Multivariate Lebesgue bounds for R-Leja lower grids
Abstract
We establish an algebraic upper bound for the Lebesgue constant of multivariate polynomial interpolation on R-Leja lower grids. For each fixed dimension d≥2 and every nonempty finite lower set A⊂ℕ0d, let PA denote the corresponding grid with N=|A| nodes. We prove Λ(PA)≤ CdNβd, where qd=d/(d−1) and βd=1+qd−1log2(1+2qd). The exponent satisfies β2=2.160964… and increases to log2 6=2.584962… as d→∞, improving both the cubic estimate and the exponent 3−1/d obtained from quadratic prefix bounds. Our proof exploits the binary structure of known univariate estimates to control sums of powers of Lebesgue constants over interpolation prefixes. A weighted Hölder inequality and a disjoint-box packing argument then transfer these prefix-moment bounds to arbitrary lower sets. The resulting estimate is uniform over all lower-set shapes and choices of disk Leja maximizers, with constants depending only on d.
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BibTeX
@misc{hofmann2026multivariate,
title = {{Multivariate Lebesgue bounds for R-Leja lower grids}},
author = {Phil-Alexander Hofmann and Michael Hecht},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40177},
eprint = {2610.40177},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40177},
}
RIS
TY - UNPB
TI - Multivariate Lebesgue bounds for R-Leja lower grids
AU - Phil-Alexander Hofmann
AU - Michael Hecht
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.40177
UR - https://arxiv.org/abs/2610.40177
AB - We establish an algebraic upper bound for the Lebesgue constant of multivariate polynomial interpolation on R-Leja lower grids. For each fixed dimension d≥2 and every nonempty finite lower set A⊂ℕ_0^d, let P_A denote the corresponding grid with N=|A| nodes. We prove Λ(P_A)≤ C_dN^β_d, where q_d=d/(d−1) and β_d=1+q_d^−1log_2(1+2^q_d). The exponent satisfies β_2=2.160964… and increases to log_2 6=2.584962… as d→∞, improving both the cubic estimate and the exponent 3−1/d obtained from quadratic prefix bounds. Our proof exploits the binary structure of known univariate estimates to control sums of powers of Lebesgue constants over interpolation prefixes. A weighted Hölder inequality and a disjoint-box packing argument then transfer these prefix-moment bounds to arbitrary lower sets. The resulting estimate is uniform over all lower-set shapes and choices of disk Leja maximizers, with constants depending only on d.
ER -