Construction of Pole Cancellation Functions at Ordinary Poles of Operator-Valued Functions

What the math gives to ML

The paper gives an explicit Laurent-coefficient construction for vector-valued functions that neutralize the singular behavior of an operator-valued function at a pole. The transferable asset is the finite algebraic constraint system on coefficients of a learned rational or resolvent-like neural layer: features can be forced to vanish in directions that would otherwise excite divergent Laurent terms. This suggests a pole-safe rational layer or resolvent branch whose output remains bounded near learned spectral singularities while retaining a nonzero limiting feature. The construction is especially suitable for spectral state-space models, implicit layers, and rational activations whose instability is caused by evaluation near poles.