Handling spurious poles when the Pick matrix fails positivity in Nevanlinna–Pick interpolation

Determine an effective and field-theoretically justified procedure to address the appearance of spurious poles that arise in Nevanlinna–Pick interpolation when statistical uncertainties cause the Pick matrix to fail to be positive semidefinite. The goal is to enable stable analytic continuation of lattice QCD Euclidean Green functions and corresponding smeared spectral reconstructions under realistic noisy data conditions.

Background

In the Nevanlinna–Pick formulation of analytic continuation for lattice QCD, Euclidean Green function data on the imaginary axis are mapped via conformal transforms so that an analytic function f: D → D interpolates transformed points in the unit disk. Existence of an interpolant is guaranteed if and only if the associated Pick matrix is positive semidefinite.

In practice, statistical uncertainties in the Euclidean data can violate the Pick matrix positivity condition. The resulting construction produces interpolants with spurious poles inside the unit disk, leading to numerical instability and undermining control over analytic continuation. The paper explicitly notes that the optimal strategy for dealing with this failure mode remains an open question, although related work on robust continuation and denoising has begun to address aspects of it.