Geometry of error amplification in solving Prony system with near-colliding nodes

Understand

We consider a reconstruction problem for ``spike-train'' signals $F$ of an a priori known form $F(x)=\sum_{j=1}^{d}a_{j}\delta\left(x-x_{j}\right),$ from their moments $m_k(F)=\int x^kF(x)dx.$ We assume that the moments $m_k(F)$, $k=0,1,\ldots,2d-1$, are known with an absolute error not exceeding $\epsilon > 0$.

• This problem is essentially equivalent to solving the Prony system $\sum_{j=1}^d a_jx_j^k=m_k(F), \ k=0,1,\ldots,2d-1.$ We study the ``geometry of error amplification'' in reconstruction of $F$ from $m_k(F),$ in situations where the nodes $x_1,\ldots,x_d$ near-collide, i.e.
• form a cluster of size $h \ll 1$.
• We show that in this case, error amplification is governed by certain algebraic varieties in the parameter space of signals $F$, which we call the ``Prony varieties''.