Journal of Computational and Applied Mathematics - Orthogonal polynomials and numerical methods
Matrix computations (3rd ed.)
On the Gibbs Phenomenon and Its Resolution
SIAM Review
Spectral methods for hyperbolic problems
Journal of Computational and Applied Mathematics - Special issue on numerical analysis 2000 Vol. VII: partial differential equations
Journal of Computational and Applied Mathematics
Towards the resolution of the Gibbs phenomena
Journal of Computational and Applied Mathematics
Parameter Optimization and Reduction of Round Off Error for the Gegenbauer Reconstruction Method
Journal of Scientific Computing
Inverse Polynomial Reconstruction of Two Dimensional Fourier Images
Journal of Scientific Computing
A note on the Gibbs phenomenon with multiquadric radial basis functions
Applied Numerical Mathematics
Multi-domain hybrid spectral-WENO methods for hyperbolic conservation laws
Journal of Computational Physics
A wavelet-based method for overcoming the Gibbs phenomenon
MATH'08 Proceedings of the American Conference on Applied Mathematics
Construction of Lanczos type filters for the Fourier series approximation
Applied Numerical Mathematics
Journal of Computational Physics
Algorithm 899: The Matlab postprocessing toolkit
ACM Transactions on Mathematical Software (TOMS)
Pseudospectral Fourier reconstruction with the modified Inverse Polynomial Reconstruction Method
Journal of Computational Physics
A Simple Regularization of the Polynomial Interpolation for the Resolution of the Runge Phenomenon
Journal of Scientific Computing
Two-Dimensional gibbs phenomenon for fractional fourier series and its resolution
AICI'12 Proceedings of the 4th international conference on Artificial Intelligence and Computational Intelligence
Gibbs phenomenon removal by adding Heaviside functions
Advances in Computational Mathematics
Hi-index | 7.32 |
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.