2021-3.md

course	course_year	question_number	tags	title	year
Analysis and Topology	IB	3	IB 2021 Analysis and Topology	Paper 2, Section II, F	2021

Let $k_{n}: \mathbb{R} \rightarrow \mathbb{R}$ be a sequence of functions satisfying the following properties:

1. $k_{n}(x) \geqslant 0$ for all $n$ and $x \in \mathbb{R}$ and there is $R>0$ such that $k_{n}$ vanishes outside $[-R, R]$ for all $n$

2. each $k_{n}$ is continuous and

$$\int_{-\infty}^{\infty} k_{n}(t) d t=1$$

3. given $\varepsilon>0$ and $\delta>0$, there exists a positive integer $N$ such that if $n \geqslant N$, then

$$\int_{-\infty}^{-\delta} k_{n}(t) d t+\int_{\delta}^{\infty} k_{n}(t) d t<\varepsilon$$

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a bounded continuous function and set

$$f_{n}(x):=\int_{-\infty}^{\infty} k_{n}(t) f(x-t) d t$$

Show that $f_{n}$ converges uniformly to $f$ on any compact subset of $\mathbb{R}$.

Let $g:[0,1] \rightarrow \mathbb{R}$ be a continuous function with $g(0)=g(1)=0$. Show that there is a sequence of polynomials $p_{n}$ such that $p_{n}$ converges uniformly to $g$ on $[0,1]$. $[$ Hint: consider the functions

$$k_{n}(t)= \begin{cases}\left(1-t^{2}\right)^{n} / c_{n} & t \in[-1,1] \ 0 & \text { otherwise }\end{cases}$$

where $c_{n}$ is a suitably chosen constant.]