Optimal functions with spectral constraints in hypercubes

Abstract

The $n$-dimensional hypercube has $n+1$ distinct eigenvalues $n-2i$, $0\le i\le n$, 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 $U_i(n)\oplus U_{i+1}(n)\oplus \dots \oplus U_j(n)$, where $0\le i\le j\le n$, 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.