Let $f$ be a periodic function of bounded variation which jumps at a point $x_0\in\mathbb{R}$. Let $S_{N}[f]$ denote the partial Fourier sum of $f$ and let $C_{N}[f]$ denote the Cesaro partial sum. It is known that, given an interval $(a,b)$ and if $x_0\notin(a,b)$, then $\lim_{N \to \infty} V_a^bS_{N}[f] = V_a^b f$.

But when $x_0\in (a,b)$, $\lim_{N \to \infty} V_a^bS_{N}[f] = \infty$, although this is some what different for Cesaro summation where the limit exists but not equal to that of the function $f$. $\lim_{N \to \infty} V_a^b C_{N}[f]$ is finite but not equal to $V_a^b f$.

Let any general mechanism of Fourier reconstruction using a finite number of Fourier coefficients (like what is done by partial sum and Cesaro partial sum) be denoted as $G_N[f]$, we know it is useful if $G_N[f] \to f$ pointwise as $N\to \infty$, Fourier partial sum and Cesaro sum being two examples which obey this. My question is how important it is for a general mechanism $G_N[f]$ to not only converge pointwise to $f$ as $N\to \infty$, but also $$\lim_{N \to \infty} V_a^bG_{N}[f] = V_a^bf$$ even when $f$ jumps atleast once in $(a,b)$.

Mathematically how important it is to search for such a Fourier reconstruction mechanism? Is it anywhere given in literature, the importance of such a thing?

PS : $ V_a^bf$ denoted the variation of the function $f$ in the interval $(a,b)$. Also assume throughout entire question, that $f$ does not jump at either $a$ or $b$.