Accuracy of reconstruction of spike-trains with two near-colliding nodes

Abstract

We consider a signal reconstruction problem for signals $F$ of the form $F(x)={\sum }_{j=1}^d{a}_j\delta \left(x-x_j\right),$ from their moments $m_k(F)=\int x^{k}F(x)dx.$ We assume $m_k(F)$ to be known for $k=0,1,\dots ,N,$ with an absolute error not exceeding $\epsilon >0$. We study the "geometry of error amplification" in reconstruction of $F$ from $m_k(F),$ in situations where two neighboring nodes $x_i$ and $x_{i+1}$ near-collide, i.e $x_{i+1}-x_i=h\ll 1$. We show that the error amplification is governed by certain algebraic curves $S_{F,i},$ in the parameter space of signals $F$, along which the first three moments $m_0,m_1,m_2$ remain constant.