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

, 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

be a decomposition of the interval

consisting of the points $a=x_0 < x_1 < \cdots < x_{n-1} < x_n = b$ . If

is function whose domain includes

, then

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

is called the variation of

with respect to

. 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

exists, then

is called the total variation of

over

. In this case

is of bounded variation.

Theorem A..10 (Weierstraß Approximation Theorem) Let

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

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

is dense in the set of the real valued continuous function on the compact interval

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

Definition A..11 A family of real valued functions

is said to separate the points of

, if for two different points

and

there is always an $f\in F$ such that $f(x)\ne f(y)$ .

Theorem A..12 (Stone-Weierstraß, Formulation 1) Let

be a compact subset of a normed space

and

be a subalgebra of

containing the function

and separating the points of

. Then

is dense in

Theorem A..13 (Stone-Weierstraß, Formulation 2) Let

be a compact subset of a normed space

and

be a closed subalgebra of

containing the function

and separating the points of

. Then

Theorem A..14 (Taylor) Let $f: G\subset\mathbb{R}^p\to\mathbb{R}$ (

open) be a

function and

be an $\epsilon$ -neighborhood of

lying in

. Then for all

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 $f: G\subset\mathbb{R}^p\to\mathbb{R}$ (

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.