I’m stuck with part b) of this exercise.
In this book the Heat Kernel on the circle is defined as $$H_t(x)=\sum_{ℤ }e^{-4π^2n^2t}e^{i2πnx}$$
The author gave this two proprieties as exercise and the respective hints (without introducing the Poisson summation formula)
a)$∫_{-1/2}^{1/2}|H_t(x)|^2dx$ has magnitude $1/\sqrt{t}$ as $t\longrightarrow0$
Hint: compare $\sum_{ℤ}e^{-4\pi^2n^2t}$ with $\int_{ℝ}e^{-4\pi^2x^2t}dx$
b)$\int_{-1/2}^{1/2}x^2|H_t(x)|^2dx=O(\sqrt{t})$ as $t\longrightarrow0$
Hint: majorize $x^2$ with $C(\sin\pi x)^2$ and apply the mean value theorem to $e^{-\pi x^2t}$
Part a) is quite easy (one may use Parserval’s identity and prove that the discrete infinite sum and the continuous infinite integral differ no more than 1 in absolute value), but for part b) I can’t really obtain the desired result.
Till now, I used Abel’s summation by parts formula and got that \begin{gather} H_t(x)\sin\pi x=\ \sum_{n≥0}(e^{-4\pi^2n^2t}-e^{-4\pi^2(n+1)^2t})[\sin(2n+1)\pi x] \end{gather}
The problem is that while I want $t\longrightarrow0$ in the series I have $n\longrightarrow\infty$ so I find it really difficult to estimate.
Please help me if you know anything. Thank you in advance.