Gibbs phenomenon for Fourier partial sums on ZpZp

It is of interest to know whether the Gibbs phenomenon occurs on a local field. A p-adic Heaviside function on the group of p  -adic integers is defined as an analogy of real variable Heaviside function and it is also shown that there exists the Gibbs phenomenon with an undershoot of at least 1/(p+1)1/(p+1) as an approximate level. As a consequence of the theorem, we see that the Fourier partial sum for a Heaviside function does not converge at the point of discontinuity.