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PreprintCC BY 4.0arXiv

A counterexample concerning weighted convolution on the integers

  • Frédéric Morneau-Guérin
  • Thomas Ransford
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Abstract

Let 1<p<∞ and let w:ℤ→(0,∞) be a weight. When is the weighted space ℓp(ℤ,w) an algebra with respect to convolution? A well-known sufficient condition is that (w−q*w−q)/w−q be bounded on ℤ, where q:=p/(p−1) is the index conjugate to p. We present an example to show that this sufficient condition is not necessary.

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Cite

BibTeXRIS
BibTeX
@misc{morneauguerin2026counterexample,
  title = {{A counterexample concerning weighted convolution on the integers}},
  author = {Frédéric Morneau-Guérin and Thomas Ransford},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.03334},
  eprint = {2611.03334},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.03334},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A counterexample concerning weighted convolution on the integers
AU  - Frédéric Morneau-Guérin
AU  - Thomas Ransford
PY  - 2026
DA  - 2026/11/02
PB  - arXiv
DO  - 10.48550/arxiv.2611.03334
UR  - https://arxiv.org/abs/2611.03334
AB  - Let 1<p<∞ and let w:ℤ→(0,∞) be a weight. When is the weighted space ℓ^p(ℤ,w) an algebra with respect to convolution? A well-known sufficient condition is that (w^−q*w^−q)/w^−q be bounded on ℤ, where q:=p/(p−1) is the index conjugate to p. We present an example to show that this sufficient condition is not necessary.
ER  -