Introduction:
Spherical harmonics are functions which are often very useful in science and engineering when trying to model things which obey some kind of spherically symmetric differential equation, especially Laplace's equation.
Discrete wavelets are families of functions which live in scale-spaces and exhibit certain properties of self-similarity. There is a sequence of scaling functions and one or several sequences of wavelet functions. These are related in a way so that the wavelet functions living in the next scale is a convolution between the scaling function on this scale and the it's "mother wavelet".
A hexagonal grid is a 2 dimensional grid which has three axes which differ by 60 degrees where a "normal" cartesian grid only has two axes which differ by 90 degrees.
Usually in 2D in applications like image processing and compression, wavelets are designed separately on a cartesian grid, by performing 1D wavelet transforms in both dimensions. However this gives very poor treatment of directional information.
To the question: Could we design wavelet filters living on a hexagonal grid, so that they (for each scale) approximate some number of spherical harmonics. It is not necessary to span a basis ( - so overcomplete frames are also of interest ).
As a follow-up question, what would be the smallest size of the filters which could be possible to reasonably approximate all spherical harmonics up to order $k$? References to research articles are also welcome.