A.4 Analysis

Definition A..8 (Lipschitz continuous)   A function $ f: \mathbb{R}^n \supset X \to \mathbb{R}$ is said to be Lipschitz continuous with Lipschitz constant $ c$, if

$\displaystyle \forall x_1, x_2 \in X:\quad
\Vert f(x_1)-f(x_2) \Vert \le c \Vert x_1-x_2\Vert.
$

Definition A..9 (Total Variation)   Let $ Z$ be a decomposition of the interval $ [a,b]$ consisting of the points $ a=x_0 < x_1 < \cdots < x_{n-1} < x_n = b$. If $ f$ is function whose domain includes $ [a,b]$, then

$\displaystyle V(f,Z) := \sum_{k=1}^n \Vert f(x_k)-f(x_{k-1}) \Vert
$

is called the variation of $ f$ with respect to $ Z$. If the upper bound $ V(f,[a,b]) := \sup_Z V(f,Z)$ of the set of all variations with respect to all decompositions of $ [a,b]$ exists, then $ V(f,[a,b])$ is called the total variation of $ f$ over $ [a,b]$. In this case $ f$ is of bounded variation.

Theorem A..10 (Weierstraß Approximation Theorem)   Let $ f$ be a real valued continuous function on the compact interval $ I:=\{
(x_1,\ldots,x_m) \mid a_k \le x_k \le b_k \}$. Then for each $ \epsilon>0$ there is a polynomial $ p$ in $ m$ indeterminates $ x_1,\ldots,x_m$ such that

$\displaystyle \vert f(x_1,\ldots,x_m) - p(x_1,\ldots,x_m) \vert < \epsilon
\qquad\forall (x_1,\ldots,x_m) \in I.
$

Put differently, the set of all polynomials on $ I$ is dense in the set of the real valued continuous function on the compact interval $ I$.

The Stone-Weierstraß Theorem is a generalization of the Weierstraß Approximation Theorem [47].

Definition A..11   A family of real valued functions $ F$ on $ X$ is said to separate the points of $ X$, if for two different points $ x$ and $ y$ of $ X$ there is always an $ f\in F$ such that $ f(x)\ne f(y)$.

Theorem A..12 (Stone-Weierstraß, Formulation 1)   Let $ X$ be a compact subset of a normed space $ E$ and $ P$ be a subalgebra of $ C(X)$ containing the function $ 1$ and separating the points of $ X$. Then $ P$ is dense in $ C(X)$.

This is equivalent to the following formulation.

Theorem A..13 (Stone-Weierstraß, Formulation 2)   Let $ X$ be a compact subset of a normed space $ E$ and $ P$ be a closed subalgebra of $ C(X)$ containing the function $ 1$ and separating the points of $ X$. Then $ P=C(X)$.

Theorem A..14 (Taylor)   Let $ f: G\subset\mathbb{R}^p\to\mathbb{R}$ ($ G$ open) be a $ C^2$ function and $ U$ be an $ \epsilon$-neighborhood of $ x_0$ lying in $ G$. Then for all $ h$ with $ x_0+h\in U$ we have

$\displaystyle f(x_0+h) = f(x_0) + f'(x_0)h
+ {1\over2} \sum_{j=1}^p\sum_{k=1}^p...
...al ^2 f(x_0)\over \partial x_j \partial x_k} h_j h_k
+ \Vert h\Vert^2 \rho(h),
$

where

$\displaystyle \lim_{h\to0} \rho(h)=0.
$

Theorem A..15 (Generalized Taylor Series)   Let $ f: G\subset\mathbb{R}^p\to\mathbb{R}$ ($ G$ open) be an analytical function. Then

$\displaystyle f({\mathbf{r}} + {\mathbf{a}}) =
\sum_{k=0}^\infty
\Bigl( {1\over...
..._{\mathbf{r'}})^k f({\mathbf{r'}}) \Bigr)
\Bigm\vert _{\mathbf{r'}=\mathbf{r}}
$

holds.

Clemens Heitzinger 2003-05-08