Problem set 2 - fouriernew

Problem 1¶

Let $D_N(x):=\sum_{k = -N}^N e^{inx}$ be the Dirichlet kernel. $F_N:=\frac{D_0+\cdots+D_{N-1}}{N}$ be the Fejer kernel. Show that $F_N(x) = \frac{1}{N} \frac{\sin^2(Nx/2)}{\sin^2(x/2)}$ .

Problem 2¶

Let $f\in L^1(\mathbb{T})$ . Show that the Cesaro sum $\sigma_N f \to f$ in $L^1$ . In other words, $\int_{-\pi}^\pi |f*F_N-f|d\theta\to 0$ when $N\to \infty$ .

Hint

The result is true for any good kernel $K_n$ .
By definition, your task is to estimate a double integral. Use Fubini theorem to change the order of integration.
Then you are going to estimate $\int_{0}^1\|f(x-y)-f(x)\|_{L^1}|K_n(y)|dy$ .
- Deal with the $L^1$ norm: We need to show that $\int_{0}^1|f(x-y)-f(x)|dx \to 0$ when $|y|\to 0$ . Choose continuous $g$ such that $\|g-f\|_{L^1}$ is small then rewrite the integral to estimate as $\|f(x-y)-g(x-y)+g(x-y)-g(x)+g(x)-f(x)\|_{L^1}$ then apply triangle inequality to reduce to the continuous case.
Now apply the property of good kernel. Choose $\delta>0$ , rewrite the integral to $\int_{-\delta}^{\delta}$ part + $\int_{(0,1)\setminus(-\delta,\delta)}$ part, then...

Problem 3¶

Show that in polar coordinate, $\Delta u = \frac{\partial^2 u}{\partial r^2} + \frac{1}{r}\frac{\partial u}{\partial r} + \frac{1}{r^2}\frac{\partial^2 u}{\partial \theta^2}$ .

Problem 4¶

Show that any continuous function on an interval $[a,b]$ can be uniformly approximated by polynomials.

Problem 5¶

Let $f(x)$ be the sawtooth function defined by the periodization of

f_0(x) = \begin{cases}0,x = 0\\\frac{\pi}{2} - \frac{x}{2}, x\in (0,2\pi)\end{cases}

(1)

Then $f$ is a periodic function with period $2\pi$ .

Show that $f$ has jump discontinunity at 0, $f(0^+) = \frac{\pi}{2}$ and $f(0^-) = -\frac{\pi}{2}$ .
Show that the Fourier series of $f$ is $\sum_{n = 1}^{+\infty}\frac{\sin nx}{n}$ .

In lecture note 1 you have seen the Gibbs phenomenon at discontinunity of $f$ . It says no matter how large $N$ is, there exists some $x_N$ near the jump (in our case, it is 0) such that $|S_N(f)(x_N)-f(x_N)|\geq c\frac{|f(0^+)-f(0^-)|}{2} = c\cdot \frac{\pi}{2}$ in our case.

Show that $S_N(f)(x) = \sum_{n = 1}^N \frac{\sin nx}{n} = \frac{1}{2}\int_{0}^x(D_N(t) - 1)dt$ .
Show that $S_N(f)$ is an increasing function on $[0,\frac{\pi}{N+1}]$ .
Show that the error term $S_N(f)(x) - f(x) = \frac{1}{2}\int_{0}^{x}D_N(t)dt - \frac{\pi}{2}$ .

Now it reduces to calculate $D_N$ . Like the last problem of problem set 1, we use $\frac{\sin((N+\frac{1}{2})t)}{\frac{1}{2}t}$ to approximate $D_N(t)$ . Let $\mathrm{sinc(t)} = \frac{\sin t}{t}$ , then $\lim_{t\to 0}\mathrm{sinc}(t) = 1$ .

Let $y_N(t) = \frac{\sin((N+\frac{1}{2})t)}{\frac{1}{2}t}$ . Then $|D_N(t)-y_N(t)|\leq| \frac{\frac{t}{2}-\sin(\frac{t}{2}) }{\frac{t}{2}\sin(\frac{t}{2})}|\to 0$ when $t\to 0$ . In particular, the error of approximation does not depend on $N$ .
Let $Y_N(x) = \frac{1}{2}\int_{0}^x \frac{(\sin(N+\frac{1}{2})t)}{\frac{t}{2}}dt$ . Show that $Y_N$ acheives its maximum at $x_N = \frac{\pi}{N+\frac{1}{2}}$ , and $Y_N(x_N) = \int_{0}^\pi \frac{\sin x}{x}$ .
Define the ‘overshoot’ of $f$ at the origin by $\lim_{\epsilon \to 0^+}\lim_{N\to \infty}\sup_{0<x<\epsilon}|f_N(x) - f(x)|$ , show that it is equal to $\int_{0}^\pi \frac{\sin x}{x}dx - \frac{\pi}{2}$ .