Generalization of the inverse polynomial reconstruction method in the resolution of the Gibbs phenomenon

Abstract

The finite Fourier representation of a function f( x) exhibits oscillations where the function or its derivatives are nonsmooth. This is known as the Gibbs phenomenon. A robust and accurate reconstruction method that resolves the Gibbs oscillations was proposed in a previous paper (J. Comput. Appl. Math. 161 (2003) 41) based on the inversion of the transformation matrix which represents the projection of a set of basis functions onto the Fourier space. If the function is a polynomial, this inverse polynomial reconstruction method (IPRM) is exact. In this paper, we develop the IPRM by requiring that the proper error be orthogonal to the Fourier or polynomial space. The IPRM is generalized to any set of basis functions. The primitive basis polynomials, nonclassical orthogonal polynomials and the Gegenbauer polynomials are used to illustrate the wide validity of the IPRM. It is shown that the IPRM yields a unique reconstruction irrespective of the basis set for any analytic function and yields spectral convergence. The ill-posedness of the transformation matrix due to the exponential growth of the condition number of the matrix is also discussed.