We know that Fourier partial sums (integrals) do not converge uniformly for BV functions with jumps due to Gibb's phenomenon. Is there any other types of sums/procedures that use only Fourier coefficients to produce a sequence of functions which converge uniformly (except isolated points) without exhibiting Gibb's phenomenon? I am not asking for approximations, but for uniform convergence except at isolated points (the exact points of jumps).
Clarify : Suppose $f$ has jump at $x_0$, then the sequence should still converge uniformly in the interval $(x_0-\epsilon,x_0)$, any $\epsilon\in \mathbb{R}$ taken such that $f$ has no jumps in $(x_0-\epsilon,x_0)$. This does not happen with Fourier partial sums.
If $\{ f_{n} \}$ is a sequence of continuous functions on $[a,b]$ that converges uniformly to a function $f$ on $[a,b]$, then $f$ is necessarily continuous on $[a,b]$. It is the general theorem that results in the non-uniform convergence of the Fourier series for a function $f$ that is not continuous on $[a,b]$.
Specifically, the Fourier series $\{ S_{n}^{f} \}_{n=0}^{\infty}$ is a sequence of continuous functions $$ S_{n}^{f}(x)=\sum_{k=-n}^{n}\left[\frac{1}{2\pi}\int_{-\pi}^{\pi}f(t)e^{-int}dt\right]e^{inx} $$ If $f$ is piecewise smooth on an interval $[a,b]$ with one discontinuity in $(a,b)$, then the Fourier series cannot converge uniformly on $[a,b]$. In that case, the logical negation of uniform convergence must be true--namely, there exists $\epsilon > 0$ such that for every $N$, there exists $n \ge N$ and $x\in [a,b]$ for which $|f(x)-S_{N}^{f}(x)| \ge \epsilon$.