We detail the Gibbs phenomenon and its resolution for the family of orthogonal expansions consisting of eigenfunctions of univariate polyharmonic operators equipped with homogeneous Neumann boundary conditions. As we establish, this phenomenon closely resembles the classical Fourier Gibbs phenomenon at interior discontinuities. Conversely, a weak Gibbs phenomenon, possessing a number of important distinctions, occurs near the domain endpoints. Nonetheless, in both cases we are able to completely describe this phenomenon, including determining exact values for the size of the overshoot.
Next, we demonstrate how the Gibbs phenomenon can be both mitigated and completely removed from such expansions using a number of different techniques. As a by-product, we introduce a generalisation of the classical Lidstone polynomials.
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