  We establish exact conditions for non triviality of all subspaces of the
standard Hardy space in the upper half plane, that consist of character
automorphic functions with respect to the action of a discrete subgroup of
$SL_2(\mathbb R)$. Such spaces are natural objects in the context of the
spectral theory of almost periodic differential operators and in asymptotics of
approximations by entire functions. A naive idea: it should be completely
parallel to the celebrated Widom characterization for Hardy spaces on Riemann
surfaces with a minor modification, namely, one has to substitute the Green
function of the domain with the Martin function. Basically, this is correct,
but...
