  We seek to find a shapelet-based scheme for deconvolving galaxy images from
the PSF which leads to unbiased shear measurements. Based on the analytic
formulation of convolution in shapelet space, we construct a procedure to
recover the unconvolved shapelet coefficients under the assumption that the PSF
is perfectly known. Using specific simulations, we test this approach and
compare it to other published approaches. We show that convolution in shapelet
space leads to a shapelet model of order $n_{max}^h = n_{max}^g + n_{max}^f$
with $n_{max}^f$ and $n_{max}^g$ being the maximum orders of the intrinsic
galaxy and the PSF models, respectively. Deconvolution is hence a
transformation which maps a certain number of convolved coefficients onto a
generally smaller number of deconvolved coefficients. By inferring the latter
number from data, we construct the maximum-likelihood solution for this
transformation and obtain unbiased shear estimates with a remarkable amount of
noise reduction compared to established approaches. This finding is
particularly valid for complicated PSF models and low $S/N$ images, which
renders our approach suitable for typical weak-lensing conditions.
