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

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

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

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

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

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

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

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

where

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

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

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

holds.