We investigate the Gibbs–Wilbraham phenomenon for generalized sampling series, and related interpolation series arising from cardinal functions. We prove the existence of the overshoot characteristic of the phenomenon for certain cardinal functions, and characterize the existence of an overshoot for sampling series.

Given a piecewise smooth function, it is possible to construct a global expansion in some complete orthogonal basis, such as the Fourier basis. However, the local discontinuities of the function will destroy the convergence of global approximations, even in regions for which the underlying function is analytic. The global expansions are contaminated by the presence of a local discontinuity, and the result is that the partial sums are oscillatory and feature non-uniform convergence. This characteristic behavior is called the Gibbs phenomenon. However, David Gottlieb and Chi-Wang Shu showed that these slowly and non-uniformly convergent global approximations retain within them high order information which can be recovered with suitable postprocessing. In this paper we review the history of the Gibbs phenomenon and the story of its resolution.

The main issue in the present paper is to comment and illustrate on new acoustic examples the method of analysis of modal series, termed “method of orthocomplement”, that has been recently proposed by the authors to improve the convergence of such series. The general method consists in a direct analysis and transformation of the remainders of ordinary series. It results in a family of “hybrid” modal representations involving an ordinary modal sum of order N, a “quasi-static” term based on the N first modes, and an “accelerated” modal series. Using the transformed modal formulae eliminates the Gibbs oscillations – that are attached in infinite dimensional models to modal boundary discontinuities – and also the consequences of such phenomena on finite element approximations. The method is applied in the present paper to plane waves in acoustic tubes and to 3D acoustic fields inside a car compartment, in view of the synthesis of acoustic receptances or impedances to be used in practical acoustic design. The main technical difficulty being the treatment of singular linear boundary problems or systems of linear equations that arise during the study of closed rigid cavities or tubes, a whole section of the paper had thus to be devoted to pseudo-inversion.

We analyse the role of Euler summation in a numerical inversion algorithm for Laplace transforms due to Abate and Whitt called the EULER algorithm. Euler summation is shown to accelerate convergence of a slowly converging truncated Fourier series; an explicit bound for the approximation error is derived that generalizes a result given by O'Cinneide. An enhanced inversion algorithm called EULER-GPS is developed using a new variant of Euler summation. The algorithm EULER-GPS makes it possible to accurately invert transforms of functions with discontinuities at arbitrary locations. The effectiveness of the algorithm is verified through numerical experiments. Besides numerical transform inversion, the enhanced algorithm is applicable to a wide range of other problems where the goal is to recover point values of a piecewise-smooth function from the Fourier series.