Diagonal and branchwise criteria for universal Fourier series

Abstract

We characterize diagonal profiles admitting universal real trigonometric Fourier series with factors in $\mathcal E_s=\bigcap_{p>s}\ell^p$ and $\ell^{s,q}$, $2\le s<\infty$, $1\le q\le\infty$. 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 $L^p$, $1\le p<2$, supplies the coefficients. A separation example admits this realization although every compatible nonincreasing majorant with bounded relative cost belongs to $\ell^1$. 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 $\{0,\pm1\}$-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.