Reconstruction of Eigenfunctions of q-ary n-dimensional Hypercube

Under study are eigenfunctions of $q$-ary $n$-dimensional hypercube. Given all values of an eigenfunction in the sphere we develop methods to reconstruct the function in full or in part. First, we obtain that all values of the function in the corresponding ball are uniquely determined under some supplementary conditions. Secondly, if the radius is equal to the eigenvalue number we obtain that all values of the eigenfunction are uniquiely determined under some supplementary conditions.