1.1 A Central Theme of Analysis

The approximation, resp. the representation, of arbitrary known or unknown functions f by means of special functions can be viewed as a central theme of analysis. "Special functions" are functions taken from a catalogue, e.g., monomials t ? t ^k, k ? ?, or functions of the form t ? e ^ct, c ? ? a parameter. As a rule special functions are well understood, very often they are easy to compute and have interesting analytical properties; in particular, they tend to incorporate and re-express the evident or hidden symmetries of the situation under consideration.

In order to fix ideas we consider a (given or unknown) function

assuming that f is sufficiently many times differentiable in a neighbourhood U of the point a ? ?. Such a function can be approximated within U by its Taylor polynomials

(1)

( jets for short), up to an error that can be quantitatively controlled, and under suitable assumptions the function f is actually represented by its Taylor series, meaning that one has

for all t in a certain neighbourhood U' ? U.

The general setup in this realm is the following: Depending on the particular situation at hand one chooses a family ( e _? ? ? I) of basis functions t ? e _?( t); the index set I may be a discrete or a "continuous"...