Taylor \& Fourier

1: These are both fundamental to math,
  and should -both- be introduced

2: They should be compared and contrasted:
 - Taylor
 - Fourier
 ...and they raise questions of convergence, namely:
 "what do you -mean- by convergence?"
   (which leads to notions of topologies on function spaces;
    also, to the notion of germ (for Taylor))
 "for -what- functions do they converge (in a given topology)?"

3: they're related to representation theory;
How does $z \mapsto z^n$ come up?
One way is: these are ``nice'' maps $\bC \to \bC$.
(or $\bC^* \to \bC^*$ for $z^{-n}$)
Better is:
$\bC$ acts naturally on $\cF(\bC, \bC)$ by
  ``multiplication on the domain'':
  $\rho_\lambda f(z) = f(\lambda z)$
it also acts by ``multiplication on target'':
  $\rho_\lambda f(z) = \lambda^n f(z)$
...and the functions for which these actions coincide are exactly
$z^n$ (or $(z-a)^n$, if you wanna move the origin around).

  - Fourier
    This is obvious; you're studying functions $S^1 \to \bC$,
      so it's immediate representation theory
    also, note that you naturally get $\widehat{S^1} = \bZ$
    [just enumerating the representations leads to
     rep'n thy of abelian groups, where it's not just a set of rep'ns,
     but a group (and yeah, more generally you get the algebra
     of representations)]
    ...and it's only a small step to see that you can go backwards.
    
  - Taylor
    This is rather more subtle;
    you're really looking at the -germ-,
    and here $\bC$ is not acting on the -plane-,
    but on the -tangent space at $a$-;
    hence you only get the infintesimal result.
    
    [The point, however, is that it also is a sort of representation
     theory.]
    
    Oh, and note that $\widehat{\bC} = \bN$, since you can't have
    $z^{-n}$ since that doesn't work for $0$;
    note that the rep'ns of an abelian -semi-group are themselves only
    a -semi-group (b/c you're lacking inverses).

    BTW, is there an integral formulation of Taylor series?
    [Also, if you're punctured, you get Laurent series,
     which look a -lot- like Taylor series;
     is it Fourier analysis on the unit circle in the tangent plane?]

4: the "special bases"
  $z^n$ for Fourier, $z^n/n!$ for Taylor;
  these correspond to ``picking out'' a ``dual basis'' or something
  ($z^n/n!$ is ``dual'' to the sequence of functionals
   $D^n(0)\from f \mapsto f^{(n)}(0)$

5: The Fourier -transform- of a function on $\bR$
   (IE, not periodic; not a function on $S^1$)
   is rather more subtle.
   - it is "simply" an operator on functions
   - well, I think it also is a duality
