Direct conditions guaranteeing a valid interpolant

Determine conditions on a bivariate potential γ that guarantee its Hermite interpolant at the five-point Fibonacci interpolation nodes lies below γ, thereby providing a direct continuous linear-programming certificate without requiring γ to have a tensor-product representation.

Background

The paper's rigorous construction assumes that the potential has the tensor-product form γ(s,t) = f1(s)f2(t), with the factors satisfying absolute-monotonicity and additional positivity conditions. The authors then apply separate one-dimensional Hermite interpolation and correct forbidden frequencies using auxiliary functions.

For a general bivariate potential, the paper considers direct Hermite interpolation at the relevant nodes but finds that natural derivative-positivity assumptions do not ensure that the interpolant remains below the potential. A satisfactory intrinsic condition on γ, possibly involving logarithmic convexity or related differential inequalities, remains unknown.