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

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.