Preliminaries on Pre-Hilbert Structures on Polynomial Spaces and Associated Laplacians

TL;DR

This paper develops a unified operator-theoretic framework to analyze stability and geometric limits of orthogonal polynomial systems using pre-Hilbert structures and associated Laplacians, with applications to classical and Sobolev geometries.

Contribution

It introduces a resolvent-based distance between polynomial Hilbert geometries and proves stability results for orthogonalization procedures under this metric.

Findings

Norm-resolvent closeness implies stability of orthogonal bases and kernels.

Explicit examples including orthogonal polynomials on the unit circle and Sobolev regularizations.

Asymptotic regimes can be interpreted as resolvent limits of polynomial geometries.

Abstract

We study orthogonal polynomial systems arising from general pre-Hilbert inner products on polynomial spaces, beyond the classical framework of measures. To each such inner product we associate a canonical Laplacian defined from an abstract derivation, and we investigate the operator-theoretic structures induced by this construction. Our main contribution is the introduction of a resolvent-based distance between polynomial Hilbert geometries, and the proof of quantitative stability results for finite-degree orthogonalization procedures. In particular, we show that norm-resolvent closeness of the associated Laplacians implies stability of Gram--Schmidt orthogonal bases, orthogonal projectors and reproducing kernels on all finite-dimensional polynomial subspaces. The general theory is illustrated by several explicit examples. We analyze in detail the case of orthogonal polynomials on the unit circle, comparing classical $L^2$ geometries associated with finite Radon measures and Sobolev-type regularizations via Fourier methods. We also revisit the thin annulus problem, showing that its asymptotic regime admits a natural interpretation as a resolvent limit of polynomial geometries. These results provide a unified operator-theoretic framework for the study of stability, degenerations and geometric limits of orthogonal polynomial systems.