Reconstruction of eigenfunctions of a $q$-ary $n$-dimensional hypercube

We prove that values of an arbitrary eigenfunction of a $q$-ary $n$-dimensional hypercube can be uniquely reconstructed at all vertices inside a ball if its values on the corresponding sphere are known; we give sufficient conditions for such reconstruction in terms of the eigenvalue and the ball radius. We show that in the case where values of an eigenfunction are given on a sphere of radius equal to the corresponding eigenvalue, all values of the eigenfunction can be reconstructed; similarly to the previous case, sufficient numerical conditions are presented.