---- > [!theorem] Theorem. ([[Weierstrauss Approximation Theorem]]) > Let $f$ be [[continuous]] on $[a,b]$. Then, there exists a [[sequence]] $\{ p_{k} \}_{k\in \nn}$ of [[polynomial]]s $p_{1},p_{2},\dots,p_{3}$ s.t. $\lim_{ k \to \infty } p_{k}(x)=f(x)$ > [[uniform convergence|uniformly]] in $x$. Equivalently, $\fa \varepsilon >0$ there is a [[polynomial]] $P$ s.t. $\sup_{x \in [a,b]} |f(x)-P(x)| \leq \varepsilon.$ > > In other words, the set of [[polynomial 4|polynomials]] on $[a,b]$ is [[dense]] in the ([[Banach space|Banach]]) [[topological space|space]] $C([a,b])$ of [[continuous]] functions on $[a,b]$ endowed with the [[uniform metric|uniform (sup) norm]]. > [!proof]- Proof. ([[Weierstrauss Approximation Theorem]]) > Idea: > ![[CleanShot 2023-04-08 at 19.39.23.jpg]] > Proof: > > Let $\text{line}$ be a function which takes in two points in $\rr ^{2}$ $(x_{1},y_{1}), (x_{2},y_{2})$ and outputs the function corresponding to the [[line]] $y=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}x$ interpolating between them. > > Define [[continuous]] $g: [a-1, b+1] \to \rr$ by $g(x)=\begin{cases} > f(x) & x \in [a,b] \\ > \text{line}\big((\textcolor{Skyblue}{0,0}), (\textcolor{Skyblue}{a,f(a)})\big)(x) & x \in [a-1, a) \\ > \text{line}\big((\textcolor{Skyblue}{b,f(b)}), (\textcolor{Skyblue}{0,0})\big)(x) & x \in (b, b+1]. > \end{cases}$Define $\textcolor{Thistle}{L:=v(\dom g)=b-a+2}$, and define the [[linear map]] $\textcolor{Apricot}{x:\dom g \to [-\pi,\pi]}$ by $\textcolor{Apricot}{x(\theta)=\frac{2\pi}{\textcolor{Thistle}{L}}(\theta-a-1})-\pi$. > > Because $g(a-1)=g(b+1)$, $g$ may be [[periodic|periodically extended]] to a [[continuous]] function $\tilde{g}: \rr \to \rr$ s.t. $\tilde{g}(x+\textcolor{Thistle}{L})=\tilde{g}(x) \ \fa x \in \rr$. Next define $\tilde{\gamma}(x):\rr \to \rr$ s.t. $\tilde{\gamma}(\theta+2\pi)=\tilde{\gamma}(\theta)$, i.e., $\tilde{\gamma}(\theta)=\tilde{g}\big(x(\theta)\big)$. > > By [[Fourier series is Cesaro summable at points of continuity|density of trigonometric polynomials on the circle]], $\tilde{\gamma}$ may be [[uniform convergence|uniformly approximated]] by [[trigonometric polynomial]]s, and so using Maclaurin expansion we get > ![[CleanShot 2023-04-13 at [email protected]]] > where the last term shows $f$ being uniformly approximated by a polynomial as required. ---- #### ----- #### References > [!backlink] > ```dataview TABLE rows.file.link as "Further Reading" FROM [[]] FLATTEN file.tags GROUP BY file.tags as Tag > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```