Spectral Methods and the Resolution of the Gibbs Phenomenon

Spectral methods involve the expansion of the solution of some partial differential equation or integral equation in a set of basis functions. The basis functions chosen are the Fourier functions, Chebyshev, Hermite and nonclassical polynomials. Pseudospectral (or collocation) methods determine the solution at the set of quadrature points for classical and nonclassical polynomials. Recent work on the use of nonclassical basis functions will be briefly dicussed.
It is well known that the expansion of an analytic nonperiodic function on a finite interval in a Fourier series leads to spurious oscillations at the interval boundaries. This result is known as the Gibbs phenomenon. The present talk describes a new method for the resolution of the Gibbs phenomenon (Shizgal and Jung, J. Comput. Appl. Math. 161, 41 (2003); Jung and Shizgal, J. Comput. Appl. Math. 172, 131-151(2004) ) which follows on the reconstruction method of Gottlieb and coworkers (SIAM Rev. 39, 644 (1997)) based on Gegenbauer polynomials. We refer to their approach as the direct method and to the new methodology as the inverse method. The direct method requires that certain conditions are met concerning lambda in the weight function (1-x^2)^{\lambda-1/2}, the number of Fourier coefficients, N and the number of Gegenbauer polynomials, m. We show that the new inverse method can give exact results for polynomials independent of lambda and with m=N. The paper presents several numerical examples applied to a single domain or to subdomains of the main domain so as to illustrate the superiority of the inverse method in comparison with the direct method. Image resolution in two dimensions will also be discussed and illustrated with several examples including a resolution of the Shepp-Logan phantom image.