  The Gibbs phenomenon is widely known for Fourier expansions of periodic
functions and refers to the phenomenon that the $n$th Fourier partial sums
overshoot a target function at jump discontinuities in such a way that such
overshoots do not die out as $n$ goes to infinity. The Gibbs phenomenon for
wavelet expansions using (bi)orthogonal wavelets has been studied in the
literature. Framelets (also called wavelet frames) generalize (bi)orthogonal
wavelets. Approximation by quasi-projection operators are intrinsically linked
to approximation by truncated wavelet and framelet expansions. In this paper we
shall establish a key identity for quasi-projection operators and then we use
it to study the Gibbs phenomenon of framelet expansions and approximation by
general quasi-projection operators. We shall also study and characterize the
Gibbs phenomenon at an arbitrary point for approximation by quasi-projection
operators. As a consequence, we show that the Gibbs phenomenon appears at all
points for every tight or dual framelet having at least two vanishing moments
and for quasi-projection operators having at least three accuracy orders. Our
results not only improve current results in the literature on the Gibbs
phenomenon for (bi)orthogonal wavelet expansions but also are new for framelet
expansions and approximation by quasi-projection operators.
