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

A case of multivariate Birkhoff interpolation using high order derivatives