Positivity loss in bandlimited spectral reproduction on spheres
Abstract
How small can the positivity loss be for an N-bandlimited spectral operator that exactly reproduces all modes up to degree L? For spherical polynomial approximation on 𝕊d, we prove that the smallest possible excess of the uniform operator norm above 1, equivalently the least positivity loss, is of sharp order (L/(N+1))2 when 1≤ L< N. The lower bound follows from a Fejér peak test and a concentration estimate for bandlimited kernels, while a matching upper bound is obtained by correcting a positive Jackson operator with a smooth filter. We illustrate the result in three settings. On the circle, taking N=sL−1, this determines the sharp order of the generalized-projection constant above 1 and identifies the gap between the s−1 excess of delayed de la Vallée--Poussin means and the optimal s−2 order. For filtered hyperinterpolation, whose operator norm has long been known to be uniformly bounded, we give a quantitative lower bound on its separation from the positivity threshold 1. Finally, we identify an operator-level obstruction to maximum principles.
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BibTeX
@misc{wu2026positivity,
title = {{Positivity loss in bandlimited spectral reproduction on spheres}},
author = {Hao-Ning Wu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02695},
eprint = {2609.02695},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02695},
}
RIS
TY - UNPB
TI - Positivity loss in bandlimited spectral reproduction on spheres
AU - Hao-Ning Wu
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.02695
UR - https://arxiv.org/abs/2609.02695
AB - How small can the positivity loss be for an N-bandlimited spectral operator that exactly reproduces all modes up to degree L? For spherical polynomial approximation on 𝕊^d, we prove that the smallest possible excess of the uniform operator norm above 1, equivalently the least positivity loss, is of sharp order (L/(N+1))^2 when 1≤ L< N. The lower bound follows from a Fejér peak test and a concentration estimate for bandlimited kernels, while a matching upper bound is obtained by correcting a positive Jackson operator with a smooth filter. We illustrate the result in three settings. On the circle, taking N=sL−1, this determines the sharp order of the generalized-projection constant above 1 and identifies the gap between the s^−1 excess of delayed de la Vallée--Poussin means and the optimal s^−2 order. For filtered hyperinterpolation, whose operator norm has long been known to be uniformly bounded, we give a quantitative lower bound on its separation from the positivity threshold 1. Finally, we identify an operator-level obstruction to maximum principles.
ER -