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Gibbs phenomenon and its removal for a class of orthogonal expansions

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Abstract

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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Authors and Affiliations

  1. Department of Mathematics, Simon Fraser University, 8888 University Drive, Burnaby, BC, V5A 1S6, Canada

    Ben Adcock

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  1. Ben Adcock
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Correspondence to Ben Adcock.

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Communicated by Lothar Reichel.

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Adcock, B. Gibbs phenomenon and its removal for a class of orthogonal expansions. Bit Numer Math 51, 7–41 (2011). https://doi.org/10.1007/s10543-010-0301-5

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