A case of multivariate Birkhoff interpolation using high order derivatives

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 $k,k=0,1,\dots$ 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.