Convergence of spectral methods for nonlinear conservation laws
SIAM Journal on Numerical Analysis
Journal of Computational and Applied Mathematics - Orthogonal polynomials and numerical methods
Accurate reconstructions of functions of finite regularity from truncated Fourier series expansions
Mathematics of Computation
On the Gibbs Phenomenon and Its Resolution
SIAM Review
Enhanced spectral viscosity approximations for conservation laws
Proceedings of the fourth international conference on Spectral and high order methods (ICOSAHOM 1998)
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
Reducing the Effects of Noise in Image Reconstruction
Journal of Scientific Computing
Distributions of zeros of discrete and continuous polynomials from their recurrence relation
Applied Mathematics and Computation - Orthogonal systems and applications
EXPONENTIALLY ACCURATE APPROXIMATIONS TO PIECE-WISE SMOOTH PERIODIC FUNCTIONS
EXPONENTIALLY ACCURATE APPROXIMATIONS TO PIECE-WISE SMOOTH PERIODIC FUNCTIONS
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
Polynomial Fitting for Edge Detection in Irregularly Sampled Signals and Images
SIAM Journal on Numerical Analysis
Determining Analyticity for Parameter Optimization of the Gegenbauer Reconstruction Method
SIAM Journal on Scientific Computing
Journal of Computational Physics
Padé-Legendre Interpolants for Gibbs Reconstruction
Journal of Scientific Computing
Determining the locations and discontinuities in the derivatives of functions
Applied Numerical Mathematics
A Hybrid Fourier---Chebyshev Method for Partial Differential Equations
Journal of Scientific Computing
An SVD analysis of equispaced polynomial interpolation
Applied Numerical Mathematics
Algorithm 899: The Matlab postprocessing toolkit
ACM Transactions on Mathematical Software (TOMS)
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Spectral series expansions of piecewise smooth functions are known to yield poor results, with spurious oscillations forming near the jump discontinuities and reduced convergence throughout the interval of approximation. The spectral reprojection method, most notably the Gegenbauer reconstruction method, can restore exponential convergence to piecewise smooth function approximations from their (pseudo-)spectral coefficients. Difficulties may arise due to numerical robustness and ill-conditioning of the reprojection basis polynomials, however. This paper considers non-classical orthogonal polynomials as reprojection bases for a general order (finite or spectral) reconstruction of piecewise smooth functions. Furthermore, when the given data are discrete grid point values, the reprojection polynomials are constructed to be orthogonal in the discrete sense, rather than by the usual continuous inner product. No calculation of optimal quadrature points is therefore needed. This adaptation suggests a method to approximate piecewise smooth functions from discrete non-uniform data, and results in a one-dimensional approximation that is accurate and numerically robust.