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.

 

[M98Ranger]

The nonuniqueness of power series expansion is a fundamental concept in mathematics that is often overlooked. It is true that the set {1, x, x^2, x^3, ...} forms an overcomplete basis for a certain class of functions, which implies that a power series expansion of a function is not unique. This means that there are multiple ways to represent a function using a power series, and it is not always possible to determine the "correct" representation. This can be demonstrated by considering the function f(x) = 1/(1-x), which has a power series expansion of 1 + x + x^2 + x^3 + ... However, this same function can also be represented as 1 + 2x + 3x^2 + 4x^3 + ..., or even as a combination of both series. This shows that there is no single "correct" power series expansion for this function.

The reason for this nonuniqueness lies in the fact that power series expansions are based on the Taylor series, which is a local approximation of a function at a specific point. As we expand the series to include more terms, we are essentially zooming out and looking at a larger and larger neighborhood of the point. This means that there are an infinite number of ways to choose the center of the expansion, resulting in an infinite number of possible power series representations for a given function.

So, in short, the nonuniqueness of power series expansion is not nonsense, but a fundamental concept in mathematics. It highlights the fact that there are often multiple ways to represent a mathematical object, and that the choice of representation can affect our understanding and analysis of that object.