  We consider a specific scheme of multivariate Birkhoff polynomial
interpolation. Our samples are derivatives of various orders $k_j$ at fixed
points $v_j$ along fixed straight lines through $v_j$ in directions $u_j$,
under the following assumption: the total number of sampled derivatives of
order $k, \ k=0,1,\ldots$ is equal to the dimension of the space homogeneous
polynomials of degree $k$. We show that this scheme is regular for general
directions. Specifically this scheme is regular independent of the position of
the interpolation nodes. In the planar case, we show that this scheme is
regular for distinct directions.
  Next we prove a "Birkhoff-Remez" inequality for our sampling scheme extended
to larger sampling sets. It bounds the norm of the interpolation polynomial
through the norm of the samples, in terms of the geometry of the sampling set.
