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

To defeat Gibbs' phenomenon in Fourier and Chebyshev series, Gottlieb et al. [D. Gottlieb, C.-W. Shu, A. Solomonoff, H. Vandeven, On the Gibbs phenomenon I: recovering exponential accuracy from the Fourier partial sum of a nonperiodic analytic function, J. Comput. Appl. Math. 43 (1992) 81-98] developed a “Gegenbauer reconstruction”. The partial sums of the Fourier or other spectral series are reexpanded as a series of Gegenbauer polynomials Cnm(x), recovering spectral accuracy even in the presence of shock waves or other discontinuities. To achieve a rate of convergence which is exponential in N, however, Gegenbauer reconstruction, requires increasing the order m of the polynomials linearly with the truncation N of the series: m = βN for some constant β > 0. When the order m is fixed, it is well-known that the Gegenbauer series converges as N → ∞ everywhere on x ∈ [−1,1] if f(x), the function being expanded, is analytic on the interval. But what happens in the diagonal limit in which m, N tend to infinity simultaneously? We show that singularities of f(x) off the real axis can destroy convergence of this diagonal approximation process in the sense that the error diverges for subintervals of x ∈ [−1,1]. Gegenbauer reconstruction must therefore be constrained to use a sufficiently small ratio of order m to truncation N. This “off-axis singularity” constraint is likely to impair the effectiveness of the reconstruction in some applications.