Problem set 4

Problem 1¶

Let $g$ be a continuous function on the circle that is differentiable at $x_0$.

(1) Let $\sigma_N = g*F_N$ be the $N$-th Cesaro mean of $g$. Show that $\frac{\partial}{\partial x}\sigma_N(x_0) = g*F_N'(x_0)$, where $F_N'$ is the derivative of $F_N$.

(2) Show that $|\sigma_N'(x_0)|\leq C \|tF_N'\|_{L^1}$ for some $C>0$.

(3) Show that $|F_N'(t)|\leq \frac{A}{|t|^2}$. Hint: Calculate the derivative of $F_N(t) = \frac{1}{N}\frac{\sin^2(Nt/2)}{\sin^2(t/2)}$, and use $|\sin(t/2)|\geq c|t|$.

(4) Show that $\|tF_N'\|_{L^1} = O(\log N)$.

Problem 2¶

Let $f$ be supported on $[-\frac{1}{2},\frac{1}{2}]$ such that $f(-\frac{1}{2}) = f(\frac{1}{2}) = 0$. The periodization $f_1(x) = \sum_{k = -\infty}^{+\infty}f(x- k)$ makes copies of $f$ to get a function on $\mathbb{T}$.

Let $f$ be $\mathscr{C}^1$ so that $f_1\in \mathscr{C}^1(\mathbb{T})$.

(1) Let $\hat f(s)$ be the Fourier transform of $f$, let $c_n$ be the Fourier coefficients of $f_1$. Show that $\hat{f}(n) = c_n$ for $n\in \mathbb{Z}$. In other words, the integral value of Fourier transform of $f$ is equal to the Fourier coefficients of periodization of $f$. This justifies our usage of notation $\hat{f}(n)$ for Fourier coefficients.

(2) Conclude that $\sum_{n = -\infty}^{+\infty}f(x+n) = \sum_{n = -\infty}^{+\infty}e^{2\pi i n x}$.

(3) Conclude that $\sum_{n = -\infty}^{+\infty}f(n) = \sum_{n = -\infty}^{+\infty} \hat f(n)$.

Problem 3¶

Show that if $f\in L^1(\mathbb{R})$, then $\mathcal F f$ is continuous.

Problem 4¶

A function $f$ on $\mathbb{R}$ is called moderate decreasing provided $|f(x)|\leq \frac{A}{1+x^2}$ for some $A>0$ and $x\in \mathbb{R}$. Let $f$ be a moderate decreasing function on $\mathbb{R}$.

(1) Show that if $f$ is moderate decreasing, then the improper integral $\int_{-\infty}^{+\infty}f(x)dx$ converges.

(2) Show that $\int_{-\infty}^{+\infty}f(x-h)dx = \int_{-\infty}^{+\infty}f(x) dx$.

(3) Show that for $a> 0$, $\int_{-\infty}^{+\infty}f(ax)dx = \frac{1}{a}\int_{-\infty}^{+\infty}f(x) dx$.

(4) Prove that let $f$ be a moderate decreasing continuous function on $\mathbb{R}$. Then $f_T(x) = \sum_{k = -\infty}^{+\infty}f(x+kT)$ defines a $T$-periodic continuous function on $\mathbb{R}$ such that $f_T(x) - f(x)\leq C \frac{\pi^2}{T^2}, \forall |x|\leq \frac{T}{2}$.

(5) If both $f$ and $\hat f$ are moderate decreasing then the Poisson summation formula holds.

Problem 5¶

Let $f,g$ be $L^1$ functions on $\mathbb{R}$. The convolution of $f,g$ is given by

Before we did convolution of periodic functions (or circular convolution). Let $f,g$ be supported on $(-\frac{1}{2},\frac{1}{2}).$ Let $f_1,g_1$ be the periodization of $f,g$ to 1-periodic functions. Show that $f_1*g_1 = f*g_1$, where the left hand side is the periodic convolution $\int_{0}^1 f_1(x - y)g_1(y)dy$, the right hand side is the convolution on $\mathbb{R}$, i.e. $\int_{-\infty}^{+\infty}f(x-y)g_1(y)dy$.