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We describe a skeletonization of the spherical harmonic connection problemthat reduces the storage and pre-computation to superoptimal complexities atthe cost of increasing the execution time by the modest multiplicative factorof $\mathcal{O}(\log n)$. One advantage of accelerating the spherical harmonicconnection problem over accelerating synthesis and analysis is thatneighbouring layers (in steps of two) may be expanded in eachother's bases. Theproposed skeletonization maximizes this interconnectivity by overlaying adyadic partitioning on the connection problem. We derive the symmetric-definitebanded generalized eigenvalue problem required to accelerate spherical harmonictransforms. We also include a full analysis of the weighted normalized Jacobiconnection problem with applications to fast harmonic polynomial transforms onthe disk, triangle, rectangle, deltoid, wedge, and any other geometry with abivariate analogue of Jacobi polynomials.

Conquering the pre-computation in two-dimensional harmonic polynomial transforms