discuss most of the results and methods in this chapter, but with less attention to proofs than in the earlier chapters. Finally, in the last week of the quarter we will take a quick look at the first half of Chapter 6 to see a third variation.

A cautionary note. This course used to be part of a 400 level "applied analysis" sequence. The workload will be comparable to our other core 400 level sequences: 402/3/4, 424/5/6, 427/8, and 441/2/3. If you have not taken any of these courses before, you should be aware that most students are surprised by the increase in difficulty from the 300 level courses to these core 400 level courses.

An added challenge is the variety of mathematics that comes into play in Fourier analysis. Indeed, the study of Fourier analysis can serve as a capstone course in undergraduate mathematics, because it builds on such a variety of mathematical topics. Solving ordinary differential equations is the most crucial prerequisite, but ideas from many other courses are useful. Some aspects of Fourier series are best understood by regarding the coefficients in the series as components of an infinite dimensional vector. Then linear algebra becomes useful, in particular ideas about eigenvalues and eigenvectors. Convergence properties for sequences and series enter the course as described above, and vector calculus occasionally makes an appearance.

The problems in Fourier Analysis are lengthy. Therefore the homework will be substantial, and the quizzes and tests will be long. In particular, the quizzes will be 40 or 50 minutes long, and the midterm will be 110 minutes. We will decide as a class the most practical way to schedule a test lasting two class periods. Possibilities include starting the test at 8:30, or scheduling a special test time on a Tuesday or Thursday or in the evening. Either way, other arrangements will be made for people who cannot make the time that works for the majority of the class. The final exam will involve substantial preparation in advance so that you can complete three or four significant problems during the standard exam period.

So, be prepared for a lot of work! But also to enjoy this great mathematical material: most students who are able to put in the work seem to really like this course.