Uniformly convergent Fourier series with universal power parts on closed subsets of measure zero

Given a closed subset $E$ of Lebesgue measure zero on the unit circle $\mathbb{T}$ there is a function $f$ on $\mathbb{T}$ with uniformly convergent symmetric Fourier series
$$ S_n(f,\zeta)=\sum_{k=-n}^n\hat{f}(k)\zeta^k\underset{\mathbb{T}}{\rightrightarrows} f(\zeta), $$
such that for every continuous function $g$ on $E$, there is a subsequence of partial power sums
$$ S^+_n(f,\zeta)=\sum_{k=0}^n\hat{f}(k)\zeta^k $$
of $f$, which converges to $g$ uniformly on $E$. Here
$$ \hat{f}(k)=\int_{\mathbb{T}}\bar{\zeta}^kf(\zeta)\, dm(\zeta), $$
and $m$ is the normalized Lebesgue measure on $\mathbb{T}$.