# Heat kernel on the unit circle $\mathbb{R}/\mathbb{Z}$

The heat kernel on the circle $\mathbb S^1\cong \mathbb{R}/\mathbb{Z}$ is given by $$(*) \qquad k_t(\theta)=\frac{1}{\sqrt{4\pi t}}\sum_{n\in \mathbb Z}e^{-\frac{(\theta-n)^2}{4 t}}, \quad \theta\in \mathbb{R}/\mathbb{Z}.$$

In a PDF, I did not understand the meaning of the following remark: $\, (*)$ is the $\mathbb{Z}$ -average of the heat kernel on $\mathbb R$ given by $$k_t(x)=\frac{1}{\sqrt{4\pi t}} e^{-\frac{x^2}{4 t}}, \quad x\in \mathbb{R}.$$ Also, the technique of unwinding $\mathbb{Z}$-averages?

Can someone explain to me more. Thank you in advance

I wouldn't call it $\mathbb{Z}$-average because "average" means dividing by the total weight, and this isn't happening here. This process is also known as "periodization" or "periodic summation" (Wikipedia). Given a function $f$ on $\mathbb{R}$, we can define a periodic function $g$ by $$g(x) = \sum_{n\in\mathbb{Z}}f(x+n) \tag1$$ provided the series converges (which is does when $f$ has sufficiently fast decay at infinity).
"Unwinding $\mathbb{Z}$-average" would mean the opposite: given a periodic function $g$, find $f$ such that (1) holds.
For example, if $g(x) = \sin 2\pi x$, then one can take $$f(x) = \begin{cases} \sin 2\pi x,\quad &0\le x\le 1, \\ 0 \quad &\text{otherwise}\end{cases}$$ This particular example doesn't seem important. But when $f$ solves some translation-invariant PDE on $\mathbb{R}$ (such as the heat equation), we get a solution of the same PDE on the circle $\mathbb{R}/\mathbb{Z}$ via (1), and that is nice to have.
Closely related: Poisson summation formula, which says that the Fourier series of $g$ can be obtained from the Fourier transform of $f$.