Generalization of the inverse polynomial reconstruction method in the resolution of the Gibbs phenomenon

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Robust reprojection methods for the resolution of the Gibbs phenomenon (2006, methodology): “Some techniques which resolve the Gibbs phenomenon up to the discontinuity include: subtracting off discontinuities to increase smoothness [6], re-expansions in terms of singular Padé approximations [5], analytic continuation methods [4], inverse methods [18], and Gegenbauer post-processing develope”
Exponentially-Convergent Strategies for Defeating the Runge Phenomenon for the Approximation of Non-Periodic Functions, Part I: Single-Interval Schemes (2009, unknown): “• Tikhonov Regularization [2]; • Gegenbauer Regularization and its generalizations; [5, 6, 13–15, 17–24, 30, 34]; • Overdetermined Least-Squares [9]; • Mock-Chebyshev sub-sampling [9]; • Gaussian Radial Basis Functions, Slowly Flattening with N [8]; • Fourier Extension [4]; • Platte’s Undetermined L”
Trouble with Gegenbauer reconstruction for defeating Gibbs' phenomenon: Runge phenomenon in the diagonal limit of Gegenbauer polynomial approximations (2005, background): “The success of Gegenbauer reconstruction has spawned an industry of theoretical papers [21,16–19,12,9,10,23–25,32,26,11] and applications in fluid mechanics [29–31,7], hyperbolic waves [15], local Fourier methods [35,36] and medical imaging [2,1,3,14].”