What is the importance of convergence of variation of Fourier reconstruction to that of variation of the function? 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 $(a,b)$,(Also assume throughout entire question, that $f$ does not jump at either $a$ or $b$) 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 (first $N$) 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 $b$.
 A: You ask for a Fourier reconstruction mechanism that avoids the Gibbs phenomenon.
One strategy in this direction was developed by David Gottlieb and collaborators, in a series of papers entitled ￼"On the Gibbs phenomenon: Recovering exponential accuracy from the Fourier partial sum of a nonperiodic analytic function" (part I, part II, part III, part IV, part V). A summary of this body of work was given in On the Gibbs phenomenon and its resolution (1997), and more recently in A review of David Gottlieb’s work on the resolution of the Gibbs phenomenon (2011).
The partial Fourier sum of order $N$ is post-processed by converting it in a series of Gegenbauer polynomials $C_n^\lambda(x)$ ($n=1,2,\ldots N$, with $\lambda=N/4$). The requirement for an exponentially accurate solution is that the location of the jump discontinuity is known in advance, and that the function is analytic elsewhere. The overall conclusion is that global expansions which are contaminated by local discontinuities still retain within them high order information, and the Gibbs phenomenon can be removed by post-processing.

The Gibbs phenomenon deals with the issue of recovering point values
  of a function from a finite set of Fourier expansion coefficients. We
  show that the knowledge of the expansion coefficients is sufficient
  for obtaining the point values of a piecewise smooth function, with
  the same order of accuracy as in the smooth case. This is done by
  using the finite Fourier expansion series to construct a different,
  rapidly convergent, approximation in terms of Gegenbauer polynomials. Thus the retrieval of the information should be done in a different basis than the storage of the information. The Fourier coefficients of a function contain enough information such that a different expansion, which is highly accurate, can be reconstructed.



EDIT in response to comments by the OP
For an application of this method to image processing, see Improving tissue segmentation of human brain MRI through preprocessing by the Gegenbauer reconstruction method (2003). In that application the jumps are due to segmentation of the magnetic resonance image, so their location is known in advance.
What if the location of the jumps is not known? Is there a way to locate the edges of the interval of analyticity using only knowledge of a partial set of Fourier coefficients? This problem has been addressed by Gelb and Tadmore in Detection of Edges in Spectral Data, part I, part II, part III. The method of edge detection followed by Gegenbauer reconstruction has been tested in this paper, for both one-dimensional and two-dimensional Fourier coefficients.
