Journal of Computational and Applied Mathematics - Orthogonal polynomials and numerical methods
Spectrally Accurate Solution of Nonperiodic Differential Equations by the Fourier--Gegenbauer Method
SIAM Journal on Numerical Analysis
On the Gibbs Phenomenon and Its Resolution
SIAM Review
Spectral methods in MatLab
Reconstruction of a piecewise constant function from noisy Fourier coefficients by Padé method
SIAM Journal on Applied Mathematics
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
Detection of Edges in Spectral Data II. Nonlinear Enhancement
SIAM Journal on Numerical Analysis
Reducing the Effects of Noise in Image Reconstruction
Journal of Scientific Computing
On inverse methods for the resolution of the Gibbs phenomenon
Journal of Computational and Applied Mathematics
Journal of Computational and Applied Mathematics
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
Journal of Computational Physics
Reconstruction of Piecewise Smooth Functions from Non-uniform Grid Point Data
Journal of Scientific Computing
Determining the locations and discontinuities in the derivatives of functions
Applied Numerical Mathematics
An accurate Fourier-spectral solver for variable coefficient elliptic equations
ISTASC'05 Proceedings of the 5th WSEAS/IASME International Conference on Systems Theory and Scientific Computation
Detection of Edges in Spectral Data III--Refinement of the Concentration Method
Journal of Scientific Computing
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
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
Discontinuous functions represented by exact, closed, continuous parametric equations
AMERICAN-MATH'10 Proceedings of the 2010 American conference on Applied mathematics
A Simple Regularization of the Polynomial Interpolation for the Resolution of the Runge Phenomenon
Journal of Scientific Computing
Detecting discontinuity points from spectral data with the quotient-difference (qd) algorithm
Journal of Computational and Applied Mathematics
Sparsity Enforcing Edge Detection Method for Blurred and Noisy Fourier data
Journal of Scientific Computing
Journal of Computational Physics
Recovering Exponential Accuracy from Non-harmonic Fourier Data Through Spectral Reprojection
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 |
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 paper introduces a new method for the resolution of the Gibbs phenomenon which follows on the reconstruction method of Gottlieb and coworkers (SIAM Rev. 39 (1997) 644) based on Gegenbauer polynomials orthogonal with respect to weight function (1 - x2)λ-1/2. We refer to their approach as the direct method and to the new methodology as the inverse method. Both methods use the finite set of Fourier coefficients of some given function as input data in the re-expansion of the function in Gegenbauer polynomials or in other orthogonal basis sets. The finite partial sum of the new expansion provides a spectrally accurate approximation to the function. In the direct method, this requires that certain conditions are met concerning the parameter λ in the weight function, 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 λ 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.