Preliminaries on Pre-Hilbert Structures on Polynomial Spaces and Associated Laplacians
Fuente:
arXiv
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866914397313564672 |
|---|---|
| author | Magnot, Jean-Pierre |
| author_facet | Magnot, Jean-Pierre |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_22394 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Preliminaries on Pre-Hilbert Structures on Polynomial Spaces and Associated Laplacians Magnot, Jean-Pierre General Mathematics 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. |
| title | Preliminaries on Pre-Hilbert Structures on Polynomial Spaces and Associated Laplacians |
| topic | General Mathematics |
| url | https://arxiv.org/abs/2512.22394 |