  In this paper, we investigate quantum Fourier analysis on subfactors and
unitary fusion categories. We prove the complete positivity of the
comultiplication for subfactors and derive a primary $n$-criterion of unitary
categorifcation of multifusion rings. It is stronger than the Schur product
criterion when $n\geq3$. The primary criterion could be transformed into
various criteria which are easier to check in practice even for noncommutative,
high-rank, high-multiplicity, multifusion rings. More importantly, the primary
criterion could be localized on a sparse set, so that it works for multifusion
rings with sparse known data. We give numerous examples to illustrate the
efficiency and the power of these criteria.
