Trouble with Gegenbauer reconstruction for defeating Gibbs' phenomenon: Runge phenomenon in the diagonal limit of Gegenbauer polynomial approximations

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A Review of David Gottlieb's Work on the Resolution of the Gibbs Phenomenon (2011, methodology): “Furthermore, singularities in the complex plane can ruin the convergence of the Gegenbauer method unless β is below some threshold [7].”
Gibbs‐ringing artifact removal based on local subvoxel‐shifts (2015, methodology): “A drawback of these methods is the requirement of an edge detection and potential instabilities in some applications (5) due to the involved choice of parameters.”
Exponentially-Convergent Strategies for Defeating the Runge Phenomenon for the Approximation of Non-Periodic Functions, Part I: Single-Interval Schemes (2009, background): “Boyd showed [5] that this shrinks the Runge Zone, that is, the region of the complex x-plane where f (x) must be free of singularities for convergence.”
Robust reprojection methods for the resolution of the Gibbs phenomenon (2006, methodology): “Using such compactly sup weights will only allow an overall root exponential accuracy in the reprojection, instead of the d true exponential accuracy which is achievable in theory through Gegenbauer reconstruction.”