TITLE: When does a real function together with its derivatives form a basis?
QUESTION [34 upvotes]: Consider real functions defined on an interval, let's say $[-\pi,\pi]$. Let $s(x)$ be an analytic, periodic function such that all of its Fourier components are non-zero. We build a set of functions from $s(x)$ and all of its derivatives:
S=$\{s(x),s'(x),s''(x),\dots\}$
What is the condition for $S$ to be a basis?
By "basis" I mean that if I take any $f(x)\in L^2([-\pi,\pi])$ and produce a series of functions $f_N(x)$ which are the projections of $f(x)$ to the subspace spanned by the first $N$ elements of $S$, then $f_N(x)\to f(x)$ in $L^2$ norm. 
What I understood is that if we want that $S$ be a basis, then it is necessary that all Fourier components are not zero. But is this enough or is there some other condition for $s(x)$?
I did not get far with any proofs. All I understand is that if I take the first few members of the set and I project to the space constructed from the first few Fourier components, then we will have a basis there. But I can't go further.

REPLY [4 votes]: It is indeed enough to assume that all Fourier coefficients of $s$ are non-zero.
The problem actually seems to become quite simple after a Fourier transform: Since $s$ is assumed to be holomorphic on a neighborhood of $[-\pi,\pi]$, we have that $\widehat{s}(k) = O(e^{-2\delta |k|})$.
In terms of the Fourier transforms, our task is to show that the vectors $(k^n \widehat{s}(k))_{k\in\mathbb Z}$, for $n=0,1,2, \ldots$, have dense linear span in $\ell^2(\mathbb Z)$. This is the case: I'm going to use the fact that we can do weighted uniform polynomial approximation on $\mathbb R$ with respect to an exponential weight $w(x)=e^{\delta |x|}$ (this is perhaps an overkill; the result is discussed in Koosis, The logarithmic integral I). In particular, for any finitely supported sequence $b_k$, we can find a polynomial $p(k)$ so that
$$
\sup_{k\in\mathbb Z} \left| \frac{b_k}{\widehat{s}(k)} - p(k) \right| e^{-\delta |k|}< \epsilon .
$$
In other words,
$$
\left| b_k - p(k)\widehat{s}(k) \right| < \epsilon\, |\widehat{s}(k)|e^{\delta |k|} ,
$$
and this means that I can make $\|b-p\widehat{s}\|_2$ arbitrarily small: first take a ridiculously small $\epsilon$ to keep things under control as long as $\widehat{s}e^{\delta |k|}$ is not small, but eventually this term will take over and make everything exponentially small.