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PreprintCC BY 4.0arXiv

Structured coordinate systems in Lp(ℝ)

  • Daniel Freeman
  • Lukas Liehr
  • Mitchell A. Taylor
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Abstract

We construct Schauder bases of Lp(ℝ) with significant structural constraints on their elements. First, we construct non-negative Schauder bases of Lp(ℝ) for every 1 ≤ p < ∞. Although such bases cannot be unconditional, their basis constants can be made arbitrarily close to one. By duality, we also obtain bases with positive biorthogonal functionals and the same bounds for all 1≤ p<∞. Finally, we prove that Lp(ℝ) admits a Schauder basis consisting of translates of a single function for every 2≤ p<∞. The generating function f can be chosen to be the restriction to ℝ of an entire function. For p>2, the translation parameters can be chosen from any prescribed unbounded subset of ℝ, a conclusion that fails for p=2.

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Cite

BibTeXRIS
BibTeX
@misc{freeman2026structured,
  title = {{Structured coordinate systems in $L^p(\mathbbR)$}},
  author = {Daniel Freeman and Lukas Liehr and Mitchell A. Taylor},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.18709},
  eprint = {2610.18709},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.18709},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Structured coordinate systems in L^p(ℝ)
AU  - Daniel Freeman
AU  - Lukas Liehr
AU  - Mitchell A. Taylor
PY  - 2026
DA  - 2026/10/13
PB  - arXiv
DO  - 10.48550/arxiv.2610.18709
UR  - https://arxiv.org/abs/2610.18709
AB  - We construct Schauder bases of L^p(ℝ) with significant structural constraints on their elements. First, we construct non-negative Schauder bases of L^p(ℝ) for every 1 ≤ p < ∞. Although such bases cannot be unconditional, their basis constants can be made arbitrarily close to one. By duality, we also obtain bases with positive biorthogonal functionals and the same bounds for all 1≤ p<∞. Finally, we prove that L^p(ℝ) admits a Schauder basis consisting of translates of a single function for every 2≤ p<∞. The generating function f can be chosen to be the restriction to ℝ of an entire function. For p>2, the translation parameters can be chosen from any prescribed unbounded subset of ℝ, a conclusion that fails for p=2.
ER  -