Optimal functions with spectral constraints in hypercubes

The $n$-dimensional hypercube has $n+1$ distinct eigenvalues $n-2i$, $0\leq i\leq n$, with corresponding eigenspaces $U_i(n)$. In 2021 it was proved by the author that if a function with non-empty support belongs to the direct sum $U_i(n)\oplus U_{i+1}(n)\oplus\ldots\oplus U_j(n)$, where $0\leq i\leq j\leq n$, then it has at least $\max(2^i,2^{n-j})$ non-zeros. In this work we give a characterization of functions achieving this bound.