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Unbounded Helson Forms with a Moment Representation
Abstract
We study the positive Helson form h0[a,b]=∑m,n≥1α(mn)anbm̅ on the finitely supported sequences c00⊂ℓ2(ℕ), where α(n)=∫ℝn−sdμ(s)<∞ for a finite positive Borel measure μ. Under the Carleman condition ∑k≥1α(4k)−1/(2k)=∞, closability of h0 is equivalent to √nα(n)→0, and hence to μ((−∞,1/2])=0. For measures concentrated on (1/2,∞), we identify the closure of the operator N0a=∑n≥1ann−s, defined on c00, with its maximal realization from ℓ2(ℕ) to L2(μ). We further prove that the sequences a∈ c00 satisfying ∑n≥1an/√n=0 form an operator core for the positive self-adjoint operator associated with the closure of h0.
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BibTeX
@misc{ma2026unbounded,
title = {{Unbounded Helson Forms with a Moment Representation}},
author = {Tengfei Ma and Yufeng Lu and Chao Zu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12252},
eprint = {2610.12252},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12252},
}
RIS
TY - UNPB
TI - Unbounded Helson Forms with a Moment Representation
AU - Tengfei Ma
AU - Yufeng Lu
AU - Chao Zu
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.12252
UR - https://arxiv.org/abs/2610.12252
AB - We study the positive Helson form h_0[a,b]=∑_m,n≥1α(mn)a_nb_m̅ on the finitely supported sequences c_00⊂ℓ^2(ℕ), where α(n)=∫_ℝn^−sdμ(s)<∞ for a finite positive Borel measure μ. Under the Carleman condition ∑_k≥1α(4^k)^−1/(2k)=∞, closability of h_0 is equivalent to √nα(n)→0, and hence to μ((−∞,1/2])=0. For measures concentrated on (1/2,∞), we identify the closure of the operator N_0a=∑_n≥1a_nn^−s, defined on c_00, with its maximal realization from ℓ^2(ℕ) to L^2(μ). We further prove that the sequences a∈ c_00 satisfying ∑_n≥1a_n/√n=0 form an operator core for the positive self-adjoint operator associated with the closure of h_0.
ER -