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Weighted Laplace Spaces for Spectral Measures and Rational Approximation
Abstract
We introduce the weighted Laplace space Hw, an RKHS of Laplace transforms on (0,∞), and study spectral measures in its dual space H'w. For conforming FEM discretizations of the Dirichlet Laplacian on bounded Lipschitz domains, we prove the dual-norm inequality |μh|H'w ≤ |μ|H'w, where μ= ∑kδλk and μh = ∑k δλk,h. The proof combines min-max monotonicity of FEM eigenvalues with a heat-trace representation of the dual norm. We then analyze Hw-adapted rational approximation of shifted symbols ϕ(x)=(x+κ2)−β and give a conditional transfer principle for estimates proved in the corresponding weighted Laplace pre-image norm. Via dual pairing, the norm inequality yields uniform bounds for finite spectral sums and related transformed observables.
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BibTeX
@misc{jakobsson2026weighted,
title = {{Weighted Laplace Spaces for Spectral Measures and Rational Approximation}},
author = {Stefan Jakobsson and Alice Kozakevicius and Stig Larsson},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.06243},
eprint = {2609.06243},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06243},
}
RIS
TY - UNPB
TI - Weighted Laplace Spaces for Spectral Measures and Rational Approximation
AU - Stefan Jakobsson
AU - Alice Kozakevicius
AU - Stig Larsson
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2609.06243
UR - https://arxiv.org/abs/2609.06243
AB - We introduce the weighted Laplace space H_w, an RKHS of Laplace transforms on (0,∞), and study spectral measures in its dual space H'_w. For conforming FEM discretizations of the Dirichlet Laplacian on bounded Lipschitz domains, we prove the dual-norm inequality |μ_h|_H'_w ≤ |μ|_H'_w, where μ= ∑_kδ_λ_k and μ_h = ∑_k δ_λ_k,h. The proof combines min-max monotonicity of FEM eigenvalues with a heat-trace representation of the dual norm. We then analyze H_w-adapted rational approximation of shifted symbols ϕ(x)=(x+κ^2)^−β and give a conditional transfer principle for estimates proved in the corresponding weighted Laplace pre-image norm. Via dual pairing, the norm inequality yields uniform bounds for finite spectral sums and related transformed observables.
ER -