I'm studying Muscalu and Schlag's Classical and Multilinear Harmonic Analysis, v. 1. One problem asks to study the heat equation on $\mathbb{T}$, i.e. $$u_{t} = u_{\theta \theta} \quad \text{on} \, \mathbb{T}, \quad u(0,\cdot) = u_{0}$$ for some $u_{0} \in \mathcal{C}(\mathbb{T}) \cup L^{p}(\mathbb{T}) \cup \mathcal{M}(\mathbb{T})$ given. So far I have established that the fundamental solution/heat kernel $G_{t}$ is a non-negative smooth function with Fourier coefficients $$\hat{G}_{t}(n) = e^{- 4 \pi^{2} n^{2} t}.$$

Part of the problem is to prove that $G_{t}$ is an approximate identity, the only non-trivial part being proving that $$ \lim_{t \to 0^{+}} \int_{|\theta| > \delta} G_{t}(\theta) \, d \theta = 0 $$ for every $\delta > 0$. Ideally, we would like to prove the stronger condition: $$\lim_{t \to 0^{+}} \sup\{ G_{t}(\theta) \, \mid \, |\theta| > \delta\} = 0. $$ Working with Fourier coefficients alone seems like a rough approach as far as proving these limits is concerned.

I understand there is a way to get a nice(er) expression for $G_{t}$ using the heat kernel on $\mathbb{R}$ and periodization. The book hasn't presented the Fourier transform up to this point, though, so I'm curious to know:

Is there a way to derive decay properties of $G_{t}$ using knowledge of $\mathbb{T}$ and Fourier series alone?

  • $\begingroup$ What is the heat equation on $\Bbb T$? The usual heat equation on $(0,1)$ with periodic boundary conditions? $\endgroup$ – Cameron Williams Mar 19 '16 at 18:56
  • $\begingroup$ I edited my question to clarify. It's not 100% clear to me this is equivalent to the problem you have in mind since I'm requiring the solution be smooth at 0 ~ 1 as well as the interior of the interval. $\endgroup$ – fourierwho Mar 19 '16 at 19:18
  • $\begingroup$ Yeah I agree. What I was going to do is probably not going to work now. Hopefully someone else can chime in. $\endgroup$ – Cameron Williams Mar 19 '16 at 19:38

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