Spectrality of convolution and product measures: a block decomposition approach

@misc{zhou2026spectrality,
  title = {{Spectrality of convolution and product measures: a block decomposition approach}},
  author = {Ting Zhou},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.33987},
  eprint = {2610.33987},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.33987},
  note = {CC0},
}

TY  - UNPB
TI  - Spectrality of convolution and product measures: a block decomposition approach
AU  - Ting Zhou
PY  - 2026
DA  - 2026/10/25
PB  - arXiv
DO  - 10.48550/arxiv.2610.33987
UR  - https://arxiv.org/abs/2610.33987
AB  - We study the spectrality of convolution and product measures. Let~μ~be a compactly supported Borel probability measure on~BbbR^d~such that ~𝒵(μ̂)∪{0}=T~is an additive subgroup, and let~ν~be a Borel probability measure on~BbbR^d. We prove that~μ*ν~is spectral if and only if~T~is a spectrum of~μ, ~ν~is spectral, and~suppν~is contained in a translate of the annihilator of~T. Moreover, every spectrum of~μ*ν~decomposes into full cosets of~T, and every choice of one point from each coset is a spectrum of~ν. We also obtain a convolution-factor criterion which strengthens a result of Wu. As applications, we characterize the spectrality of~L_[0,1]*ν~and prove that the spectrality of~L_[0,1]^d*ν~implies that of~ν~without a compact support assumption. For product measures, we give necessary and sufficient conditions for an orthogonal set to be a spectrum by relating its block structure to spectra of the factors. No compact support assumptions are required.
ER  -