Generalized Hardy identity and relations to Gibbs–Wilbraham and Pinsky phenomena

We consider the Fourier series of the indicator functions of several dimensional balls. For the spherical partial sum of the Fourier series, we extract the Gibbs–Wilbraham (or Gibbs), Pinsky and the third phenomena as an extension of Hardy's identity. The third phenomenon has been shown by Kuratsubo recently. We prove the Gibbs–Wilbraham phenomenon for all dimensions and give another proof of the Pinsky phenomenon. Pinsky gave the order of the divergence for the Fourier inversion at the origin. We give the order of the divergence of the Fourier series at the origin and show that both orders coincide. We also investigate the uniform convergence for the Fourier series and the Fourier inversion.