Heat equation on a ring

[stripes]

Homework Statement

I have to give a seminar to my math class about the heat equation on the ring. I will be introducing the heat kernel on the circle to the class, which is as follows:

$H_{t} (x) = \sum ^{-\infty}_{\infty} e^{-4 \pi ^{2} n^{2} t} e^{2 \pi i n x} .$

I basically will decide what I am presenting to the class, so I don't have any "homework" questions in that sense. My questions stem from what I would like to present to the class.

After introducing the heat kernel on the circle, I will also state that if we are given an initial condition at time = 0, u(x, 0) = f(x), then the convolution

$u(x, t) = \sum ^{-\infty}_{\infty} a_{n} e^{-4 \pi ^{2} n^{2} t} e^{2 \pi i n x} = (f \ast H_{t}) (x).$

How do I show u(x,t) = the convolution of f and H? Basically I would like to derive the formula but because the convolution involves introducing a new variable y, and then integrating w.r.t. it, I can't really "show" that this statement is true. How do I go about proving this?

I would also like to show that the heat kernel is a good kernel. The definition of a good kernel is given in the image I have attached. Now my textbook gives a proof of this. However, it is located in the next chapter. My seminar specifically says I am presenting on section 4.4, the heat equation on the circle. In section 4.4, it says I require the use of "celebrated Poisson summation" which is taken up in chapter 5. So strictly speaking, I am not required to show that H is a good kernel. But this is the "meat of the sandwich"; if I present this without a proof, I know I won't get a good mark.

I read the section in the following chapter. I contemplated using these tools to show it is a good kernel. First I would need to define the heat kernel on the real numbers, which appears to be a lot different than the one for a circle. Then I would need to introduce the concept of, and define the Poisson summation formula. This definition says that a Poisson summation of a function is called the periodization of that function. Then I would need to show that the heat kernel for the circle is a periodization of the heat kernel for the real numbers. Once I have these on the table, I can show that the heat kernel for the circle is a good kernel.

These above concepts are out of the scope of section 4.4, the section I am to present. Is there any way to show it is a good kernel without the use of the concepts in the above paragraph? The textbook states specifically in an exercise for chapter 4 that "The fact that the kernel Ht(x) is a good kernel ... is not easy to prove."

Also, the textbook constantly says that if a kernel is greater than or equal to zero, then (b) from the image attached is a consequence of (a). But wouldn't (b) be a consequence of (a) no matter what? Because if the kernel is integrable from (a), then it must be bounded. Isn't (b) a redundant criterion?

My seminar is tomorrow so I doubt I will get any answers in time. It's my fault for leaving things so late but I also have 3 assignments due tomorrow so there's not really enough time for anything in my life these days.

If anyone can help out, I would appreciate it. Even if you read this far, thank you.

Homework Equations

The Attempt at a Solution

 

[lippyka]

I have not been able to attempt a solution as I am still confused about the concept of proving the heat kernel is a good kernel without the use of the concepts from the next chapter.