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.
Abdul's fourth book is on a fascinating (but more specialized) research topic: The Gibbs Phenomenon in Fourier Analysis, Splines and Wavelet Approximations, [B 4].
Advances in Gibbs Phenomenon [B 5] is an edited collection of papers by major contributors to the subject, invited by Abdul, who has also contributed three long introductory chapters in this collection.
Reducing the Gibbs phenomenon in a Fourier-Bessel series, Hankel and Fourier transform.
The Gibbs phenomenon in sampling and interpolation.
Jerri, The Gibbs Phenomenon in Fourier Analysis, Splines, and Wavelet Application.
Jerri, Advances in the Gibbs Phenomenon, Sampling Publishing, Potsdam, New York, 2008, $94.
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.
Key words and phrases : Gibbs phenomenon, Gegenbauer polynomials, spherical harmonics, edge detection.
This behavior is known as the Gibbs phenomenon [7, 18].
In effort to combat the Gibbs phenomenon for both the modeling and reconstruction of geophysical data, some sort of smoothing filter is often employed [7, 27].
Other methods have been effectively used to combat the Gibbs phenomenon when the given information is Fourier data [8, 9, 29].