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On the Gram matrix of standard inner products of asymmetrically-weighted Hermite functions

  • Ruiyang Dai
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Abstract

Let A denote the infinite Gram matrix associated with the standard L2 inner product of asymmetrically-weighted (AW) Hermite functions. We derive an explicit representation of its entries and its Cholesky factorization. We further show that this factorization admits a natural interpretation on a scaled Bargmann-Fock basis. An explicit formula for the inverse of A is also obtained. We then consider the corresponding finite Gram matrix and analyze its asymptotic property, as well as that of its Schur complement. The analysis is motivated by numerical methods for plasma physics, in particular Galerkin spectral methods applied to the Vlasov-Poisson (VP) system. As an application, we demonstrate how the derived Gram matrix formulas and asymptotic results can be exploited in the analysis and implementation of a Galerkin spectral method for the VP system.

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Cite

BibTeXRIS
BibTeX
@misc{dai2026gram,
  title = {{On the Gram matrix of standard inner products of asymmetrically-weighted Hermite functions}},
  author = {Ruiyang Dai},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.01144},
  eprint = {2609.01144},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.01144},
}
RIS
TY  - UNPB
TI  - On the Gram matrix of standard inner products of asymmetrically-weighted Hermite functions
AU  - Ruiyang Dai
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.01144
UR  - https://arxiv.org/abs/2609.01144
AB  - Let A denote the infinite Gram matrix associated with the standard L2 inner product of asymmetrically-weighted (AW) Hermite functions. We derive an explicit representation of its entries and its Cholesky factorization. We further show that this factorization admits a natural interpretation on a scaled Bargmann-Fock basis. An explicit formula for the inverse of A is also obtained. We then consider the corresponding finite Gram matrix and analyze its asymptotic property, as well as that of its Schur complement. The analysis is motivated by numerical methods for plasma physics, in particular Galerkin spectral methods applied to the Vlasov-Poisson (VP) system. As an application, we demonstrate how the derived Gram matrix formulas and asymptotic results can be exploited in the analysis and implementation of a Galerkin spectral method for the VP system.
ER  -