On reconstruction of eigenfunctions of Johnson graphs

In the present work we consider the problem of a reconstruction of eigenfunctions of the Johnson graph $J(n,w)$. We give necessary and sufficient numerical conditions for a unique reconstruction of an eigenfunction with given eigenvalue by its values on a sphere of given radius $r$ for $n$ big enough. We also provide examples of functions equal on the sphere but not equal on the full vertex set in the case of a failure of these conditions.