Power series for the sawtooth wave

Question

This wikipedia article described a Fourier expansion of the sawtooth wave. Does this wave have a power series expansion (around any point)? If so, what is it? Does every function with a Fourier expansion also has a power series expansion?

Answer 1

If the sawtooth wave is defined by, say

$$f(x)=x, 0<x\leq 1$$ $$f(x)=f(x+1)$$

Then yes, it has a power series expansion around any point that isn't a discontinuity, but it's not particularly exciting:

$$x$$

Since what we're dealing with on any interval around a non-jump point is merely a line, and so has constant derivative and $0$ higher ones. Power series are only affected by local behaviour, and since areas within some positive radius of convergence are locally straight lines, that's what you get. The radius of convergence is just the distance to the nearest discontinuity.

I'm not quite sure about the last result (so I suppose this is an incomplete answer), but I can say that the converse isn't true. Just take any unbounded function, like $e^x$, which is entire but has no fourier transform.