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Independent powers and bounded Fourier partial sums in L1

  • Przemysław Ohrysko
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Abstract

We construct a probability measure on the circle with independent convolution powers and uniformly bounded symmetric Fourier partial sums in L1. Every positive convolution power is therefore singular with respect to Haar measure. The measure is Rajchman. The construction answers a question of Andersson concerning measures with bounded Fourier partial sums outside the radical of L1, and contrasts with his theorem that the classical examples of Weiss, Katznelson, and Brown-Hewitt have convolution squares in L2. Our proof combines sparse probability blocks, an endpoint Llog L estimate, and a diagonal generalized Riesz product. Exact factorization of local moments permits the simultaneous separation of all convolution powers by Brown's singularity criterion. By the classical spectral theory of independent-power measures, its spectrum is the closed unit disc. In particular, it has non-natural spectrum and induces a non-decomposable convolution operator on L1. The image of the measure in M(𝕋)/L1(𝕋) has spectral radius one, and the closed unital algebra it generates in M(𝕋) is isometrically isomorphic to ℓ1(ℕ0).

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Cite

BibTeXRIS
BibTeX
@misc{ohrysko2026independent,
  title = {{Independent powers and bounded Fourier partial sums in $L^1$}},
  author = {Przemysław Ohrysko},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.34587},
  eprint = {2610.34587},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.34587},
}
RIS
TY  - UNPB
TI  - Independent powers and bounded Fourier partial sums in L^1
AU  - Przemysław Ohrysko
PY  - 2026
DA  - 2026/10/25
PB  - arXiv
DO  - 10.48550/arxiv.2610.34587
UR  - https://arxiv.org/abs/2610.34587
AB  - We construct a probability measure on the circle with independent convolution powers and uniformly bounded symmetric Fourier partial sums in L^1. Every positive convolution power is therefore singular with respect to Haar measure. The measure is Rajchman. The construction answers a question of Andersson concerning measures with bounded Fourier partial sums outside the radical of L^1, and contrasts with his theorem that the classical examples of Weiss, Katznelson, and Brown-Hewitt have convolution squares in L^2. Our proof combines sparse probability blocks, an endpoint Llog L estimate, and a diagonal generalized Riesz product. Exact factorization of local moments permits the simultaneous separation of all convolution powers by Brown's singularity criterion. By the classical spectral theory of independent-power measures, its spectrum is the closed unit disc. In particular, it has non-natural spectrum and induces a non-decomposable convolution operator on L^1. The image of the measure in M(𝕋)/L^1(𝕋) has spectral radius one, and the closed unital algebra it generates in M(𝕋) is isometrically isomorphic to ℓ^1(ℕ_0).
ER  -