Characterizing infinitely oscillating functions on $L_2[0,1]$

Question

Descarte's Rule of Signs tells me, as a function of the (sign pattern of the) vector $\mathbf{a}\in\mathbb{R}^{n+1}$, a bound on the number of roots of the polynomial $p(k)=\sum_{k=0}^n a_k x^k$. Is there an infinite dimensional version? I.e. for a given functional basis (Fourier, wavelet, etc.) of $L_2[0,1]$ are there conditions one can place on the basis coordinates to check that a function has only finitely many zero-crossings? (I wouldn't expect this to be easy for all bases, but perhaps it emerges naturally in a particular one).