Tensorial Fourier expansion of orientation distribution function defined on the orthogonal group O(3)

Recently, Man and Du, in the context of classical texture analysis where the orientation distribution function (ODF) is defined on the rotation group SO(3), presented a systematic procedure by which the classical expansion of an ODF truncated at the order [Formula: see text] can be directly rewritten as a tensorial Fourier expansion truncated at the same order. In classical texture analysis, the groups of crystallite and texture symmetries are assumed to be subgroups of SO(3), which is unreasonable, say, for aggregates of crystallites in a crystal class defined by an improper point group. In this paper, we consider ODFs defined on the orthogonal group O(3) and extend the aforementioned procedure to write down tensorial Fourier expansions for polycrystals with crystallite symmetry defined by any of the 21 improper point groups. Examples that illustrate the general procedure are given. This article is part of the theme issue 'Foundational issues, analysis and geometry in continuum mechanics'.