Gibbs phenomenon for Fourier series on finite sets (Q611966)

The classical Gibbs phenomenon describes the behaviour of the partial sums of the Fourier series at the discontinuity points of the function whose Fourier series is studied. The author is interested in the Gibbs phenomenon over the finite circle \(\mathbb Z/p\mathbb Z\), with \(p\) an odd prime. He studies the `jumping function' \(F\) defined by \(F(\varepsilon^m)=m\), at the \(p\)-th roots of unity, where \(-q\leq m\leq q\), \(p=2q+1\) and \(\varepsilon=\exp(2\pi i/p)\). The jump of the function is the large difference \(F(\varepsilon^q)-F(\epsilon^{q+1})=2q\). The Fourier series is in this case, the representation of (any) function \(f:\mathbb Z_p\rightarrow \mathbb C\) in the form of a linear combination of the characters \(f=\sum_{k=0}^{p-1}c_kf_k\), where \(f_k(z)=z^k\). For the function \(F\) the author considers the approximation obtained by removing the ``highest harmonics'' \(f_q\) and \(f_{q+1}\). He then shows that, as in the classical Gibbs phenomenon, the approximation has in some sense a larger jump than the original function (being approximately \(1+1/p\) times greater in this setup).