Two-dimensional polynomial interpolation from nonuniform samples

A number of results are presented concerning sufficient conditions under which the two-dimensional (2-D) polynomial interpolation problem has a unique or nonunique solution. It is found that unless an appropriate number of interpolation points are chosen on an appropriate number of irreducible curves, the resulting problem might become singular. Specifically, if the sum of the degrees of the irreducible curves on which the interpolation points are chosen is small compared to the degree of the interpolating polynomial, then the problem becomes singular. Similarly, if there are too many points on any of the irreducible curves on which the interpolation points are chosen, the interpolation problem runs into singularity. Examples of geometric distributions of interpolation points satisfying these conditions are shown. The examples include polynomial interpolation of polar samples, and samples on straight lines. The author proposes a recursive algorithm for a class of interpolation points