Bivariate interpolation with integrals

The bivariate interpolation problem, where the interpolation parameters are integrals over bounded regions, is considered in this paper. H. Hakopian posed a hypothesis for this problem in the case, when regions are obtained from intersection of lines in general position [2]. Till now the hypothesis is proved for polynomials of degree $\leq1$. In this paper we bring a new proof. Meanwhile we solve the problem in more general setting – in the case of arbitrary regions.