Optimal functions with spectral constraints in hypercubes

Abstract

The n-dimensional hypercube has n+1 distinct eigenvalues n−2i, 0≤i≤n, with corresponding eigenspaces Ui(n). In 2021 it was proved by the author that if a function with non-empty support belongs to the direct sum Ui(n)⊕Ui+1(n)⊕⋯⊕Uj(n), where 0≤i≤j≤n, then it has at least max⁡(2i,2n−j) non-zeros. In this work we give a characterization of functions achieving this bound.