# Does the Total variation of the Fourier partial sum of a bv function with jumps converge to TV of the function as $N\to\infty$

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.

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