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Interpolation by the Exact Inversion of the Gram Matrix

John Spitzer

TL;DR

This work proposes a framework for exact kernel interpolation by inverting Gram matrices through Davis's lemma and classical orthogonal polynomials on canonical domains (Laguerre, Legendre, Hermite). By constructing reproducing kernels that separate into even/odd components on symmetric domains, the method yields polynomial approximants with significantly improved error variances over local Taylor expansions, albeit at the cost of rapidly increasing condition numbers. The approach provides explicit kernel forms and inverse coefficients, enabling interpolants and kernel-based estimates up to the degree of the Gram matrix, with an emphasis on how domain symmetry and transformability affect the kernels. The results suggest practical gains for 1D interpolation on standard domains and point toward extensions to higher dimensions via Kronecker products, while acknowledging unisolvence-related challenges that warrant further investigation.

Abstract

Using a lemma of Davis on Gram matrices applied to the classical Orthogonal Polynomials to generate reproducing kernel interpolation over the classical domains for polynomials. These kernels have terms which are exact over the rational ring. The Condition Numbers are readily shown to get very large with the size of the Gram matrices as expected. The calculation of the error variances for trigonometric functions and the exponential show a significant improvement over the equivalent Taylor expansion variances.

Interpolation by the Exact Inversion of the Gram Matrix

TL;DR

This work proposes a framework for exact kernel interpolation by inverting Gram matrices through Davis's lemma and classical orthogonal polynomials on canonical domains (Laguerre, Legendre, Hermite). By constructing reproducing kernels that separate into even/odd components on symmetric domains, the method yields polynomial approximants with significantly improved error variances over local Taylor expansions, albeit at the cost of rapidly increasing condition numbers. The approach provides explicit kernel forms and inverse coefficients, enabling interpolants and kernel-based estimates up to the degree of the Gram matrix, with an emphasis on how domain symmetry and transformability affect the kernels. The results suggest practical gains for 1D interpolation on standard domains and point toward extensions to higher dimensions via Kronecker products, while acknowledging unisolvence-related challenges that warrant further investigation.

Abstract

Using a lemma of Davis on Gram matrices applied to the classical Orthogonal Polynomials to generate reproducing kernel interpolation over the classical domains for polynomials. These kernels have terms which are exact over the rational ring. The Condition Numbers are readily shown to get very large with the size of the Gram matrices as expected. The calculation of the error variances for trigonometric functions and the exponential show a significant improvement over the equivalent Taylor expansion variances.
Paper Structure (20 sections, 1 theorem, 32 equations)

This paper contains 20 sections, 1 theorem, 32 equations.

Table of Contents

  1. Introduction
  2. The Lemma of Davis
  3. The Reproducing Kernels
  4. A Non-symmetric Orthogonal Polynomial
  5. Even Orthogonal Polynomials
  6. Odd Orthogonal Polynomials
  7. Notes
  8. Examples of condition numbers
  9. The Laguerre polynomial case
  10. The Odd Legendre polynomial case
  11. The Even Legendre polynomial case
  12. The Odd Hermite polynomial case
  13. The Even Hermite polynomial case
  14. Examples of error variance numbers
  15. Trigonometric Functions over a finite domain
  16. ...and 5 more sections

Key Result

Lemma 1

According to (Davis Dav) we have Let Let A designate the matrix $(a_{ij})$ and $\tilde{A}$ be its conjugate transpose $(\tilde{a_{ji}})$. Then and

Theorems & Definitions (2)

  • Definition 1: Davis
  • Lemma 1: Davis