Deep zero problems

In Hilbert spaces of holomorphic functions, we are typically concerned with zero sets, interpolation, and at times sampling. Here, I will introduce a seemingly unexplored direction, which considers in a sense multiple zeros but not in the classical sense. We begin with (say) finitely many points, and add conditions on higher derivatives at those points. If all higher derivatives $f^{(n)}(z_0)$ vanish at one of the given point $z_0$, for $n=0,1,2,...$, then by Taylor's formula the holomorphic function $f$ vanishes. But what if only a fraction of those derivatives vanish? If there is only one point, then clearly we can offer an immediate example of such a function ($(z-z_0)^k$ for a suitable $k$ will do). But now we have several such points and it is not clear what actually happens. It may depend on the density of vanishing derivatives and of course on the given Hilbert space of holomorphic functions. We take a look at specific examples to see what can happen. It appears not even clear what happens if we have the space of polynomials of degree $<N$. A natural conjecture is that $N$ vanishing conditions on higher derivatives of order $<N$ should be exactly the condition for uniqueness. Once we have the concept of deep zeros it is not hard to develop the concept of deep interpolation. We will discuss this as well.