We introduce fusion bialgebras and their duals and systematically study their
Fourier analysis. As an application, we discover new efficient analytic
obstructions on the unitary categorification of fusion rings. We prove the
Hausdorff-Young inequality, uncertainty principles for fusion bialgebras and
their duals. We show that the Schur product property, Young's inequality and
the sum-set estimate hold for fusion bialgebras, but not always on their duals.
If the fusion ring is the Grothendieck ring of a unitary fusion category, then
these inequalities hold on the duals. Therefore, these inequalities are
analytic obstructions of categorification. We classify simple integral fusion
rings of Frobenius type up to rank 8 and of Frobenius-Perron dimension less
than 4080. We find 34 ones, 4 of which are group-like and 28 of which can be
eliminated by applying the Schur product property on the dual. In general,
these inequalities are obstructions to subfactorize fusion bialgebras.Comment: 39 pages; 8 figures; addition of a classification in Subsection 9.2;
the long lists in Subsection 9.3 are now more pleasant to read; addition of
Section 7 providing a categorical proof of Schur Product Theore