On the Gibbs phenomenon for linear canonical series expansion

The Gibbs phenomenon describes the behavior of the partial sums of a Fourier series (FS) in a neighborhood of a simple discontinuity of the function expanded. In this paper, we investigate the Gibbs phenomenon for linear canonical series (LCS) in linear canonical transform (LCT) domain. We show that a similar phenomenon is observed for the LCS of a function with jump discontinuities. First, the convergence and Gibbs phenomenon of the LCS are discussed for the real parts of the LCS. It is showed that if a function is continuous, then there is no Gibbs phenomenon. Moreover, we proved that the uniform convergence of the LCS for a non-periodic analysis function in the smooth region. Then, we present a theorem that the Gibbs constant that arise for the LCS appears to be the same to that occurring in standard FS.