A case of multivariate Birkhoff interpolation using high order derivatives

TL;DR

This paper introduces a multivariate Birkhoff interpolation scheme using high-order derivatives along fixed lines, demonstrating its regularity and establishing a Birkhoff-Remez inequality that bounds polynomial norms based on sample geometry.

Contribution

The paper develops a new multivariate Birkhoff interpolation method with regularity results and extends the Birkhoff-Remez inequality to this scheme, linking polynomial bounds to sampling set geometry.

Findings

Scheme is regular for general directions

Scheme is regular for distinct directions in the planar case

Establishes a Birkhoff-Remez inequality for the sampling scheme

Abstract

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.