  We study functions defined on the vertices of the Hamming graphs $H(n,q)$.
The adjacency matrix of $H(n,q)$ has $n+1$ distinct eigenvalues $n(q-1)-q\cdot
i$ with corresponding eigenspaces $U_{i}(n,q)$ for $0\leq i\leq n$. In this
work, we consider the problem of finding the minimum possible support (the
number of nonzeros) of functions belonging to a direct sum $U_i(n,q)\oplus
U_{i+1}(n,q)\oplus\ldots\oplus U_j(n,q)$ for $0\leq i\leq j\leq n$. For the
case $n\geq i+j$ and $q\geq 3$ we find the minimum cardinality of the support
of such functions and obtain a characterization of functions with the minimum
cardinality of the support. In the case $n<i+j$ and $q\geq 4$ we also find the
minimum cardinality of the support of functions, and obtain a characterization
of functions with the minimum cardinality of the support for $i=j$, $n<2i$ and
$q\geq 5$. In particular, we characterize eigenfunctions from the eigenspace
$U_{i}(n,q)$ with the minimum cardinality of the support for cases $i\le
\frac{n}{2}$,$q\ge 3$ and $i> \frac{n}{2}$,$\,q\ge 5$.
