Two-dimensional polynomial interpolation from nonuniform samples

The authors derive a number of results on sufficient conditions under which the 2-D polynomial interpolation problem has a unique or nonunique solution. It is found that, 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, then the interpolation problem runs into singularity. Examples of geometric distribution of interpolation points satisfying these conditions are shown. The examples include polynomial interpolation of polar samples, and samples on straight lines. The authors propose a recursive algorithm for computing 2-D polynomial coefficients for the nonsingular case where all the interpolation points are chosen on lines passing through the origin. The result is applied to the problem of nonuniform frequency sampling design for 2-D FIR filter design, and a few examples of such design are shown