# A.4 Analysis

Definition A..8 (Lipschitz continuous)   A function is said to be Lipschitz continuous with Lipschitz constant , if

Definition A..9 (Total Variation)   Let be a decomposition of the interval consisting of the points . If is function whose domain includes , then

is called the variation of  with respect to . If the upper bound 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 . Then for each there is a polynomial  in indeterminates such that

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  on  is said to separate the points of , if for two different points  and  of  there is always an such that .

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 .

This is equivalent to the following formulation.

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 ( open) be a function and be an -neighborhood of  lying in . Then for all with we have

where

Theorem A..15 (Generalized Taylor Series)   Let ( open) be an analytical function. Then

holds.

Clemens Heitzinger 2003-05-08