ABSTRACT
The approximation, resp. the representation, of arbitrary known or unknown functions / 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 »—► h e N, or functions of the form ί h eci, c E 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 / is sufficiently many times differentiable in a neighbourhood U of the point a e R. Such a function can be approximated within U by its Taylor polynomials
(jets for short), up to an error that can be quantitatively controlled, and under suitable assumptions the function / is actually represented by its Taylor series, meaning that one has
for all t in a certain neighbourhood U' C U. The general setup in this realm is the following: Depending on the particular situation at hand one chooses a family (eQ | a £ /) of basis functions t »—► ea (£); the index set I may be a discrete or a “continuous” set. An approximation of a more or less arbitrary function / by means of the ea then has the form
with coefficients Ck to be determined, and a representation of / has the form
or it appears as an integral over the index set I:
(3) In the ideal case there are exactly as many basis functions at our disposal as are needed to represent any function / of the considered kind in exactly one way in the form (2) resp. (3). The operation that assigns a given function / the corresponding coefficient vector or array (ca | a G I) is called the analysis of / with respect to the family (ea | a e I). The coefficients ca are particularly easy to determine, if the basis functions ea are orthonormal (see below). In the case of the Taylor expansion (1) the coefficients have to be determined by computing recursively ever higher derivatives of /; and in the case of the so-called Tchebycheff approximation there are no formulas for the coefficients Cfc, even though they are uniquely determined. The inverse operation that takes a given coefficient vector (ca | a £ I) as input and returns the function itself as output is called the synthesis of / by means of the ea.
