Gibbs phenomenon on sampling series based on Shannon's and Meyer's wavelet analysis
Suppose \(f\in L^2(\mathbb R)\) is continuous except in one point \(t= \lambda 2^{- \mu}\), where \(\mu \in {\mathbb N}\), \(\lambda \in {\mathbb Z}\). Assume further that \(f(t^-) \leq f(t) \leq f(t^+)\). Choose \(\alpha \in [0,1]\) such that \(f(t)= \alpha f(t^-)+(1- \alpha)f(t^+)\). It is proved that in this situation, the classical Shannon sampling formula always exhibits Gibbs phenomenon. For Meyer sampling series it is proved that Gibbs phenomenon appears when \(\alpha < 0.12495\) and when \(\alpha > 0.306853\). For some values of \(\alpha\), the overshoot is proved to be shorter than for the Shannon sampling formula.
scientific article; zbMATH DE number 1441220