This deficiency of the spectral approximation results in the so-called Gibbs phenomenon which shows nonvanishing spikes near the discontinuity [1,2].
In this work, focusing on the Fourier partial sum approximation for a piecewise smooth function f having a jump-discontinuity [xi], we aim to develop a constructive approximation procedure which is available for eliminating the Gibbs phenomenon near the discontinuity.
. The complex amplitude of a wavefront obtained by a uniformly collimated monochromatic beam of light reflected from an object is given by
Summability techniques were also applied on some engineering problems; for example, Chen and Jeng  implemented the Cesaro sum of order (C, 1) and (C, 2), in order to accelerate the convergence rate to deal with the Gibbs phenomenon
, for the dynamic response of a finite elastic body subjected to boundary traction.
But Gibbs phenomenon
occurs when the Fourier representation method is applied to a non-periodic or discontinuous function .
Using Mathematica students were able to visualise Fourier series of functions and explore Gibbs phenomenon
which is usually a part of college mathematics.
This effect is called the Gibbs phenomenon
and caused by the very slow convergence of a truncated Fourier series near a discontinuity.
Over the years Abdul's wide-ranging research interests have been in integral and discrete transforms where he was an innovator of iterative methods for nonlinear problems, sampling expansions, the Gibbs phenomenon
and operational sum methods for difference equations.
One mathematical effect, the Gibbs phenomenon
, can add anomalies to a signal.
Jerri, The Gibbs Phenomenon
in Fourier Analysis, Splines, and Wavelet Application.
In this case, the well-known Gibbs phenomenon
reduces the order of accuracy to first order and produces spurious oscillations, particularly ill regions near the discontinuities.