Problem set 3

Let $S^1$ be the unit circle. We work on $\mathscr{C}^{\infty}(S^1)$, the smooth functions on the circle. In this note we identify $S^1$ with $[0,2\pi]$ for simplicity. Let $(f,g) = \frac{1}{2\pi}\int_{0}^{2\pi} f(x)\overline{g(x)} dx$ be the inner product.

Let $D:\mathscr{C}^{\infty}(S^1)\mapsto \mathscr{C}^{\infty}(S^1)$ be the differential operator given by $f\mapsto -i\frac{d}{dx}f$, it is a linear operator on $\mathscr{C}^{\infty}(S^1)$. $D$ is called the Dirac operator, it satisfies the property that $D^2 = \Delta$.

Problem 1.¶

(1) Show that $D$ is self adjoint, i.e. $(Df,g) = (f,Dg)$ for $f,g\in \mathscr{C}^{\infty}(S^1)$.

(2) Show that $D^2 = \frac{d}{dx^2}$, which is the 1-dimensional Laplacian operator. Show that $e^{ i n x}$ are eigenfunctions of $D$, with eigenvalue $n$.

Problem 2.¶

For $s\in \mathbb{C}$, let $D+s$ denote the operator $D+sI$, so that $(D+s)(f) = Df + sf$. Show that for $s\in \mathbb{C}/\mathbb{Z}$, $D+s$ is invertible.

Problem 3.¶

If you have done right in problem 2, you will get

Find a function $g$ such that $(D+s)^{-1}(f) = f*g$.

Problem 4.¶

If you are comfortable with convolution theorem, you would find $g(x) = \sum_{n = -\infty}^{+\infty}\frac{1}{n+s}e^{inx}$ in problem 3.

(1) Show that $\sum_{n\in \mathbb{Z}}\frac{1}{n+s}e^{inx}$ is the Fourier series of $\frac{\pi}{\sin \pi s}e^{i(\pi-x)s}$, conclude that $g(x) = \frac{\pi}{\sin \pi s} e^{i(\pi -x)s}$.

(2) Show that $\sum_{n=-\infty}^{+\infty}\frac{1}{n+s} = \pi \cot(\pi s) = \pi i - 2\pi i \sum_{k= 0}^{+\infty}e^{2\pi i ms}$. Note that this is a key identity to derive the Fourier series of Eisenstein series. See also page 5 of Diamond-Shurman A first course to modular forms.

(3) Show that if $s\in \mathbb{T}$ and $s \neq 0$, then $\sum_{n = -\infty}^{+\infty}\frac{1}{|n+s|^2} = \frac{\pi^2}{|\sin(\pi s)|^2}$.

Problem 5¶

Let $f$ be continuous at everywhere on the circle except for a jump discontinunity at 0.

(1) Let $\sigma(x) = \lim_{N\to \infty}\sigma_Nf(x)$. Show that $\sigma(x) = \begin{cases} f(x), x\neq 0\\ \frac{f(0^+)+f(0^-)}{2}, x = 0 \end{cases}$

(2) We know that if $S_N(f)$ converges pointwise to $F$, then $\sigma_N f$ converges pointwise to the same $F$. Show that the reverse holds when $|c_n| = o(1/n)$, where $c_n$ is the Fourier coefficients of $f$.

(3) Conclude that when $f$ is piecewise $\mathscr{C}^1$ on the circle with jump discontinunity at 0, then $S_N(f)(0)\to \frac{f(0^-)+f(0^+)}{2}$. Apply this result to problem 4(2).