  The method for calculation of valence quark distributions at intermediate $x$
is presented. The imaginary part of the virtual photon forward scattering
amplitude on the quark current with meson quantum number is considered. Initial
and final virtualities $p^2_1$ and $p^2_2$ of the currents are assumed to be
large, negative and different, $p^2_1 \not= p^2_2$. The operator product
expansion (OPE) in $p^2_1, p^2_2$ up to dimension 6 operators is performed.
Double dispersion representations in $p^2_1, p^2_2$ of the amplitude in terms
of physical states contributions are used. Equalling them to those calculated
in QCD by OPE the desired sum rules for quark distributions in mesons are
found. The double Borel transformations are applied to the sum rules, killing
non-diagonal transition terms, which deteriorated the accuracy in the previous
calculations of quark distributions in nucleon. Leading order perturbative
corrections are accounted. Valence quark distributions in pion, longitudinally
and transversally polarized $\rho$-mesons are calculated at intermediate $x$,
$0.2 \la x \la 0.7$ and normalization points $Q^2 = 2-4 GeV^2$ with no fitting
parameters. The use of the Regge behaviour at small $x$ and quark counting
rules at large $x$ allows one to find the first and the second moments of
valence quark distributions. The obtained quark distributions may be used as an
input for evolution equations. In the case of pion the quark distribution is in
agreement with those found from the data on the Drell-Yan process.
