A Riemann-Hilbert representation for Sobolev orthogonal polynomials

What the math gives to ML

The paper supplies a principled Sobolev-orthogonal basis whose geometry penalizes both function magnitude and derivative magnitude under a confining weight. This is transferable to neural networks as a fixed or learnable feature basis for MLPs, replacing highly correlated polynomial features with features that are orthogonal in the exact metric relevant to smoothness. The associated second-order ODE also provides a constructive route for generating numerically stable basis functions and for monitoring how the derivative penalty changes feature concentration.