A corner point Gibbs phenomenon for Fourier series in two dimensions
Let \[ A=\biggl\{(x,y): 0\leq y\leq \pi,\;{y\over c}\leq x\leq {y\over c}+\pi \biggr\}\quad (0\neq c\in {\mathbb R}) \] and \(\chi_{_A}\) be the indicator function of \(A\). Denote by \(f\) the function with period \(2\pi\) in each variable which extends \(\chi_{_{A}}\). Denote by \(S_m(f)(x,y)\) the square partial sum of the Fourier series of \(f\), i.e., \[ S_m(f)(x,y)=\sum_{|\mu|\leq m, |\nu|\leq m} c_{\mu,\nu}(f)e^{i(\mu x+\nu y)}\quad (m\in {\mathbb Z}_+) \] where \(c_{\mu,\nu}(f)\) denote the Fourier coefficients of \(f\). The author shows the Gibbs phenomenon of the Fourier series of \(f\) at the corner point \(O=(0,0)\) of \(A\) by finding the limit of \( S_{2m+1}(f)(P_m)\) as \(m\to \infty\) where \(P_m={\pi\over 2m+1}(x+{y\over c}, y)\). The calculation is very careful and rather complicated.
scientific article; zbMATH DE number 1356329