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

Jean-Pierre Magnot

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.

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 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.
Paper Structure (47 sections, 30 theorems, 240 equations)

This paper contains 47 sections, 30 theorems, 240 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Polynomial spaces and pre-Hilbert structures
  4. Gram--Schmidt orthogonalization and polynomial projectors
  5. Derivations, closability, and associated Laplacians
  6. Resolvents, norm-resolvent distance, and finite-degree compressions
  7. Resolvents and operator topologies
  8. The Laplacian associated with a polynomial inner product
  9. Degree growth and admissible derivations
  10. Closability of derivations and definition of $\Delta$
  11. Matrix representation in an orthonormal polynomial basis
  12. Banded recurrences and banded Laplacians
  13. Example: thin annuli and mode-by-mode block structure
  14. The Laplacian in the classical one-dimensional weighted $L^2$ setting
  15. Weighted $L^2$ space and adjoint of the derivative
  16. ...and 32 more sections

Key Result

Proposition 3.3

Let $(P_n)_{n\ge 0}$ be an orthonormal polynomial basis of $H$ and let $B=(B_{k,n})_{k,n\ge 0}$ be the matrix of $D$ in this basis, Assume that $B$ is banded with half-bandwidth $r\ge 1$, i.e. Then the matrix of $\Delta=D^*D$ in the same basis, is banded with half-bandwidth at most $2r$, i.e. Moreover $[\Delta]$ is Hermitian and positive semidefinite.

Theorems & Definitions (82)

  • Remark 2.1
  • Remark 2.2: Variable Hilbert spaces and quasi-unitary equivalence
  • Remark 3.1: Positive definite case
  • Remark 3.2
  • Proposition 3.3: Bandwidth propagation
  • proof
  • Remark 3.4
  • Proposition 3.5: Block diagonalization by angular modes
  • Lemma 3.6: Angular decomposition induced by rotational invariance
  • proof
  • ...and 72 more