Problem set 1

Fourier series¶

Problem 1. (1) Let $f(x) = \frac{(\pi - x)^2}{4}$, $x\in [0,2\pi]$. Calculate the Fourier series of $f$. (2) Prove that the Fourier series of $f$ converges uniformly on $[0,2\pi]$, conclude that $\sum_{n = 1}^{\infty}\frac{1}{n^2} = \frac{\pi^2}{6}$.

Problem 2. Calculate $\sum_{n\text{ odd }\geq 1}\frac{1}{n^2}$ by examing the function $f(\theta) = \theta(\pi - \theta)$.

Problem 3. (Exercise 2 of chapter 2 in Stein-Sharkarchi) Let $f$ be a $2\pi$-periodic function on $\mathbb{R}$.

(a). Show that the Fourier series of $f$ can be written as $S(f)(\theta) = f(0)+\sum_{n\geq 1}[\hat{f}(n)+\hat{f}(-n)]\cos(n\theta) + i [\hat{f}(n)-\hat{f}(-n)]\sin(n\theta)$.

(b). Show that if $f$ is even, then $\hat{f}(-n) = \hat{f}(n)$, so there is no sine term.

(c) Show that if $f(\theta + \pi) = f(\theta)$ for all $\theta \in \mathbb{R}$, then $\hat{f}(n) = 0$ for all odd $n$.

(d) Show that if $f$ is real-valued, then $\overline{\hat{f}(n)} = \hat{f}(-n)$ for all $n\in \mathbb{Z}$.

Approximation and convolution¶

Problem 4.

Let $\chi_{(a,b)}$ be the function given by

Let Λ be the function determined by $\Lambda(x)=4(\frac{1}{2}-|x|)$ on $[-\frac{1}{2},\frac{1}{2}]$ and 0 elsewhere. The function does look like its name “Λ”.

(a) Show that $\int_{-\pi}^\pi \Lambda(x)dx = 1$. Let $\epsilon > 0$ be a small positive number. Let $\Lambda_{\epsilon} = \frac{1}{2\epsilon}\Lambda(\frac{x}{2\epsilon})$. Show that $\int_{-\pi}^\pi \Lambda_\epsilon(x)dx$ is still 1. Observe that when $\Lambda_\epsilon(y)$ is non-zero only when $y\in (-\epsilon,\epsilon)$.

(b) Consider the function $\chi_{(a,b)}* \Lambda_{\epsilon}$, what does it look like? Show that when ε is small enough, $\chi_{(a,b)}*\Lambda_\epsilon(x) = \chi_{(a,b)}(x)$ when $x\notin (a-\epsilon, a+\epsilon)\cup(b-\epsilon,b+\epsilon)$.

(c) Show that $\chi_{(a,b)}*\Lambda$ is continuous.

(d) Conclude that there exists continuous function $g$ such that $\|\chi_{(a,b)}-g\|:=\int_{-\pi}^{\pi}|\chi_{(a,b)}(x)-g(x)|dx<\epsilon$ (the “:=” means " is defined by"), see problem 5.

(e) Let $f$ be a Riemann integrable function on $[-\pi,\pi]$. Show that for $\epsilon > 0$, there exists a continuous function $g$ such that $\|f-g\|_{L^1}<\epsilon$. (See Lemma 1.5 in Appendix: Integration of Stein-Shakarchi).

Dirichlet Kernel¶

Problem 5. Let $D_N(x) = \frac{\sin((N+\frac{1}{2})x)}{\sin(\frac{1}{2}x)}$ be the Dirichlet kernel. Let $\|D_N\|:=\frac{1}{2\pi}\int_{0}^{2\pi}|D_N(x)|dx$.

(a) Prove $\|D_N\|\geq c\ln N$ for some $c>0$.

(b) Show that there exists a continuous function $f_N$ such that $|f_N|\leq 1$ and $|S_N(f_N)(0)|\geq c'\ln N$. (Hint: Let $A_+$ be the set $\{x\in[-\pi,\pi]:D_N(x)>0\}$, $A_{-}$ be the set $\{x\in [-\pi,\pi]:D_N(x)< 0\}$. Let $g_N(x) = 1\cdot \chi_{A_+}(x)+(-1)\cdot \chi_{A_-}(x)$. Apply problem 4 to $g_N$).