Restriction of Fourier Transform to Fuzzy Subgroups

In this paper, we study the restriction of Fourier transform to fuzzy subgroups of finite groups. For finite abelian groups, we prove that the Fourier transform on a fuzzy subgroup is again (after normalization) a fuzzy subgroup of the dual group, thereby establishing a duality between fuzzy subgroups of [Formula: see text] and those of [Formula: see text]. We further extend the restriction–projection duality and Donoho–Stark uncertainty principle to the fuzzy setting. In the non-abelian case, we show that Fourier transform on fuzzy subgroups can be realized as orthogonal projections onto a fuzzy vector subspace of representation spaces of the irreducible representations of the group.