Diagonal and branchwise criteria for universal Fourier series
Abstract
We characterize diagonal profiles admitting universal real trigonometric Fourier series with factors in ℰs=⋂p>sℓp and ℓs,q, 2≤ s<∞, 1≤ q≤∞. For nonincreasing profiles we obtain exact dyadic criteria and logarithmic endpoints. For independently regular odd/even sine/cosine branches we characterize simultaneous subseries, sign, and rearrangement universality for finite almost-everywhere targets. One function in every Lp, 1≤ p<2, supplies the coefficients. A separation example admits this realization although every compatible nonincreasing majorant with bounded relative cost belongs to ℓ1. Residue-class decompositions extend the criterion; a weakest-model condition permits extended-real rearrangement targets. A Walsh--Paley counterpart yields the same diagonal multiplier thresholds for ternary {0,±1}-universality, together with a Fourier--Walsh realization. Trigonometric realization uses Pogosyan's majorant theorem; the multiplier, factorization, projection-transfer, separation, and Walsh Fourier-origin arguments are proved here.
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BibTeX
@misc{episkoposian2026diagonal,
title = {{Diagonal and branchwise criteria for universal Fourier series}},
author = {Sergo A. Episkoposian},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27348},
eprint = {2610.27348},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27348},
}
RIS
TY - UNPB
TI - Diagonal and branchwise criteria for universal Fourier series
AU - Sergo A. Episkoposian
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.27348
UR - https://arxiv.org/abs/2610.27348
AB - We characterize diagonal profiles admitting universal real trigonometric Fourier series with factors in ℰ_s=⋂_p>sℓ^p and ℓ^s,q, 2≤ s<∞, 1≤ q≤∞. For nonincreasing profiles we obtain exact dyadic criteria and logarithmic endpoints. For independently regular odd/even sine/cosine branches we characterize simultaneous subseries, sign, and rearrangement universality for finite almost-everywhere targets. One function in every L^p, 1≤ p<2, supplies the coefficients. A separation example admits this realization although every compatible nonincreasing majorant with bounded relative cost belongs to ℓ^1. Residue-class decompositions extend the criterion; a weakest-model condition permits extended-real rearrangement targets. A Walsh--Paley counterpart yields the same diagonal multiplier thresholds for ternary {0,±1}-universality, together with a Fourier--Walsh realization. Trigonometric realization uses Pogosyan's majorant theorem; the multiplier, factorization, projection-transfer, separation, and Walsh Fourier-origin arguments are proved here.
ER -