  To a good approximation, on large cosmological scales the evolved two-point
correlation function of biased tracers is related to the initial one by a
convolution. For Gaussian initial conditions, the smearing kernel is Gaussian,
so if the initial correlation function is parametrized using simple polynomials
then the evolved correlation function is a sum of generalized Laguerre
functions of half-integer order. This motivates an analytic Laguerre
reconstruction algorithm which previous work has shown is fast and accurate.
This reconstruction requires as input the width of the smearing kernel. We show
that the method can be extended to estimate the width of the smearing kernel
from the same dataset. This estimate, and associated uncertainties, can then be
used to marginalize over the distribution of reconstructed shapes, and hence
provide error estimates on the value of the distance scale which are not tied
to a particular cosmological model. We also show that if, instead, we
parametrize the evolved correlation function using simple polynomials, then the
initial one is a sum of Hermite polynomials, again enabling fast and accurate
deconvolution. If one is willing to use constraints on the smearing scale from
other datasets, then marginalizing over its value is simpler for Hermite
reconstruction, potentially providing further speed-up in cosmological
analyses.
