Nonuniqueness of power series expansion

[Timbuqtu]

A couple of days ago one of my teachers mentioned (when discussing the completeness of spherical harmonics) that {1,x,x^2,x^3,...} forms an overcomplete basis for (a certain class of) functions. This implies that a power series expansion of a function is not unique. And you can for instance write x as a sum over higher powers of x^n.

I tried to find something on the internet about it, because it's seems really odd to me. But I didn't find anything. Has anyone of you made this observation and maybe seen a proof of it? (Or is it just nonsense?)

[matt grime]

I think the answer is Yes And No.

It is important to remember that you're talking about this being a basis in the analytic sense, so there is a distance notion involved (or a kernel if you know about that).

The Taylor series of a function about a point is unique, which is what one usually means when talking about a (smooth) function having a power series.

But in an analytic sense with a different notion of "distance", then we may be able to apporximate arbitrarily closely in that notion of "distance" with different series of powers of x. (But these aren't what we refer to as power series expansions).

An example. Consider the set, C, of continuous functions from [0,1] to R.

Let E be the set of even powers of x, let O be the set of odd powers of x plus the zeroeth power of x, ie 1.

E= {1,x^2,x^4,x^6...}
O={1,x,x^3,x^5...}

Then I believe I can invoke something called the Stone-Weierstrass theorem to conclude that both E and O span C in the sup norm.