  Consider $L$ groups of point sources or spike trains, with the
$l^{\text{th}}$ group represented by $x_l(t)$. For a function $g:\mathbb{R}
\rightarrow \mathbb{R}$, let $g_l(t) = g(t/\mu_l)$ denote a point spread
function with scale $\mu_l > 0$, and with $\mu_1 < \cdots < \mu_L$. With $y(t)
= \sum_{l=1}^{L} (g_l \star x_l)(t)$, our goal is to recover the source
parameters given samples of $y$, or given the Fourier samples of $y$. This
problem is a generalization of the usual super-resolution setup wherein $L =
1$; we call this the multi-kernel unmixing super-resolution problem. Assuming
access to Fourier samples of $y$, we derive an algorithm for this problem for
estimating the source parameters of each group, along with precise
non-asymptotic guarantees. Our approach involves estimating the group
parameters sequentially in the order of increasing scale parameters, i.e., from
group $1$ to $L$. In particular, the estimation process at stage $1 \leq l \leq
L$ involves (i) carefully sampling the tail of the Fourier transform of $y$,
(ii) a \emph{deflation} step wherein we subtract the contribution of the groups
processed thus far from the obtained Fourier samples, and (iii) applying
Moitra's modified Matrix Pencil method on a deconvolved version of the samples
in (ii).
