Not sure if this is a 'mathematical' problem or a numerical/computational problem, but I thought I might post it here and hope someone can point me in the right direction.
Consider a function $g(t)$ in the time-domain that consists of a sum of time-scaled versions of an unknown basis function $f(t)$ ie.
$g(t) = \sum_{i=1}^{N} f(\alpha_i t)$,
where $N \in \mathbb{N}$ and the time scaling coefficients $\alpha_i \in \mathbb{R}^+, i \in \{1,..,N\}$ are known. We can assume $0 < \alpha_i \leq 1$.
Observations of $g(t)$ are made at uniformly sampled discrete points $g(n), n \in [0,1,...,M-1]$. It is assumed that $g(t)$ and $f(t)$ are continuous and otherwise "well behaved" for $t \in \mathbb{R}$.
The objective is to obtain an estimate of the function $f(t)$ at the sample points ie. $f(n), n \in [0,1,...,M-1]$.
One approach I have tried is to express both $f(t)$ and $g(t)$ as polynomials of degree $K$, equate polynomial coefficients and solve; ie.
$g(t) = \sum_{k=0}^{K} g_{k}.t^{k}$
$f(t) = \sum_{k=0}^{K} f_{k}.t^{k}$
and hence
$f_K = g_K . \left(\sum_{i=1}^{N} \alpha_{i}^{K} \right)^{-1}$
$f_{K-1} = g_{K-1} .\left(\sum_{i=1}^{N} \alpha_{i}^{K-1} \right)^{-1}$
$\ldots$ etc $\ldots$
$f_0 = g_0.\left(\sum_{i=1}^{N} \alpha_{i}^{0}\right)^{-1}$
Directly implementing this approach (naively?) suffers from the sensitivity of estimating $g(t)$ with a degree $K$ polynomial. Does anyone know of other approaches to this problem that may be more numerically stable?