Gibbs' phenomenon and arclength (Q1588373)

Gibbs' phenomenon shows that the partial sums \(s_N(f)\) of the Fourier series of a piecewise smooth function \(f\) with jump discontinuities are poorly behaved. One simple consequence is that the arclengths of the graphs \(\Gamma(s_N(f))\) of the partial sums do not converge to the arclength of the graph \(\Gamma(f)\) of \(f\) as \(N\to\infty\). Among others, the following two propositions are proved. Proposition 1. If \(f\) is a piecewise \(C^1\) function on the circle with a finite number of jump discontinuities, then \[ \text{length}(\Gamma(s_N(f)))= O(\log N)\quad\text{as}\quad N\to\infty. \] Proposition 2. If \(f\) is a continuous piecewise \(C^1\) function on the circle, then \[ \text{length}(\Gamma(s_N(f)))\to \text{length}(\Gamma(f))\quad\text{as}\quad N\to\infty. \]