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Does the total variation of the Fourier partial sum of a piecewise continuous bv function converge to the total variation of the function as $N\to\infty$. To explain briefly,

Let $f$ be a periodic BV function with atleast one jump, and let $S^f_N(x)$ be its Fourier partial sum, $N = 1,2,3,...$ Let $t_N = TV(S^f_N(x))$ where $TV(f)$ denotes the total variation of the function $f$. Does $$\lim_{N\to\infty}\{t_N-TV(f)\}$$ exists and goes to zero or to $\infty$? I think it goes to $\infty$, but not sure. Request your answers.

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  • $\begingroup$ To put it simply, Does the Fourier expansion preserves the Total variation of the function? $\endgroup$
    – Rajesh D
    Jun 8, 2015 at 14:28
  • $\begingroup$ math.stackexchange.com/a/698775/2987 $\endgroup$
    – Rajesh D
    Jun 8, 2015 at 15:52
  • $\begingroup$ the overshoots and undershoots do not seem to dampen that quickly. $\endgroup$
    – Rajesh D
    Jun 8, 2015 at 15:53
  • $\begingroup$ $t_N$ does not converge to $TV(f)$; try an example such as $f=\chi_{(0,\pi)}$. $\endgroup$
    – Christian Remling
    Jun 8, 2015 at 18:43

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