A note on the convergence analysis of Laguerre approximations for analytic functions
Abstract
In a recent paper (H. Wang, Math. Comp. 93, 2861-2884, 2024), Wang has presented a number of results concerning the convergence of approximations based on generalised Laguerre polynomials, claiming to have provided "the first rigorous proof of root-exponential convergence of Laguerre approximations for analytic functions". In this note we argue that the proofs of the main results of Wang's paper are incomplete, because they rely on taking a limit under an integral sign, and this is not properly justified. We explain how the proofs could be completed using the dominated convergence theorem, provided that certain nontrivial inequalities involving generalised Laguerre polynomials and their weighted Cauchy transforms hold. We make a related conjecture, which, if true, would complete the proofs of Wang's results. This conjecture remains unproven, but appears plausible, based on numerical experiments.
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BibTeX
@misc{caussade2026note,
title = {{A note on the convergence analysis of Laguerre approximations for analytic functions}},
author = {Thomas Caussade and David P. Hewett},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11259},
eprint = {2609.11259},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11259},
}
RIS
TY - UNPB
TI - A note on the convergence analysis of Laguerre approximations for analytic functions
AU - Thomas Caussade
AU - David P. Hewett
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.11259
UR - https://arxiv.org/abs/2609.11259
AB - In a recent paper (H. Wang, Math. Comp. 93, 2861-2884, 2024), Wang has presented a number of results concerning the convergence of approximations based on generalised Laguerre polynomials, claiming to have provided "the first rigorous proof of root-exponential convergence of Laguerre approximations for analytic functions". In this note we argue that the proofs of the main results of Wang's paper are incomplete, because they rely on taking a limit under an integral sign, and this is not properly justified. We explain how the proofs could be completed using the dominated convergence theorem, provided that certain nontrivial inequalities involving generalised Laguerre polynomials and their weighted Cauchy transforms hold. We make a related conjecture, which, if true, would complete the proofs of Wang's results. This conjecture remains unproven, but appears plausible, based on numerical experiments.
ER -