The estimation of gravitational radiations multipole moments is a central problem in gravitational wave theory, with essential applications in gravitational wave signal modeling and data analysis. This problem is complicated by most astrophysically relevant systems not having angular modes that are analytically understood. A ubiquitous workaround is to use spin weighted spherical harmonics to estimate multipole moments; however, these are only related to the natural modes of non-spinning spacetimes, thus obscuring the behavior of radiative modes when the source has angular momentum. In such cases, radiative modes are spheroidal in nature. Here, common approaches to the estimation of spheroidal harmonic multipole moments are unified under a simple framework. This framework leads to a new class of spin weighted spheroidal harmonic functions. Adjoint-spheroidal harmonics are introduced and used to motivate the general estimation of spheroidal harmonic multipole moments via bi-orthogonal decomposition with overtone subsets. In turn, the adjoint-spheroidal harmonics are used to construct a single linear operator for which all spheroidal harmonics are eigenfunctions. Implications of these results on gravitational wave theory are discussed.
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