  Gauss-Hermite and Gauss-Laguerre ("shapelet") decompositions of images have
become important tools in galaxy modeling, particularly for the purpose of
extracting ellipticity and morphological information from astronomical data.
However, the standard shapelet basis functions cannot compactly represent
galaxies with high ellipticity or large Sersic index, and the resulting
underfitting bias has been shown to present a serious challenge for
weak-lensing methods based on shapelets. We present here a new convolution
relation and a compound "multi-scale" shapelet basis to address these problems,
and provide a proof-of-concept demonstration using a small sample of nearby
galaxies.
