Approximation quality of k_λ for θ_x

Show that the function k_λ(u) = (h_λ)(u), where h_λ is the unique linear combination of the even prolate spheroidal eigenfunctions h_{0,λ} and h_{4,λ} with vanishing integral, provides a sufficiently accurate approximation to θ_x(u) = η_x(x^{1/2}u) on [x^{-1/2}, x^{1/2}] for λ = √x.

Background

The paper proposes approximating the minimizer θ_x by replacing Hermite functions with their prolate spheroidal counterparts and applying a Poisson-type summation. Numerics suggest excellent agreement for small eigenvectors of the Weil quadratic form.

A rigorous proof that k_λ is a sufficiently good approximation to θ_x (with λ = √x) remains to be supplied, and is a necessary step to promote the numerical evidence to a theorem.